SWI-Prolog coding
The Art of Prolog
Leon Sterling Ehud Shapiro with a foreword by David H. D. Warren
The Art of Prolog Advanced Programming Techniques Second Edition
The MIT Press Cambridge, Massachusetts London, England
© 1986, 1994 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.
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Library of Congress Cataloging-in-Publication Data Sterling, Leon
The art of Prolog : advanced programming techniques / Leon Sterling, Ehud Shapiro ; with a foreword by David H. D. Warren.
p. cm. - (MIT Press series in logic programming) Includes bibliographical references and index. ISBN 978-O-262-19338-2 (hardcover: alk. paper), 978-O-262-69163-5 (paperback) 1. Prolog (Computer program language) I. Shapiro, Ehud Y.
II. Title. III. Series. QA76.73.P76S74 1994 OOS.13'3dc2O 93-49494
lo CIP
To Ruth, Miriam, Micha!, Dan ya, and Sara
Contents
Figures xiii
Programs xvii
Series Foreword xxv
Foreword xxvii
Preface xxxi
Preface to First Edition
I Logic Programs 9
Introduction i
Basic Constructs 11 Li Facts 11 1.2 Queries 12
1.3 The Logical Variable, Substitutions, and Instances 13
1.4 Existential Queries 14
1.5 UniversalFacts 15
1.6 Conjunctive Queries and Shared Variables 16
1.7 Rules 18
viii Contents
1.8 A Simple Abstract Interpreter 22 1.9 The Meaning of a Logic Program 25
1.10 Summary 27
2 Database Programming 29 2.1 Simple Databases 29
2.2 Structured Data and Data Abstraction 35 2.3 Recursive Rules 39
2.4 Logic Programs and the Relational Database Model 42
2.5 Background 44
3 Recursive Programming 45 3.1 Arithmetic 45
3.2 Lists 56
3.3 Composing Recursive Programs 65 3.4 Binary Trees 72
3.5 Manipulating Symbolic Expressions 78 3.6 Background 84
4 The Computation Model of Logic Programs 87 4.1 Unification 87
4.2 An Abstract Interpreter for Logic Programs 91 4.3 Background 98
S Theory of Logic Programs 101 5.1 Semantics 101
5.2 Program Correctness 105
5.3 Complexity 108
5.4 Search Trees 110
5.5 Negation in Logic Programming 113
5.6 Background 115
ix Contents
II The Prolog Language 117
6 Pure Prolog 119
6.1 The Execution Model of Prolog 119
6.2 Comparison to Conventional Programming Languages 124
6.3 Background 127
7 Programming in Pure Prolog 129 7.1 Rule Order 129
7.2 Termination 131
7.3 Goal Order 133
7.4 Redundant Solutions 136
7.5 Recursive Programming in Pure Prolog 139
7.6 Background 147
8 Arithmetic 149 8.1 System Predicates for Arithmetic 149
8.2 Arithmetic Logic Programs Revisited 152
8.3 Transforming Recursion into Iteration 154 8.4 Background 162
9 Structure Inspection 163 9.1 Type Predicates 163 9.2 Accessing Compound Terms 167
9.3 Background 174
10 Meta-Logical Predicates 175
10.1 Meta-Logical Type Predicates 176
10.2 Comparing Nonground Terms 180
10.3 Variables as Objects 182
10.4 The Meta-Variable Facility 185
10.5 Background 186
x Contents
11 Cuts and Negation 189 11.1 Green Cuts: Expressing Determinism 189 11.2 Tail Recursion Optimization 195
11.3 Negation 198
11.4 Red Cuts: Omitting Explicit Conditions 202
11.5 Default Rules 206
11.6 Cuts for Efficiency 208
11.7 Background 212
12 Extra-Logical Predicates 215
12.1 Input/Output 215 12.2 Program Access and Manipulation 219
12.3 Memo-Functions 221
12.4 Interactive Programs 223
12.5 Failure-Driven Loops 229
12.6 Background 231
13 Program Development 233 13.1 Programming Style and Layout 233
13.2 Reflections on Program Development 235
13.3 Systematizing Program Construction 238 13.4 Background 244
HI Advanced Prolog Programming Techniques 247
14 Nondeterministic Programming 249 14.1 Generate-and-Test 249
14.2 Don't-Care and Don't-Know Nondeterminism 263
14.3 Artificial Intelligence Classics: ANALOGY, ELIZA, and McSAM 270
14.4 Background 280
15 Incomplete Data Structures 283 15.1 Difference-Lists 283
xi Contents
15.2 Difference-Structures 291
15.3 Dictionaries 293
15.4 Queues 297
15.5 Background 300
16 Second-Order Programming 301
16.1 All-Solutions Predicates 301
16.2 Applications of Set Predicates 305
16.3 Other Second-Order Predicates 314
16.4 Background 317
17 Interpreters 319 17.1 Interpreters for Finite State Machines 319
17.2 Meta-Interpreters 323
17.3 Enhanced Meta-Interpreters for Debugging 331
17.4 An Explanation Shell for Rule-Based Systems 341
17.5 Background 354
18 Program Transfoutiation 357 18.1 Unfold/Fold Transformations 357
18.2 Partial Reduction 360 18.3 Code Walking 366
18.4 Background 373
19 Logic Grammars 375 19.1 Definite Clause Grammars 375
19.2 A Grammar Interpreter 380
19.3 Application to Natural Language Understanding 382
19.4 Background 388
20 Search Techniques 389
20.1 Searching State-Space Graphs 389
20.2 Searching Game Trees 401
20.3 Background 407
xii Contents
IV Applications 409
21 Game-Playing Programs 411
21.1 Mastermind 411
21.2 Nim 415
21.3 Kalah 420
21.4 Background 423
22 A Credit Evaluation Expert System 429
22.1 Developing the System 429
22.2 Background 438
23 An Equation Solver 439 23.1 An Overview of Equation Solving 439 23.2 Factorization 448
23.3 Isolation 449
23.4 Polynomial 452
23.5 Homogenization 454 23.6 Background 457
24 A Compiler 439 24.1 Overview of the Compiler 459
24.2 The Parser 466 24.3 The Code Generator 470
24.4 The Assembler 475
24.5 Background 478
A Operators 479
References 483
Index 497
Figure s
1.1 An abstract interpreter to answer ground queries with respect to logic programs 22
1.2 Tracing the interpreter 23 1.3 A simple proof tree 25 2.1 Defining inequality 31
2.2 A logical circuit 32
2.3 Still-life objects 34
2.4 A simple graph 41
3.1 Proof trees establishing completeness of programs 47
3.2 Equivalent forms of lists 57 3.3 Proof tree verifying a list 58 3.4 Proof tree for appending two lists 61 3.5 Proof trees for reversing a list 63 3.6 Comparing trees for isomorphism 74 3.7 A binary tree and a heap that preserves the tree's shape 77 4.1 A unification algorithm 90
4.2 An abstract interpreter for logic programs 93
4.3 Tracing the appending of two lists 94
4.4 Different traces of the same solution 95 4.5 Solving the Towers of Hanoi 97
4.6 A nonterminating computation 97
xiv Figures
5.1 A nonterminating computation 107 5.2 Two search trees 111
5.3 Search tree with multiple success nodes 112 5.4 Search tree with an infinite branch 113
6.1 Tracing a simple Prolog computation 121 6.2 Multiple solutions for splitting a list 122
6.3 Tracing a quicksort computation 123 7.1 A nonterminating computation 132 7.2 Variant search trees 139
7.3 Tracing a reverse computation 146 8.1 Computing factorials iteratively 155 9.1 Basic system type predicates 164
9.2 Tracing the substitute predicate 171 11.1 Theeffectofcut 191 13.1 Template for a specification 243
14.1 A solution to the 4 queens problem 253 14.2 A map requiring four colors 255 14.3 Directed graphs 265 14.4 Initial and final states of a blocks world problem 267 14.5 A geometric analogy problem 271
14.6 Sample conversation with ELIZA 273
14.7 AstoryfilledinbyMcSAM 276 14.8 Three analogy problems 279
15.1 Concatenating difference-lists 285
15.2 Tracing a computation using difference-lists 287
15.3 Unnormalized and normalized sums 292 16.1 Power of Prolog for various searching tasks 307
16.2 The problem of Lee routing for VLSI circuits 308
16.3 Input and output for keyword in context (KWIC) problem 312
16.4 Second-order predicates 315
17.1 A simple automaton 321
xv Figures
17.2 Tracing the meta-interpreter 325 17.3 Fragment of a table of builtin predicates 327
17.4 Explaining a computation 351
18.1 A context-free grammar for the language a*b*c* 371
20.1 The water jugs problem 393 20.2 A simple game tree 405
21.1 A starting position for Nim 415
21.2 Computing nim-sums 419
21.3 Board positions for Kalah 421
23.1 Test equations 440
23.2 Position of subterms in terms 449 24.1 A PL program for computing factorials 460
24.2 Target language instructions 460
24.3 Assembly code version of a factorial program 461
24.4 The stages of compilation 461
24.5 Output from parsing 470
24.6 The generated code 475 24.7 The compiled object code 477
Programs
1.1 A biblical family database 12
1.2 Biblical family relationships 23
2.1 Defining family relationships 31
2.2 A circuit for a logical and-gate 33
2.3 The circuit database with names 36
2.4 Course rules 37
2.5 The ancestor relationship 39 2.6 A directed graph 41 2.7 The transitive closure of the edge relation 41 3.1 Defining the natural numbers 46 3.2 The less than or equal relation 48 3.3 Addition 49
3.4 Multiplication as repeated addition 51
3.5 Exponentiation as repeated multiplication 51
3.6 Computing factorials 52 3.7 The minimum of two numbers 52
3.8a A nonrecursive definition of modulus 53
3.8b A recursive definition of modulus 53
3.9 Ackermann's function 54 3.10 The Euclidean algorithm 54
3.11 Defining a list 57
xviii Programs
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
5.1
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Membership of a list 58
Prefixes and suffixes of a list 59
Determining sublists of lists 60
Appending two lists 60
Reversing a list 62
Determining the length of a list 64
Deleting all occurrences of an element from a list 67
Selecting an element from a list 67
Permutation sort 69 Insertion sort 70 Quicksort 70
Defining binary trees 73
Testing tree membership 73 Determining when trees are isomorphic 74
Substituting for a term in a tree 75 Traversals ofabinary tree 76 Adjusting a binary tree to satisfy the heap property Recognizing polynomials 79
Derivative rules 80
Towers of Hanoi 82
Satisfiability of Boolean formulae 83 Yet another family example 102
Yet another family example 130
Merging ordered lists 138
Checking for list membership 139
Selecting the first occurrence of an element from a list Nonmembership of a list 141
77
140
Testing for a subset 142
Testing for a subset 142
Translating word for word 143
Removing duplicates from a list 145
xix Programs
710 Reversing with no duplicates 146 8.1 Computing the greatest common divisor of two integers 152 8.2 Computing the factorial of a number 153
8.3 An iterative factorial 155
8.4 Another iterative factorial 156
8.5 Generating a range of integers 157
8.6a Summing a list of integers 157
8.6b Iterative version of summing a list of integers using an accumu- lator 157
8.7a Computing inner products of vectors 158 8.7b Computing inner products of vectors iteratively 158
8.8 Computing the area of polygons 159
8.9 Finding the maximum of a list of integers 160 8.10 Checking the length of a list 160 811 Finding the length of a list 161 8.12 Generating a list of integers in a given range 161 9.la Flattening a list with double recursion 165 9,lb Flattening a list using a stack 166 9.2 Finding subterms of a term 168 9.3 A program for substituting in a term 170 9.4 Subtermdefinedusinguniv 172 9,5a Constructing a list corresponding to a term 173 9.5b Constructing a term corresponding to a list 174 10.1 Multiple uses for plus 176
10.2 A multipurpose length program 177 10.3 A more efficient version of grandparent 178 10.4 Testing if a term is ground 178
10.5 Unification algorithm 180
10.6 Unification with the occurs check 181
10.7 Occurs in 182
10.8 Numbering the variables in a term 185
xx Programs
10.9 Logical disjunction 186
11.1 Merging ordered lists 190 11.2 Merging with cuts 192
11.3 ininimuinwith cuts 193
11.4 Recognizing polynomials 193
11.5 Interchange sort 195 11.6 Negation as failure 198 11.7 Testing if terms are variants 200
11.8 Implementing 201
11.9a Deleting elements from a list 204
i 1.9b Deleting elements from a list 204 11.10 If-then-else statement 205
11.1 la Determining welfare payments 207
ll.11b Determining welfare payments 207 12.1 Writing a list of terms 216 12.2 Reading in a list of words 217
12.3 Towers of Hanoi using a memo-function 222 12.4 Basic interactive loop 223 12.5 A line editor 224
12.6 An interactive shell 226 12.7 Logging a session 228 12.8 Basic interactive repeat loop 230 12.9 Consulting a file 230
13.1 Finding the union of two lists 241
13.2 Finding the intersection of two lists 241
13.3 Finding the union and intersection of two lists 241 14.1 Finding parts of speech in a sentence 251 14.2 Naive generate-and-test program solving N queens 253
14.3 Placing one queen at a time 255 14.4 Map colormg 256
14.5 Test data for map coloring 257
xxi Programs
14.6 A puzzle solver 259
14.7 A description of a puzzle 260 14.8 Connectivity in a finite DAG 265
14.9 Finding a path by depth-first search 266 14.10 Connectivity in a graph 266 14.11 A depth-first planner 268
14.12 Testing the depth-first planner 269 14.13 A program solving geometric analogies 272
14.14 Testing ANALOGY 273
14.15 ELIZA 275
14.16 McSAM 277
14.17 Testing McSAM 278
15.1 Concatenating difference-lists 285 15.2 Flattening a list of lists using difference-lists 286
15.3 Reverse with difference-lists 288
15.4 Quicksort using difference-lists 289
15.5 A solution to the Dutch flag problem 290 15.6 Dutch flag with difference-lists 291
15.7 Normalizing plus expressions 292 15.8 Dictionary lookup from a list of tuples 294 15.9 Dictionary lookup in a binary tree 295 15.10 Meltingaterm 296 15.11 Aqueueprocess 297 15.12 Flattening a list using a queue 298 16.1 Sample data 302
16.2 Applying set predicates 303
16.3 Implementing an all-solutions predicate using difference-lists, assert, and retract 304
16.4 Testing connectivity breadth-first in a DAG 306
16.5 Testing connectivity breadth-first in a graph 307
16.6 Lee routing 310
16.7 Producing a keyword in context (KWIC) index 313
xxii Programs
16.8 Second-order predicates in Prolog 316
17.1 An interpreter for a nondeterministic finite automaton (NDFA) 320
17.2 An NDFA that accepts the language (ab)* 321
17.3 An interpreter for a nondetermimstic pushdown automaton (NPDA) 322
17.4 An NPDA for palindromes over a finite alphabet 322
17.5 A meta-interpreter for pure Prolog 324
17.6 A meta-interpreter for pure Prolog in continuation style 326
17.7 AtracerforProlog 328 17.8 A meta-interpreter for building a proof tree 329
17.9 A meta-interpreter for reasoning with uncertainty 330 17.10 Reasoning with uncertainty with threshold cutoff 331
17.11 A meta-interpreter detecting a stack overflow 333 17.12 A nonterminating insertion sort 334 17.13 An incorrect and incomplete insertion sort 335
17.14 Bottom-up diagnosis of a false solution 336 17.15 Top-down diagnosis of a false solution 338 17.16 Diagnosing missing solution 340 17.17 Oven placement rule-based system 342 17.18 A skeleton two-level rule interpreter 343 17.19 An interactive rule interpreter 345 17.20 A two-level rule interpreter carrying rules 347
17.21 A two-level rule interpreter with proof trees 348
17.22 Explaining aproof 350 17.23 An explanation shell 352
18.1 A program accepting palindromes 359
18.2 A meta-interpreter for determining a residue 361 18.3 A simple partial reduction system 362 18.4 Specializing an NPDA 363
18.5 Specializing a rule interpreter 364
18.6 Composing two enhancements of a skeleton 368
xxiii Programs
18.7 Testing program composition 370 18.8 A Prolog program parsing the language a*b*c* 371
18.9 Translating grammar rules to Prolog clauses 372
19.1 Enhancing the language a*b*c* 377
19.2 Recognizing the language dt1cN 377 19.3 Parsing the declarative part of a Pascal block 378 19.4 A definite clause grammar (DCG) interpreter 381
19.5 A DCG interpreter that counts words 382
19.6 A DCG context-free grammar 383
19.7 A DCG computing a parse tree 384
19.8 A DCG with subject/object number agreement 385
19.9 A DCG for recognizing numbers 387
20.1 A depth-first state-transition framework for problem solving 390
20.2 Solving the wolf, goat, and cabbage problem 392
20.3 Solving the water jugs problem 394 20.4 Hill climbing framework for problem solving 397 20.5 Test data 398 20.6 Best-first framework for problem solving 399 20.7 Concise best-first framework for problem solving 400 20.8 Framework for playing games 402
20.9 Choosing the best move 403
20.10 Choosing the best move with the minimax algorithm 406 20.11 Choosing a move using minimax with alpha-beta pruning 407 21.1 Playing mastermind 413 21.2 A program for playing a winning game of Nim 417
21.3 A complete program for playing Kalah 424
22.1 A credit evaluation system 432 22.2 Test data for the credit evaluation system 437
23.1 A program for solving equations 442
24.1 A compiler from PL to machine language 462
24.2 Test data 465
Series Foreword
The logic programming approach to computing investigates the use of logic as a programming language and explores computational models based on controlled deduction.
The field of logic programming has seen a tremendous growth in the last several years, both in depth and in scope. This growth is reflected in the number of articles, journals, theses, books, workshops, and confer- ences devoted to the subject. The MIT Press series in logic programming was created to accommodate this development and to nurture it. lt is dedicated to the publication of high-quality textbooks, monographs, col- lections, and proceedings in logic programming.
Ehud Shapiro The Weizmann Institute of Science Rehovot, Israel
Foreword
Programming in Prolog opens the mind to a new way of looking at com- puting. There is a change of perspective which every Prolog programmer experiences when first getting to know the language.
I shall never forget my first Prolog program. The time was early 1974. I had learned about the abstract idea of logic programming from Bob Kowaiski at dinburgh, although the name "logic programming" had not yet been coined. The main idea was that deduction could be viewed as a form of computation, and that a declarative statement of the form
P if Q and R and S.
could also be interpreted procedurally as
To solve P, solve Q and R and S.
Now I had been invited to Marseilles. Here, Alain Colmerauer and his col- leagues had devised the language Prolog based on the logic programming concept. Somehow, this realization of the concept seemed to me, at first sight, too simpleminded. However, Gerard Battani and Henri Meloni had implemented a Prolog interpreter in Fortran (their first major exercise in programming, incidentally). Why not give Prolog a try?
I sat at a clattering teletype connected down an ordinary telephone line to an IBM machine far away in Grenoble. I typed in some rules defining how plans could be constructed as sequences of actions. There was one important rule, modeled on the SRI planner Strips, which described how a plan could be elaborated by adding an action at the end. Another rule, necessary for completeness, described how to elaborate a plan by insert- ing an action in the middle of the plan. As an example for the planner to
xxviii Foreword
work on, I typed in facts about some simple actions in a "blocks world" and an initial state of this world. I entered a description of a goal state to be achieved. Prolog spat back at me:
7
meaning it couldn't find a solution. Could it be that a solution was not deducible from the axioms I had supplied? Ah, yes, I had forgotten to enter some crucial facts. I tried again. Prolog was quiet for a long time and then responded:
DEBORDEMENT DE PILE
Stack overflow! I had run into a loop. Now a loop was conceivable since the space of potential plans to be considered was infinite. However, I had taken advantage of Prolog's procedural semantics to organize the axioms so that shorter plans ought to be generated first. Could something else be wrong? After a lot of head scratching, I finally realized that I had mistyped the names of some variables. I corrected the mistakes, and tried again.
Lo and behold, Prolog responded almost instantly with a correct plan to achieve the goal state. Magic! Declaratively correct axioms had assured a correct result. Deduction was being harnessed before my very eyes to produce effective computation. Declarative programming was truly programming on a higher plane! I had dimly seen the advantages in theory. Now Prolog had made them vividly real in practice. Never had I experienced such ease in getting a complex program coded and running.
Of course, I had taken care to formulate the axioms and organize them in such a way that Prolog could use them effectively. I had a general idea of how the axioms would be used. Nevertheless it was a surprise to see how the axioms got used in practice on particular examples. It was a delightful experience over the next few days to explore how Prolog actually created these plans, to correct one or two more bugs in my facts and rules, and to further refine the program.
Since that time, Prolog systems have improved significantly in terms of debugging environments, speed, and general robustness. The techniques of using Prolog have been more fully explored and are now better un- derstood. And logic programming has blossomed, not least because of its adoption by the Japanese as the central focus of the Fifth Generation project.
xxix Foreword
After more than a decade of growth of interest in Prolog, it is a great pleasure to see the appearance of this book. Hitherto, knowledge of how to use Prolog for serious programming has largely been communicated by word of mouth. This textbook sets down and explains for the first time in an accessible form the deeper principles and techniques of Prolog programming
The book is excellent for not only conveying what Prolog is but also ex- plaining how it should be used. The key to understanding how to use Prolog is to properly understand the relationship between Prolog and logic programming. This book takes great care to elucidate the relation- ship.
Above all, the book conveys the excitement of using Prologthe thrill of declarative programming As the authors put it, "Declarative program- ming clears the mind" Declarative programming enables one to concen- trate on the essentials of a problem without gettmg bogged down in too much operational detail. Programming should be an intellectually rewarding activity. Prolog helps to make it so. Prolog is indeed, as the authors contend, a tool for thinking.
David H. D. Warren Manchester, England, September 1986
Preface
Seven years have passed since the first edition of The Art of Prolog was published. In that time, the perception of Prolog has changed markedly. While not as widely used as the language C, Prolog is no longer regarded as an exotic language. An abundance of books on Prolog have appeared. Prolog is now accepted by many as interesting and useful for certain applications. Articles on Prolog regularly appear in popular magazines. Prolog and logic programming are part of most computer science and engineering programs, although perhaps in a minor role in an artificial intelligence or programming languages class. The first conference on Practical Applications of Prolog was held in London in April 1992. A standard for the language is likely to be in place in 1994. A future for Prolog among the programming languages of the world seems assured.
In preparing for a second edition, we had to address the question of how much to change. I decided to listen to a request not to make the new edition into a new book. This second edition is much like the first, al- though a number of changes are to be expected in a second edition. The typography of the book has been improved: Program code is now in a dis- tinctive font rather than in italics. Figures such as proof trees and search trees are drawn more consistently. We have taken the opportunity to be more precise with language usage and to remove minor inconsistencies with hyphenation of words and similar details. All known typographi- cal errors have been fixed. The background sections at the end of most chapters have been updated to take into account recent, important re- search results. The list of references has been expanded considerably. Extra, more advanced exercises, which have been used successfully in my Prolog classes, have been added.
xxxii Preface
Let us take an overview of the specific changes to each part in turn. Part IV, Applications, is unchanged apart from minor corrections and tidying. Part I, Logic Programs, is essentially unchanged. New programs have been added to Chapter 3 on tree manipulation, including heapifying a binary tree. Extra exercises are also present.
Part II, The Prolog Langauge, is primarily affected by the imminence of a Prolog standard. We have removed all references to Wisdom Prolog in the text in preparation for Standard Prolog. It has proved impossible to guarantee that this book is consistent with the standard. Reaching a stan- dard has been a long, difficult process for the members of the committee. Certain predicates come into favor and then disappear, making it difficult for the authors of a text to know what to write. Furthermore, some of the proposed I/O predicates are not available in current Prologs, so it is im- possible to run all the code! Most of the difficulties in reaching a Prolog standard agreeable to all interested parties have been with builtin or sys- tem predicates. This book raises some of the issues involved in adding builtins to Prolog but largely avoids the concerns by using pure Prolog as much as possible. We tend not to give detailed explanations of the con- troversial nonlogical behaviors of some of the system predicates, and we certainly do not use odd features in our code.
Part III, Advanced Programming Tecimiques, is the most altered in this second edition, which perhaps should be expected. A new chapter has been added on program transformation, and many of the other chapters have been reordered. The chapters on Interpreters and Logic Grammars have extensive additions.
Many people provided us feedback on the first edition, almost all of it very positive. I thank you all. Three people deserve special thanks for taking the trouble to provide long lists of suggestions for improve- ments and to point out embarrassingly long lists of typos in the first edition: Norbert Fuchs, Harald Søndergaard, and Stanley Selkow. The following deserve mention for pointing out mistakes and typos in the various printings of the first edition or making constructive comments about the book that led to improvements in later printings of the first edition and for this second edition. The list is long, my memory some- times short, so please forgive me if I forget to mention anyone. Thanks to Ham Assiryani, Tim Boemker, Jim Brand, Bill Braun, Pu Chen, Yves Deville, George Ernst, Claudia Günther, Ann Halbran, Sundar Iyengar, Gary Kacmarcik, Mansoor Khan, Sundeep Kumar, Arun Lakhotia, Jean-
xxxiii Preface
Louis Lassez, Charlie Linville, Per Ljung, David Maier, Fred Mailey, Martin Marshall, Andre Mesarovic, Dan Oldham, Scott Pierce, Lynn Pierce, David Pedder, S. S. Ramakrishnan, Chet Ramey, Marty Silverstein, Bill Sloan, Ron Taylor, Rodney Topor, R. J. Wengert, Ted Wright, and Nan Yang. For the former students of CMPS41Ì, I hope the extra marks were sufficient re- ward.
Thanks to Sarah Fliegelmann and Venkatesh Srinivasan for help with entering changes to the second edition and TeXing numerous drafts. Thanks to Phil Gannon and Zoë Sterling for helpful discussions about the figures, and to Joe Geiles for drawing the new figures. For proofreading the second edition, thanks to Kathy Kovacic, David Schwartz, Ashish Jam, and Venkatesh Srinivasan. Finally, a warm thanks to my editor, Terry Ehling, who has always been very helpful and very responsive to queries.
Needless to say, the support of my family and friends is the most important and most appreciated.
Leon Sterling Cleveland, January 1993
Preface to First Edition
The origins of this book lie in graduate student courses aimed at teach- ing advanced Prolog programming A wealth of techniques has emerged in the fifteen years since the inception of Prolog as a programming lan- guage. Our intention in this book has been to make accessible the pro- grammmg techniques that kindled our own excitement, imagination, and involvement in this area.
The book fills a general need. Prolog, and more generally logic pro- gramming, has received wide publicity in recent years. Currently avail- able books and accounts, however, typically describe only the basics. All but the simplest examples of the use of Prolog have remained essentially inaccessible to people outside the Prolog community.
We emphasize throughout the book the distinction between logic pro- gramming and Prolog programming Logic programs can be understood and studied, using two abstract, machine-independent concepts: truth and logical deduction. One can ask whether an axiom in a program is true, under some interpretation of the program symbols; or whether a logical statement is a consequence of the program. These questions can be answered independently of any concrete execution mechanism.
On the contrary, Prolog is a programming language, borrowing its basic constructs from logic. Prolog programs have precise operational mean- ing: they are instructions for execution on a computera Prolog ma- chine. Prolog programs in good style can almost always be read as log- ical statements, thus inheriting some of the abstract properties of logic programs. Most important, the result of a computation of such a Pro- log program is a logical consequence of the axioms in it. Effective Prolog
xxxvi Preface to First Edition
progranmiing requires an understanding of the theory of logic program- ming.
The book consists of four parts: logic programming, the Prolog lan- guage, advanced techniques, and applications. The first part is a self- contained introduction to logic programming. It consists of five chapters. The first chapter introduces the basic constructs of logic programs. Our account differs from other introductions to logic programming by ex- plaining the basics in terms of logical deduction. Other accounts explain the basics from the background of resolution from which logic program- ming originated. We have found the former to be a more effective means of teaching the material, which students find intuitive and easy to under- stand.
The second and third chapters of Part I introduce the two basic styles of logic programming: database programming and recursive program- ming. The fourth chapter discusses the computation model of logic pro- gramming, introducing unification, while the fifth chapter presents some theoretical results without proofs. In developing this part to enable the clear explanation of advanced techniques, we have introduced new con- cepts and reorganized others, in particular, in the discussion of types and termination. Other issues such as complexity and correctness are concepts whose consequences have not yet been fully developed in the logic programming research community.
The second part is an introduction to Prolog. It consists of Chapters 6 through 13. Chapter 6 discusses the computation model of Prolog in contrast to logic programming, and gives a comparison between Prolog and conventional programming languages such as Pascal. Chapter 7 dis- cusses the differences between composing Prolog programs and logic programs. Examples are given of basic programming techniques.
The next five chapters introduce system-provided predicates that are essential to make Prolog a practical programming language. We clas- sify Prolog system predicates into four categories: those concerned with efficient arithmetic, structure inspection, meta-logical predicates that discuss the state of the computation, and extra-logical predicates that achieve side effects outside the computation model of logic pro- gramming. One chapter is devoted to the most notorious of Prolog extra-logical predicates, the cut. Basic techniques using these system predicates are explained. The final chapter of the section gives assorted pragmatic programming tips.
xxxvii Preface to First Edition
The main part of the book is Part III. We describe advanced Prolog programming techniques that have evolved in the Prolog programming community, illustrating each with small yet powerful example programs. The examples typify the applications for which the technique is useful. The six chapters cover nondeterministic programming, incomplete data structures, parsing with DCGs, second-order programming, search tech-. niques, and the use of meta-interpreters.
The final part consists of four chapters that show how the material in the rest of the book can be combined to build application programs. A common request of Prolog newcomers is to see larger applications. They understand how to write elegant short programs but have difficulty in building a major program. The applications covered are game-playing programs, a prototype expert system for evaluating requests for credit, a symbolic equation solver, and a compiler.
During the development of the book, it has been necessary to reorga- nize the foundations and basic examples existing in the folklore of the logic programming community. Our structure constitutes a novel frame- work for the teaching of Prolog.
Material from this book has been used successfully for several courses on logic programming and Prolog: in Israel, the United States, and Scot- land. The material more than suffices for a one-semester course to first- year graduate students or advanced undergraduates. There is consider- able scope for instructors to particularize a course to suit a special area of interest.
A recommended division of the book for a 13-week course to senior un- dergraduates or first-year graduates is as follows: 4 weeks on logic pro- gramming, encouraging students to develop a declarative style of writing programs, 4 weeks on basic Prolog programming, 3 weeks on advanced techniques, and 2 weeks on applications. The advanced techniques should include some discussion of nondeterminism, incomplete data structures, basic second-order predicates, and basic meta-interpreters. Other sections can be covered instead of applications. Application areas that can be stressed are search techniques in artificial intellígence, build- ing expert systems, writing compilers and parsers, symbol manipulation, and natural language proces sing.
There is considerable flexibility in the order of presentation. The ma- terial from Part I should be covered first. The material in Parts III and IV can be interspersed with the material in Part lito show the student how
xxxviii Preface to First Edition
larger Prolog programs using more advanced techniques are composed in the same style as smaller examples.
Our assessment of students has usually been 50 percent by homework assignments throughout the course, and 50 percent by project. Our expe- rience has been that students are capable of a significant prograniming task for their project. Examples of projects are prototype expert systems, assemblers, game-playing programs, partial evaluators, and implementa- tions of graph theory algorithms.
For the student who is studying the material on her own, we strongly advise reading through the more abstract material iii Part I. A good Pro- log progranirning style develops from thinking declaratively about the logic of a situation. The theory in Chapter 5, however, can be skipped until a later reading.
The exercises in the book range from very easy and well defined to difficult and open-ended. Most of them are suitable for homework exer- cises. Some of the more open-ended exercises were submitted as course projects.
The code in this book is essentially in Edinburgh Prolog. The course has been given where students used several different variants of Edinburgh Prolog, and no problems were encountered. All the examples run on Wisdom Prolog, which is discussed in the appendixes.
We acknowledge and thank the people who contributed directly to the book. We also thank, collectively and anonymously, all those who indi- rectly contributed by influencing our programming styles in Prolog. Im- provements were suggested by Lawrence Byrd, Oded Maler, Jack Minker, Richard O'Keefe, Fernando Pereira, and several anonymous referees.
We appreciate the contribution of the students who sat through courses as material from the book was being debugged. The first author acknowledges students at the University of Edinburgh, the Weizmann Institute of Science, Tel Aviv University, and Case Western Reserve Uni- versity. The second author taught courses at the Weizmanri Institute and Hebrew University of Jerusalem, and in industry.
We are grateful to many people for assisting in the technical aspects of producing a book. We especially thank Sarah Fliegelmann, who pro- duced the various drafts and camera-ready copy, above and beyond the call of duty. This book might not have appeared without her tremendous efforts. Arvind Bansal prepared the index and helped with the references. Yehuda Barbut drew most of the figures. Max Goldberg and Shmuel Safra
xxxix Preface to First Edition
prepared the appendix. The publishers, MIT Press, were helpful and sup- portive.
Finally, we acknowledge the support of family and friends, without which nothing would get done.
Leon Sterling 1986
Introduction
The inception of logic is tied with that of scientific thinking. Logic pro- vides a precise language for the explicit expression of one's goals, knowl- edge, and assumptions. Logic provides the foundation for deducing consequences from premises; for studying the truth or falsity of state- ments given the truth or falsity of other statements; for establishing the consistency of one's claims; and for verifying the validity of one's argu- ments.
Computers are relatively new in our intellectual history. Similar to logic, they are the object of scientific study and a powerful tool for the advancement of scientific endeavor. Like logic, computers require a precise and explicit statement of one's goals and assumptions. Un- like logic, which has developed with the power of human thinking as the only external consideration, the development of computers has been gov- erned from the start by severe technological and engineering constraints. Although computers were intended for use by humans, the difficul- ties in constructing them were so dominant that the language for expressing problems to the computer and instructing it how to solve them was designed from the perspective of the engineering of the com- puter alone.
Almost all modern computers are based on the early concepts of von Neumann and his colleagues, which emerged during the 1940s. The von Neumann machine is characterized by a large uniform store of memory cells and a processing unit with some local cells, called registers. The processing unit can load data from memory to registers, perform arith- metic or logical operations on registers, and store values of registers back into memory. A program for a von Neumann machine consists of
2 Introduction
a sequence of instructions to perform such operations, and an additional set of control instructions, which can affect the next instruction to be executed, possibly depending on the content of some register.
As the problems of building computers were gradually understood and solved, the problems of using them mounted. The bottleneck ceased to be the inability of the computer to perform the human's instructions but rather the inability of the human to instruct, or program, the computer. A search for programming languages convenient for humans to use be- gan. Starting from the language understood directly by the computer, the machine language, better notations and formalisms were developed. The main outcome of these efforts was languages that were easier for humans to express themselves in but that still mapped rather directly to the underlying machine language. Although increasingly abstract, the languages in the mainstream of development, starting from assembly language through Fortran, Algol, Pascal, and Ada, all carried the mark of the underlying machinethe von Neumann architecture.
To the uninitiated intelligent person who is not familiar with the en- gineering constraints that led to its design, the von Neumann machine seems an arbitrary, even bizarre, device. Thinking in terms of its con- strained set of operations is a nontrivial problem, which sometimes stretches the adaptiveness of the human mind to its limits.
These characteristic aspects of programming von Neumann computers led to a separation of work: there were those who thought how to solve the problem, and designed the methods for its solution, and there were the coders, who performed the mundane and tedious task of translating the instructions of the designers to instructions a computer can use.
Both logic and programming require the explicit expression of one's knowledge and methods in an acceptable formalism. The task of making one's knowledge explicit is tedious. However, formalizing one's knowl- edge in logic is often an intellectually rewarding activity and usually reflects back on or adds insight to the problem under consideration. In contrast, formalizing one's problem and method of solution using the von Neumann instruction set rarely has these beneficial effects.
We believe that programming can be, and should be, an intellectu- ally rewarding activity; that a good programming language is a powerful conceptual toola tool for organizing, expressing, experimenting with, and even communicating one's thoughts; that treating programming as
Introduction
"coding," the last, mundane, intellectually trivial, time-consuming, and tedious phase of solving a problem using a computer system, is perhaps at the very root of what has been known as the "software crisis."
Rather, we think that programming can be, and should be, part of the problem-solving process itself; that thoughts should be organized as programs, so that consequences of a complex set of assumptions can be investigated by "running" the assumptions; that a conceptual solution to a problem should be developed hand-in-hand with a working program that demonstrates it and exposes its different aspects. Suggestions in this direction have been made under the title "rapid prototyping."
To achieve this goal in its fullestto become true mates of the human thinking processcomputers have still a long way to go. However, we find it both appropriate and gratifying from a historical perspective that logic, a companion to the human thinking process since the early days of human intellectual history, has been discovered as a suitable stepping- stone in this long journey.
Although logic has been used as a tool for designing computers and for reasoning about computers and computer programs since almost their beginning, the use of logic directly as a prograniming language, termed logic programming, is quite recent.
Logic programming, as well as its sister approach, functional program- ming, departs radically from the mainstream of computer languages. Rather then being derived, by a series of abstractions and reorganiza- tions, from the von Neumann machine model and instruction set, it is derived from an abstract model, which has no direct relation to or de- pendence on to one machine model or another. It is based on the belief that instead of the human learning to think in terms of the operations of a computer that which some scientists and engineers at some point in history happened to find easy and cost-effective to build, the com- puter should perform instructions that are easy for humans to provide. In its ultimate and purest form, logic programming suggests that even explicit instructions for operation not be given but rather that the knowl- edge about the problem and assumptions sufficient to solve it be stated explicitly, as logical axioms. Such a set of axioms constitutes an alterna- tive to the conventional program. The program can be executed by pro- viding it with a problem, formalized as a logical statement to be proved, called a goal statement. The execution is an attempt to solve the prob-
4 Introduction
lem, that is, to prove the goal statement, given the assumptions in the logic program.
A distinguishing aspect of the logic used in logic programming is that a goal statement typically is existentially quantified: it states that there exist some individuals with some property. An example of a goal state- ment is, "there exists a list X such that sorting the list 13, 1,21 gives X." The mechanism used to prove the goal statement is constructive. If suc- cessful, it provides the identity of the unknown individuals mentioned in the goal statement, which constitutes the output of the computation. In the preceding example, assuming that the logic program contains appro- priate axioms defining the sort relation, the output of the computation wouldbeX= [1,2,3].
These ideas can be summarized in the following two metaphorical equations:
program = set of axioms.
computation = constructive proof of a goal statement from the program.
The ideas behind these equations can be traced back as far as intuition- istic mathematics and proof theory of the early twentieth century. They are related to Hilbert's program, to base the entire body of mathemati- cal knowledge on logical foundations and to provide mechanical proofs for its theories, starting from the axioms of logic and set theory alone. It is interesting to note that the failure of this program, from which en- sued the incompleteness and undecidability results of Gödel and Turing, marks the beginning of the modern age of computers.
The first use of this approach in practical computing is a sequel to Robinson's unification algorithm and resolution principle, published in 1965. Several hesitant attempts were made to use this principle as a basis of a computation mechanism, but they did not gain any momentum. The beginning of logic programming can be attributed to Kowalski and Colmerauer. Kowalski formulated the procedural interpretation of Horn clause logic. He showed that an axiom
A if B1 and B2 and... and B
can be read and executed as a procedure of a recursive programming language, where A is the procedure head and the B are its body. In
Introduction
addition to the declarative reading of the clause, A is true if the B are true, it can be read as follows: To solve (execute) A, solve (execute) B1 and B2 and. . . and B. In this reading, the proof procedure of Horn clause logic is the interpreter of the language, and the unification algorithm, which is at the heart of the resolution proof procedure, performs the basic data manipulation operations of variable assignment, parameter passing, data selection, and data construction.
At the same time, in the early 1970s, Colmerauer and his group at the University of Marseilles-Aix developed a specialized theorem prover, written in Fortran, which they used to implement natural language pro- cessing systems. The theorem prover, called Prolog (for Programmation en Logique), embodied Kowaiski's procedural interpretation. Later, van Emden and Kowalski developed a formal semantics for the language of logic programs, showing that its operational, model-theoretic, and fix- point semantics are the same.
In spite of all the theoretical work and the exciting ideas, the logic pro- gramming approach seemed unrealistic. At the time of its inception, re- searchers in the United States began to recognize the failure of the "next- generation Al languages," such as Micro-Planner and Conniver, which de- veloped as a substitute for Lisp. The main claim against these languages was that they were hopelessly inefficient, and very difficult to control. Given their bitter experience with logic-based high-level languages, it is no great surprise that U.S. artificial intelligence scientists, when hearing about Prolog, thought that the Europeans were over-excited over what they, the Americans, had already suggested, tried, and discovered not to work.
In that atmosphere the Prolog-lo compiler was almost an imaginary being. Developed in the mid to late 1970s by David H. D. Warren and his colleagues, this efficient implementation of Prolog dispelled all the myths about the impracticality of logic programming. That compiler, still one of the finest implementations of Prolog around, delivered on pure list-processing programs a performance comparable to the best Lisp sys- tems available at the time. Furthermore, the compiler itself was written almost entirely in Prolog, suggesting that classic programming tasks, not just sophisticated AI applications, could benefit from the power of logic programming.
6 Introduction
The impact of this implementation cannot be overemphasized. Without it, the accumulated experience that has led to this book would not have existed.
In spite of the promise of the ideas, and the practicality of their im- plementation, most of the Western computer science and AI research community was ignorant, openly hostile, or, at best, indifferent to logic programming. By 1980 the number of researchers actively engaged in logic programming were only a few dozen in the United States and about one hundred around the world.
No doubt, logic programming would have remained a fringe activity in computer science for quite a while longer hadit not been for the an- nouncement of the Japanese Fifth Generation Project, which took place in October 1981. Although the research program the Japanese presented was rather baggy, faithful to their tradition of achieving consensus at almost any cost, the important role of logic programming in the next generation of computer systems was made clear.
Since that time the Prolog language has undergone a rapid transition from adolescence to maturity. There are numerous commercially avail- able Prolog implementations on most computers. A large number of Pro- log programming books are directed to different audiences and empha- size different aspects of the language. And the language itself has more or less stabilized, having a de facto standard, the Edinburgh Prolog fam- ily.
The maturity of the language means that it is no longer a concept for scientists yet to shape and define but rather a given object, with vices and virtues. lt is time to recognize that, on the one hand, Prolog falls short of the high goals of logic programming but, on the other hand, is a powerful, productive, and practical programming formalism. Given the standard life cycle of computer programming languages, the next few years will reveal whether these properties show their merit only in the classroom or prove useful also in the field, where people pay money to solve problems they care about.
What are the current active subjects of research in logic programming and Prolog? Answers to this question can be found in the regular sci- entific journals and conferences of the field; the Logic Programming Journal, the Journal of New Generation Computing, the International Conference on Logic Programming, and the IEEE Symposium on Logic
7 Introduction
Programming as well as in the general computer science journals and conf erences.
Clearly, one of the dominant areas of interest is the relation between logic programming, Prolog, and parallelism. The promise of parallel com- puters, combined with the parallelism that seems to be available in the logic programming model, have led to numerous attempts, still ongoing, to execute Prolog in parallel and to devise novel concurrent program- ming languages based on the logic programming computation model. This, however, is a subject for another book.
r
-- &J 2è - L
Leonardo Da Vinci. Old Man thinking. Pen and ink (slightly enlarged). About 1510. Windsor Castle, Royal Library.
i
I Logic Programs
A logic program is a set of axioms, or rules, defining relations between objects. A computation of a logic program is a deduction of conse- quences of the program. A program defines a set of consequences, which is its meaning The art of logic programming is constructing concise and elegant programs that have the desired meaning.
Li Facts
The simplest kind of statement is called a fact. Facts are a means of stating that a relation holds between objects. An example is
father(abrahani,isaac).
This fact says that Abraham is the father of Isaac, or that the relation f a- ther holds between the individuals named abraham and isaac. Another name for a relation is a predicate. Names of individuals are known as atoms. Similarly, plus(2,3,5) expresses the relation that 2 plus 3 is 5. The familiar plus relation can be realized via a set of facts that defines the addition table. An initial segment of the table is
plus(O,O,O). plus(O,1,1). plus(O,2,2). plus(O,3,3).
Basic Constructs
The basic constructs of logic programming, terms and statements, are inherited from logic. There are three basic statements: facts, rules, and queries. There is a single data structure: the logical term.
plus(1,O,1). plus(1,1,2). plus(1,2,3). plus(1,3,4).
A sufficiently large segment of this table, which happens to be also a legal logic program, will be assumed as the definition of the plus relation throughout this chapter.
The syntactic conventions used throughout the book are introduced as needed. The first is the case convention. It is significant that the names
12 Chapter 1
father(terach,abraham). male(terach).
father (terach,nachor). male (abraham).
father (terach,haran). male (nachor).
f ather(abraham,isaac). male (haran).
father (haran,lot). male(isaac).
father (haran,milcah). male (lot).
father(haran,yiscah).
female(sarah).
mother(sarah,isaac). female (milcah).
female(yiscah).
Program 1.1 A biblical family database
of both predicates and atoms in facts begin with a lowercase letter rather than an uppercase letter.
A finite set of facts constitutes a program. This is the simplest form of logic program. A set of facts is also a description of a situation. This insight is the basis of database programming, to be discussed in the next chapter. An example database of family relationships from the Bible is given as Program 1.1. The predicates father, mother, male, and female express the obvious relationships.
1.2 Queries
The second form of statement in a logic program is a query. Queries are a means of retrieving information from a logic program. A query asks whether a certain relation holds between objects. For example, the query father (abraham, isaac)? asks whether the father relationship holds between abraham and isaac. Given the facts of Program 1.1, the answer to this query is yes.
Syntactically, queries and facts look the same, but they can be distin- guished by the context. When there is a possibility of confusion, a term!- nating period will indicate a fact, while a terminating question mark will indicate a query. We call the entity without the period or question mark a goal. A fact P. states that the goal P is true. A query P? asks whether the goal P is true. A simple query consists of a single goal.
Answering a query with respect to a program is determining whether the query is a logical consequence of the program. We define logical
13 Basic Constructs
consequence incrementally through this chapter. Logical consequences are obtained by applying deduction rules. The simplest rule of deduction is identity: from P deduce P. A query is a logical consequence of an identical fact.
Operationally, answermg simple queries using a program containing facts like Program 1.1 is straightforward. Search for a fact in the program that implies the query. If a fact identical to the query is found, the answer is yes.
The answer no is given if a fact identical to the query is not found, because the fact is not a logical consequence of the program. This answer does not reflect on the truth of the query; it merely says that we failed to prove the query from the program. Both the queries f emale(abrahazn)? and plus(1 ,1,2)? will be answered no with respect to Program 1.1.
1.3 The Logical Variable, Substitutions, and Instances
A logical variable stands for an unspecified individual and is used ac- cordingly. Consider its use in queries. Suppose we want to know of whom abraham is the father. One way is to ask a series of queries, father(abraham,lot)?, father(abraham,milcah)?, ..., father
(abraham,isaac)?,. . . until an answer yes is given. A variable allows a better way of expressing the query as f ather(abraham,X)?, to which the answer is X=isaac. Used in this way, variables are a means of sum- marizing many queries. A query containing a variable asks whether there is a value for the variable that makes the query a logical consequence of the program, as explained later.
Variables in logic programs behave differently from variables in con- ventional programming languages. They stand for an unspecified but sin- gle entity rather than for a store location in memory.
Having introduced variables, we can define a term, the single data structure ¡ii logic programs. The definition is inductive. Constants and variables are terms. Also compound terms, or structures, are terms. A compound term comprises a functor (called the principal functor of the term) and a sequence of one or more arguments, which are terms. A functor is characterized by its name, which is an atom, and its arity, or number of arguments. Syntactically, compound terms have
14 Chapter 1
the form f(t1,t2,. . .,t), where the functor has name f and is of arity n, and the t are the arguments. Examples of compound terms include s(0), hot(milk), name(john,doe), list(a,list(b,nil)), foo(X), and tree(tree(nil,3,nil) ,5,R).
Queries, goals, and more generally terms where variables do not occur are called ground. Where variables do occur, they are called nonground. For example, f oo (a, b) is ground, whereas bar (X) is nonground.
Definition A substitution is a finite set (possibly empty) of pairs of the form X = t, where X is a variable and t is a term, and X X1 for every i j, and X does not occur in t1, for any i and j. .
An example of a substitution consisting of a single pair is {X=isaac}. Substitutions can be applied to terms. The result of applying a substi- tution G to a term A, denoted by AO, is the term obtained by replacing every occurrence of X by t in A, for every pair X = t in O.
The result of applying {X=isaac} to the term f ather(abraham,X) is the term father(abrahani,isaac).
Definition A is an instance of B if there is a substitution O such that A = BO. .
The goal f ather(abraham,isaac) is an instance of father (abraham, X) by this definition. Similarly, mother(sarah, isaac) is an instance of mother(X,Y) under the substitution {X=sarah,Y=isaac}.
1.4 Existential Queries
Logically speaking, variables in queries are existentially quantified, which means, intuitively, that the query father (abraham, X)? reads: "Does there exist an X such that abraham is the father of X?" More generally, a query p(T1,T2,. . .,T)?, which contains the variables X1,X2,. . .,Xk reads: "Are there X1,X2,. . .,Xk such that p(T1,T2,. . .,T)?" For convenience, exis- tential quantification is usually omitted.
The next deduction rule we introduce is generalization. An existential query P is a logical consequence of an instance of it, PO, for any substi- tution 6. The fact f ather(abraham, isaac) implies that there exists an X such that f ather(abraham,X) is true, namely, X=isaac.
15 Basic Constructs
Operationally, to answer a nonground query using a program of facts, search for a fact that is an instance of the query. If found, the answer, or solution, is that instance. A solution is represented in this chapter by the substitution that, if applied to the query, results in the solution. The answer is no if there is no suitable fact in the program.
In general, an existential query may have several solutions. Program 1.1 shows that Haran is the father of three children. Thus the query father (haran,X)? has the solutions (X=lot}, {X=milcah}, {X=yiscah}. Another query with multiple solutions is plus (X, Y, 4)? for finding num- bers that add up to 4. Solutions are, for example, {X=O, Y=4} and {X=1, Y=3}. Note that the different variables X and Y correspond to (possibly) different objects.
An interesting variant of the last query is plus(X,X,4)?, which insists that the two numbers that add up to 4 be the same. It has a unique answer {X=2}.
LS Universal Facts
Variables are also useful in facts. Suppose that all the biblical characters like pomegranates. Instead of including in the program an appropriate fact for every individual,
likes(abraham,pomegranates).
likes(sarah,pomegranates).
a fact likes (X,pomegranates) can say it all. Used in this way, variables are a means of summarizing many facts. The fact tirnes(O,X,O) summa- rizes all the facts stating that o times some number is O.
Variables in facts are implicitly universally quantified, which means, intuitively, that the fact likes(X,pomegranates) states that for all X, X likes pomegranates. In general, a fact p(T1,. . .,T) reads that for all X1,. . .,Xk, where the X1 are variables occurring in the fact, p(T1,. . is true. Logically, from a universally quantified fact one can deduce any instance of it. For example, from likes(X,poniegranates), deduce likes (abraham, pomegranates).
16 Chapter 1
This is the third deduction rule, called instantiation. From a universally quantified statement P, deduce an instance of it, PO, for any substitution 0.
As for queries, two unspecified objects, denoted by variables, can be constrained to be the same by using the same variable name. The fact plus (0, X, X) expresses that O is a left identity for addition. It reads that for all values of X, O plus X is X. A similar use occurs when translating the English statement "Everybody likes himself" to likes (X, X).
Answering a ground query with a universally quantified fact is straight- forward. Search for a fact for which the query is an instance. For example, the answer to plus(0,2,2)? is yes, based on the fact plus(0,X,X). An- swering a nonground query using a nonground fact involves a new defi- nition: a common instance of two terms.
Definition C is a common instance of A and B if it is an instance of A and an instance of B, in other words, if there are substitutions 0 and 02 such that C=A01 is syntactically identical to BO2.
For example, the goals plus(0,3,Y) and plus(0,X,X) have a com- mon instance plus (0,3,3). When the substitution {Y=3} is applied to plus (0,3, Y) and the substitution {X=3} is applied to plus (0, X, X), both yield plus(0,3,3).
In general, to answer a query using a fact, search for a common in- stance of the query and fact. The answer is the common instance, if one exists. Otherwise the answer is no.
Answering an existential query with a universal fact using a common instance involves two logical deductions. The instance is deduced from the fact by the rule of instantiation, and the query is deduced from the instance by the rule of generalization.
1.6 Conjunctive Queries and Shared Variables
An important extension to the queries discussed so far is conjunctive queries. Conjunctive queries are a conjunction of goals posed as a query, for example, f ather(terach,X) ,father(X,Y)? or in general, Q,. . Simple queries are a special case of conjunctive queries when there is a
17 Basic Constructs
single goal. Logically, it asks whether a conjunction is deducible from the program. We use '," throughout to denote logical and. Do not confuse the comma that separates the arguments in a goal with commas used to separate goals, denoting conjunction.
In the simplest conjunctive queries all the goals are ground, for exam- ple, f ather(abraham,isaac) ,male(lot)?. The answer to this query us- ing Program 1.1 is clearly yes because both goals in the query are facts in the program. In general, the query Q,. .,Q,?, where each Q is a ground goal, is answered yes with respect to a program P if each Q is implied by P. Hence ground conjunctive queries are not very interesting.
Conjunctive queries are interesting when there are one or more shared variables, variables that occur in two different goals of the query. An ex- ample is the query f ather(haran,X) ,male(X)?. The scope of a variable in a conjunctive query, as in a simple query, is the whole conjunction. Thus the query p(X),q(X)? reads: "Is there an X such that both p(X) and
Shared variables are used as a means of constraining a simple query by restricting the range of a variable. We have already seen an example with the query plus (X , X, 4)?, where the solution of numbers adding up to 4 was restricted to the numbers being the same. Consider the query f ather(harari,X) ,male(X)?. Here solutions to the query f a- ther(haran,X)? are restricted to children that are male. Program 1.1 shows there is only one solution, {X1ot}. Alternatively, this query can be viewed as restricting solutions to the query male (X)? to individuals who have Haran for a father.
A slightly different use of a shared variable can be seen in the query father (terach,X) ,father(X,Y)?. On the one hand, it restricts the sons of terach to those who are themselves fathers. On the other hand, it con- siders individuals Y, whose fathers are sons of terach. There are several solutions, for example, (Xabrahain, Y=isaac J and {X=haran, Y=lot J.
A conjunctive query is a logical consequence of a program P if all the goals in the conjunction are consequences of P, where shared variables are instantiated to the same values in different goals. A sufficient condi- tion is that there be a ground instance of the query that is a consequence of P. This instance then deduces the conjuncts in the query via general- ization.
The restriction to ground instances is unnecessary and will be lifted in Chapter 4 when we discuss the computation model of logic programs.
18 Chapter 1
We employ this restriction in the meantime to simplify the discussion in the coming sections.
Operationally, to solve the conjunctive query A1,A2 ,...,A? using a pro- gram P, find a substitution O such that A10 and.. . and AO are ground instances of facts in P. The same substitution applied to all the goals en- sures that instances of variables are common throughout the query. For example, consider the query f ather(haran,X) ,male(X)? with respect to Program 1.1. Applying the substitution {X=lot} to the query gives the ground instance father (haran,lot) ,male (lot)?, which is a conse- quence of the program.
1.7 Rules
Interesting conjunctive queries are defining relationships in their own right. The query father (haran,X) ,male(X)? is asking for a son of Ha- ran. The query father(terach,X) ,father(X,Y)? is asking about grand- children of Terach. This brings us to the third and most important state- ment in logic programming, a rule, which enables us to define new rela- tionships in terms of existing relationships.
Rules are statements of the form:
A - B1,B2,. .
where n O. The goal A is the head of the rule, and the conjunction of goals B1,. . .,B is the body of the rule. Rules, facts, and queries are also called Horn clauses, or clauses for short. Note that a fact is just a special case of a rule when n = O. Facts are also called unit clauses. We also have a special name for clauses with one goal in the body, namely, when n = 1. Such a clause is called an iterative clause. As for facts, variables appearing in rules are universally quantified, and their scope is the whole rule.
A rule expressing the son relationship is
son(X,Y) - father(Y,X), male(X).
Similarly one can define a rule for the daughter relationship:
daughter(X,Y) - father(Y,X), female(X).
19 Basic Constructs
A rule for the grandfather relationship is
grandfather(X,Y) father(X,Z), father(Z,Y).
Rules can be viewed in two ways. First, they are a means of ex- pressing new or complex queries in terms of simple queries. A query son(X,haran)? to the program that contains the preceding rule for son is translated to the query f ather(haran,X) ,male(X)? according to the rule, and solved as before. A new query about the son relationship has been built from simple queries involving father and male relationships. Interpreting rules in this way is their procedural reading. The procedural reading for the grandfather rule is: "To answer a query is X the grand- father of Y?, answer the conjunctive query Is X the father of Z and Z the father of Y?."
The second view of rules comes from interpreting the rule as a logical axiom. The backward arrow is used to denote logical implication. The son rule reads: "X is a son of Y if Y is the father of X and X is male." In this view, rules are a means of defining new or complex relationships using other, simpler relationships. The predicate son has been defined in terms of the predicates father and male. The associated reading of the rule is known as the declarative reading. The declarative reading of the grandfather rule is: "For all X, Y, and Z, X is the grandfather of Y if X is the father of Z and Z is the father of Y."
Although formally all variables in a clause are universally quantified, we will sometimes refer to variables that occur in the body of the clause, but not in its head, as if they are existentially quantified inside the body. For example, the grandfather rule can be read: "For all X and Y, X is the grandfather of Y if there exists a Z such that X is the father of Z and Z is the father of Y." The formal justification of this verbal transformation will not be given, and we treat it just as a convenience. Whenever it is a source of confusion, the reader can resort back to the formal reading of a clause, in which all variables are universally quantified from the outside.
To incorporate rules into our framework of logical deduction, we need the law of modus ponens. Modus ponens states that from B and A B we can deduce A,
Definition The law of universal modus ponens says that from the rule
R = (A - B1,B2,. .
20 Chapter 1
and the facts
B. B.
B.
A' can be deduced if
A' - B'1,B,. .
is an instance of R.
Universal modus ponens includes identity and instantiation as special cases.
We are now in a position to give a complete definition of the concept of a logic program and of its associated concept of logical consequence.
Definition A logic program is a finite set of rules.
Definition An existentially quantified goal G is a logical consequence of a program P if there is a clause in P with a ground instance A - B1,. . . ,B, n O such that B1.....B are logical consequences of P, and A is an instance of G. .
Note that the goal G is a logical consequence of a program P if and only if G can be deduced from P by a finite number of applications of the rule of universal modus pollens.
Consider the query son (S, haran)? with respect to Program 1.1 aug- mented by the rule for son. The substitution {X=lot , Y=haran} applied to the rule gives the instance son(lot,haran) f ather(haran,lot), male(lot). Both the goals in the body of this rule are facts in Pro- gram 1.1. Thus universal modus ponens implies the query with answer {S=lot}.
Operationally, answering queries reflects the definition of logical con- sequence. Guess a ground instance of a goal, and a ground instance of a rule, and recursively answer the conjunctive query corresponding to the body of that rule. To solve a goal A with program P, choose a rule A1 B1,B2,. . .,B in P, and guess substitution O such that A = A10, and
21 Basic Constructs
BO is ground for i í n. Then recursively solve each BLO. This pro- cedure can involve arbitrarily long chains of reasoning. lt is difficult in general to guess the correct ground instance and to choose the right rule. We show in Chapter 4 how the guessing of an instance can be removed.
The rule given for son is correct but is an incomplete specification of the relationship. For example, we cannot conclude that Isaac is the son of Sarah. What is missing is that a child can be the son of a mother as well as the son of a father. A new rule expressing this relationship can be added, namely,
son(X,Y) - niother(Y,X), male(X).
To define the relationship grandparent correctly would take four rules to include both cases of father and mother:
grandparent (X, Y)
grandparent (X ,Y)
grandparent (X, Y)
grandparent (X ,Y)
- father(X,Z)
- father(X,Z)
- mother(X,Z)
- mother(X,Z)
father(Z,
mother (Z
father(Z,
mother (Z
Y).
Y).
Y).
Y).
There is a better, more compact, way of expressing these rules. We need to define the auxiliary relationship parent as being a father or a mother. Part of the art of logic programming is deciding on what intermediate predicates to define to achieve a complete, elegant axiomatization of a relationship. The rules defining parent are straightforward, capturing the definition of a parent being a father or a mother. Logic programs can incorporate alternative definitions, or more technically disjunction, by having alternative rules, as for parent:
parent(X,Y) - f ather(X,Y).
parent(X,Y) mother(X,Y).
Rules for son and grandparent are now, respectively,
son(X,Y) - parent(Y,X), male(X).
grandparent(X,Y) - parent(X,Z), parent(Z,Y).
A collection of rules with the same predicate in the head, such as the pair of parent rules, is called a procedure. We shall see later that under the operational interpretation of these rules by Prolog, such a collection of rules is indeed the analogue of procedures or subroutines in conventional programming languages,
22 Chapter 1
1.8 A Simple Abstract Interpreter
An operational procedure for answering queries has been informally de- scribed and progressively developed in the previous sections. In this section, the details are fleshed out into an abstract interpreter for logic programs. In keeping with the restriction of universal modus ponens to ground goals, the interpreter only answers ground queries.
The abstract interpreter performs yes/no computations. It takes as input a program and a goal, and answers yes if the goal is a logi- cal consequence of the program and no otherwise. The interpreter is given in Figure 1.1. Note that the interpreter may fail to terminate if the goal is not deducible from the program, in which case no answer is given.
The current, usually conjunctive, goal at any stage of the computation is called the resolvent. A trace of the interpreter is the sequence of resol- vents produced during the computation. Figure 1.2 is a trace of answer- ing the query son(1ot,harn)? with respect to Program 1.2, a subset of the facts of Program 1.1 together with rules defining son and daughter. For clarity, Figure 1.2 also explicitly states the choice of goal and clause made at each iteration of the abstract interpreter.
Each iteration of the while loop of the abstract interpreter corresponds to a single application of modus ponens. This is called a reduction.
Input:
Output:
Algorithm
A ground goal G and a program P
yes if G is a logical consequence of P, no otherwise
Initialize the resolvent to G. while the resolvent is not empty do
choose a goal A from the resolvent choose a ground instance of a clause A' B1.....B,, from P
such that A and A' are identical (if no such goal and clause exist, exit the while loop)
replace A by B1.....B,, in the resolvent If the resolvent is empty, then output yes, else output no.
Figure 1.1 An abstract interpreter to answer ground queries with respect to logic programs
23 Basic Constructs
Input: sori(lot,haran)? and Program L2 Resolvent is son(lot ,haran) Resolvent is not empty
choose son(lot,haran) (the only choice) choose son(lot,haran) father(haran,lot), male(lot) new resolvent is father(haran,lot), male(lot)
Resolvent is not empty choose father (hamo ,lot) choose father (haran,lot). new resolvent is inale(lot)
Resolvent is not empty choose male(lot) choose male(lot) new resolvent is empty
Output: yes
Figure 1.2 Tracing the interpreter
father(abraham,isaac) male(isaac)
father(haran,lot). male(lot).
father (haran,mïlcah). female (milcah).
father(haran,yiscah) female(yiscah).
son(X,Y) - father(Y,X), male(X).
daughter(X,Y) father(Y,X), female(X).
Program 1.2 Biblical family relationships
Definition A reduction of a goal G by a program P is the replacement of G by the body of an instance of a clause in P, whose head is identical to the chosen goal.
A reduction is the basic computational step in logic programming. The goal replaced in a reduction is reduced, and the new goals are derived. In this chapter, we restrict ourselves to ground reductions, where the goal and the instance of the clause are ground. Later, in Chapter 4, we consider more general reductions where unification is used to choose the instance of the clause and make the goal to be reduced and the head of the clause identical.
24 Chapter 1
The trace in Figure 1.2 contains three reductions. The first reduces the goal son(lot,haran) and produces two derived goals, f ather(haran, lot) and male (lot). The second reduction is of father (har,lot) and produces no derived goals. The third reduction also produces no derived goals in reducing male (lot).
There are two unspecified choices in the interpreter in Figure 1.1. The first is the goal to reduce from the resolvent. The second choice is the clause (and an appropriate ground instance) to reduce the goal. These two choices have very different natures.
The selection of the goal to be reduced is arbitrary. In any given resol- vent, all the goals must be reduced. It can be shown that the order of reductions is immaterial for answering the query.
In contrast, the choice of the clause and a suitable instance is criti- cal. In general, there are several choices of a clause, and infinitely many ground instances. The choice is made nondeterministically. The concept of nondeterministic choice is used in the definition of many computa- tion models, e.g., finite automata and Turing machines, and has proven to be a powerful theoretic concept. A nondeterrninistic choice is an un- specified choice from a number of alternatives, which is supposed to be made in a "clairvoyant" way. If only some of the alternatives lead to a successful computation, then one of them is chosen. Formally, the con- cept is defined as follows. A computation that contains nondeterministic choices succeeds if there is a sequence of choices that leads to success. Of course, no real machine can directly implement this definition. How- ever, it can be approximated in a useful way, as done in Prolog. This is explained in Chapter 6.
The interpreter given in Figure 1.1 can be extended to answer non- ground existential queries by an initial additional step. Guess a ground instance of the query. This is identical to the step in the interpreter of guessing ground instances of the rules. It is difficult in general to guess the correct ground instance, since that means knowing the result of the computation before performing it.
A new concept is needed to lift the restriction to ground instances and remove the burden of guessing them. In Chapter 4, we show how the guess of ground instances can be eliminated, and we introduce the com- putational model of logic programs more fully. Until then it is assumed that the correct choices can be made.
25 Basic Constructs
Figure 1.3 A simple proof tree
A trace of a query implicitly contains a proof that the query follows from the program. A more convenient representation of the proof is with a proof tree. A proof tree consists of nodes and edges that represent the goals reduced during the computation. The root of the proof tree for a simple query is the query itself. The nodes of the tree are goals that are reduced during the computation. There is a directed edge from a node to each node corresponding to a derived goal of the reduced goal. The proof tree for a conjunctive query is just the collection of proof trees for the individual goals in the conjunction. Figure 1.3 gives a proof tree for the program trace in Figure 1.2.
An important measure provided by proof trees is the number of nodes in the tree. lt indicates how many reduction steps are performed in a computation. This measure is used as a basis of comparison between different programs in Chapter 3.
1.9 The Meaning of a Logic Program
How can we know if a logic program says what we wanted it to say? If it is correct, or incorrect? In order to answer such questions, we have to define what is the meaning of a logic program. Once defined, we can examine if the program means what we have intended it to mean.
Definition The meaning of a logic program P, M(P), is the set of ground goals deducible from P. u
From this definition it follows that the meaning of a logic program composed just of ground facts, such as Program 1.1, is the program it- self. In other words, for simple programs, the program "means just what
26 Chapter 1
it says." Consider Program 1.1 augmented with the two rules defining the parent relationship. What is its meaning? It contains, in addition to the facts about fathers and mothers, mentioned explicitly in the pro- gram, all goals of the form parent(X,Y) for every pair X and Y such that father (X, Y) or mother (X, Y) is in the program. This example shows that the meaning of a program contains explicitly whatever the program states implicitly.
Assuming that we define the intended meaning of a program also to be a set of ground goals, we can ask what is the relation between the actual and the intended meanings of a program. We can check whether everything the program says is correct, or whether the program says everything we wanted it to say.
Informally, we say that a program is correct with respect to some intended meaning M if the meaning of P, M(P), is a subset of M. That is, a correct program does not say things that were not intended. A program is complete with respect to M if M is a subset of M(P). That is, a complete program says everything that is intended. It follows that a program P is correct and complete with respect to an intended meaning M if M = M(P).
Throughout the book, when meaningful predicate and constant names are used, the intended meaning of the program is assumed to be the one intuitively implied by the choice of names.
For example, the program for the son relationship containing only the first axiom that uses father is incomplete with respect to the in- tuitively understood intended meaning of son, since it cannot deduce son(isaac,sarah). If we add to Program 1.1 the rule
son(X,Y) - rnother(X,Y), male(Y).
it would make the program incorrect with respect to the intended mean- mg, since it deduces son(sarah,isaac).
The notions of correctness and completeness of a logic program are studied further in Chapter 5.
Although the notion of truth is not defined fully here, we will say that a ground goal is true with respect to an intended meaning if it is a member of it, and false otherwise. We will say it is simply true if it is a member of the intended meaning implied by the names of the predicate and constant symbols appearing in the program.
27 Basic Constructs
1.10 Summary
We conclude this section with a summary of the constructs and concepts introduced, filling in the remaining necessary definitions.
The basic structure in logic programs is a term. A term is a constant, a variable, or a compound term. Constants denote particular individuals such as integers and atoms, while variables denote a single but unspec- ified individual. The symbol for an atom can be any sequence of char- acters, which is quoted if there is possibility of confusion with other symbols (such as variables or integers). Symbols for variables are distin- guished by beginning with ari uppercase letter.
A compound term comprises a functor (called the principal functor of the term) and a sequence of one or more terms called arguments. A functor is characterized by its name, which is an atom, and its arity or number of arguments. Constants are considered functors of arity O. Syn- tactically, compound terms have the form f(t1,t2.....t) where the functor has name f and is of arity n, and the t are the arguments. A functor f of arity n is denoted f/n. Functors with the same name but different arities are distinct. Terms are ground if they contain no variables; other- wise they are nonground. Goals are atoms or compound terms, and are generally nonground.
A substitution is a finite set (possibly empty) of pairs of the form X = t, where X is a variable and t is a term, with no variable on the left-hand side of a pair appearing on the right-hand side of another pair, and no two pairs having the same variable as left-hand side. For any substitution O {X1 = ti,X2 t2,... ,X = t} and term s, the term sO denotes the result of simultaneously replacing in s each occurrence of the variable X by t, i j <n; the term sO is called an instance of s. More will be said on this restriction on substitutions in the background to Chapter 4.
A logic program is a finite set of clauses. A clause or rule is a univer- sally quantified logical sentence of the form
A - B1,B2,. . .,Bk. k O,
where A and the B are goals. Such a sentence is read declaratively: "A is implied by the conjunction of the B1," and is interpreted procedurally "To answer query A, answer the conjunctive query B1,B2,. . .,Bk." A is called the clause's head and the conjunction of the B the clause's body. If k = O,
28 Chapter 1
the clause is known as a fact or unit clause and written A., meaning A is true under the declarative reading, and goal A is satisfied under the procedural interpretation. If k = 1, the clause is known as an iterative clause.
A query is a conjunction of the form
A1,...,A? n>O,
where the A are goals. Variables in a query are understood to be existen- tially quantified.
A computation of a logic program P finds an instance of a given query logically deducible from P. A goal G is deducible from a program P if there is an instance A of G where A B1,.. .,B, n O, is a ground instance of a clause in P, and the B are deducible from P. Deduction of a goal from an identical fact is a special case.
The meaning of a program P is inductively defined using logical de- duction. The set of ground instances of facts in P are in the meaning A ground goal G is in the meaning if there is a ground instance G B1,. . of a rule in P such that B1,. . .,B are in the meaning. The meaning consists of the ground instances that are deducible from the program.
An intended meaning M of a program is also a set of ground unit goals. A program P is correct with respect to an intended meaning M if M(P) is a subset of M. It is complete with respect to M if M is a subset of M(P). Clearly, it is correct and complete with respect to its intended meaning, which is the desired situation, if M = M(P).
A ground goal is true with respect to an intended meaning if it is a member of it, and false otherwise.
Logical deduction is defined syntactically here, and hence also the meaning of logic programs. In Chapter 5, alternative ways of describing the meaning of logic programs are presented, and their equivalence with the current definition is discussed.
2 Database Programming
2.1 Simple Databases
We begin by revising Program 1.1, the biblical database, and its aug- mentation with rules expressing family relationships. The database itself had four basic predicates, f ather/2, mother/2, male/i, and f e- male/i. We adopt a convention from database theory and give for each relation a relation scheme that specifies the role that each po- sition in the relation (or argument in the goal) is intended to repre- sent. Relation schemes for the four predicates here are, respectively, f ather(Father,Chilcl), mother(Mother,Child), male(Person), and female (Person). The mnemonic names are intended to speak for them- selves.
Variables are given mnemonic names in rules, but usually X or Y when discussing queries. Multiword names are handled differently for vari- ables and predicates. Each new word ¡n a variable starts with an upper- case letter, for example, NieceOrNephew, while words are delimited by
There are two basic styles of using logic programs: defining a logical database, and manipulating data structures. This chapter discusses data- base programming. A logic database contains a set of facts and rules. We show how a set of facts can define relations, as in relational data- bases. We show how rules can define complex relational queries, as in relational algebra. A logic program composed of a set of facts and rutes of a rather restricted format can express the functionalities associated with relational databases.
30 Chapter 2
underscores for predicate and function names, for example, schedule_ conf lict.
New relations are built from these basic relationships by defining suit- able rules. Appropriate relation schemes for the relationships introduced in the previous chapter are son(Son,Parent), daughter(Daughter, Parent), parent (Parent ,Child), and grandparent(Grandparent, Grandchild). From the logical viewpoint, it is unimportant which re- lationships are defined by facts and which by rules. For example, if the available database consisted of parent, male and female facts, the rules defining son and grandparent are still correct. New rules must be writ- ten for the relationships no longer defined by facts, namely, father and mother. Suitable rules are
father (Dad, Child) - parent(Dad, Child), male(Dad).
mother (Mum, Child) - parent(Mum, Child), f emale(Mum).
Interesting rules can be obtained by making relationships explicit that are present in the database only implicitly. For example, since we know the father and mother of a child, we know which couples produced off- spring, or to use a Biblical term, procreated. This is not given explicitly in the database, but a simple rule can be written recovering the information. The relation scheme is procreated(Man,Woman).
procreated(Man,Woman) -
father (Man,Child), mother(Woman,Child).
This reads: "Man and Woman procreated if there is a Child such that Man is the father of Child and Woman is the mother of Child."
Another example of information that can be recovered from the simple information present is sibling relationships - brothers and sisters. We give a rule for brother (Brother,Sibling).
brother (Brother ,Sib) -
parent (Parent ,Brother), parent(Parent ,Sib), male(Brother).
This reads: "Brother is the brother of Sib if Parent is a parent of both Brother and Sib, and Brother is male."
There is a problem with this definition of brother. The query brother (X,X)? is satisfied for any male child X, which is not our understanding of the brother relationship.
In order to preclude such cases from the meaning of the program,
31 Database Programming
abraham isaac.
abraham milcah.
isaac lot.
liaran lot.
lot milcah.
Figure 2.1 Defining inequality
uncle (Uncle Person)
brother(Uncle,Parent), parent(Parent,Person).
sibling(Sibl,Sib2) -
parent(Parent,Sibl), parent(Parent,Sib2), Sibi Sib2.
cousin(Cousini ,Cousin2) -
parent (Parenti ,Cousini),
parent (Parent2 ,Cousin2),
sibling(Parentl,Parent2).
Program 2.1 Defining family relationships
we introduce a predicate (Termi , Term2). It is convenient to write this predicate as an infix operator. Thus Termi Term2 is true if Terni and Term2 are different. For the present it is restricted to constant terms. lt can be defined, in principle, by a table X Y for every two different individuals X and Y in the domain of interest. Figure 2.1 gives part of the appropriate table for Program 1.1.
The new brother rule is
brother (Brother,Sib) parent (Parent ,Brother), parent (Parent ,Sib), male (Brother) Brother Sib.
The more relationships that are present, the easier it is to define com- plicated relationships. Program 2.1 defines the relationships uncle(Uncle,Person), sibling(Sibl,Sib2), and cousin(Cousinl, Cousin2). The definition of uncle in Program 2.1 does not define the husband of a sister of a parent to be an uncle. This may or may not be the intended meaning. In general, different cultures define these family relationships differently. In any case, the logic makes clear exactly what the programmer means by these family relationships.
abraham haran.
abraham yiscah.
isaac milcah.
haran milcah.
lot yiscah.
abraham lot.
isaac liaran.
isaac yiscah.
haran 4 yiscah.
mïlcah yiscah.
32 Chapter 2
Another relationship implicit in the family database is whether a woman is a mother. This is determined by using the mother/2 relation- ship. The new relation scheme is mother (Woman), defined by the rule
mother(Woman) - mother(Woman,Child).
This reads: "Woman is a mother if she is the mother of some Child." Note that we have used the same predicate name, mother, to describe two different mother relationships. The mother predicate takes a different number of arguments, i.e., has a different arity, in the two cases. In general, the same predicate name denotes a different relation when it has a different arity.
We change examples, lest the example of family relationships become incestuous, and consider describing simple logical circuits. A circuit can be viewed from two perspectives. The first is the topological layout of the physical components usually described in the circuit diagram. The second is the interaction of functional units. Both views are easily ac- commodated in a logic program. The circuit diagram is represented by a collection of facts, while rules describe the functional components.
Program 2.2 is a database giving a simplified view of the logical and- gate drawn in Figure 2.2. The facts are the connections of the particular resistors and transistors comprising the circuit. The relation scheme for resistors is resistor(Endl,End2) and for transistors transis- tor (Gate ,Source, Drain).
Power
Figure 2.2 A logical circuit
n3 o
n5 o
33 Database Programming
rosistor(power,nl).
resistor(power,n2).
transistor(n2,ground,nl).
transistor(n3,n4,ri2)
transistor(n5,ground,n4).
inverter(Input,Output) - Output is the inversion of Input.
inverter(Input ,Output) -
transistor(Input ,ground,Uutput),
resistor (power ,Output).
nand_gate(Inputl,Input2,Output) - Output is the logical nand of Inputi and Input2.
nand_gate(Inputl , Input2,Uutput)
transïstor(Inputl ,X,Output),
transistor(Input2 ,ground,X),
resistor (power ,Output).
and_gate(Inputl,Iriput2,Output) - Output is the logical and of Inputi and Iriput2.
and_gate(Inputl ,Input2,Output) -
nand_gate(Inputl ,Iriput2,X),
inverter (X ,Output).
Program 2.2 A circuit for a logical and-gate
The program demonstrates the style of commenting of logic programs we will follow throughout the book. Each interesting procedure is pre- ceded by a relation scheme for the procedure, shown in italic font, and by English text defining the relation. We recommend this style of comment- ing, which emphasizes the declarative reading of programs, for Prolog programs as well.
Particular configurations of resistors and transistors fulfill roles cap- tured via rules defining the functional components of the circuit. The circuit describes an and-gate, which takes two input signals and pro- duces as output the logical and of these signals. One way of building an and-gate, and how this circuit is composed, is to connect a nand-gate with an inverter. Relation schemes for these three components are and_ gate(Inputl,Input2,Output), nand_gate(Inputl,Input2,Output),
and inverter(Input ,Output).
34 Chapter 2
To appreciate Program 2.2, let us read the inverter rule. This states that an inverter is built up from a transistor with the source connected to the ground, and a resistor with one end connected to the power source. The gate of the transistor is the input to the inverter, while the free end of the resistor must be connected to the dram of the transistor, which forms the output of the inverter. Sharing of variables is used to insist on the common connection.
Consider the query and_gate(Inl , 1n2 ,Out)? to Program 2.2. It has the solution {Ini=n3, 1n2=n5 , Out=nl}. This solution confirms that the circuit described by the facts is an and-gate, and indicates the inputs and output.
2.1.1 Exercises for Section 2.1
Modify the rule for brother on page 21 to give a rule for sister, the rule for uncle in Program 2.1 to give a rule for niece, and the rule for sibling in Program 2.1 so that it only recognizes full siblings, i.e., those that have the same mother and father.
Using a predicate married_couple(Wife,Husband), define the rela- tionships mother_in_law, brother_in_law, and son_in_law.
Describe the layout of objects in Figiire 2.3 with facts using the predicates left_of(Objectl,Object2) and above(Objecti3Ob- ject2). Define predicates right_of (Objectl,Object2) and below (Obj ect 1. .Obj ect2) in terms of left_of and above, respectively.
Figure 2.3 Still-life objects
35 Database Programming
22 Structured Data and Data Abstraction
A limitation of Program 2.2 for describing the and-gate is the treatment of the circuit as a black box. There is no indication of the structure of the circuit in the answer to the and_gate query, even though the structure has been implicitly used in finding the answer. The rules tell us that the circuit represents an and-gate, but the structure of the and-gate is present only implicitly. We remedy this by adding an extra argument to each of the goals in the database. For uniformity, the extra argument becomes the first argument. The base facts simply acquire an identifier. Proceeding from left to right in the diagram of Figure 2.2, we label the resistors rl and r2, and the transistors ti, t2, and t3.
Names of the functional components should reflect their structure. An inverter is composed of a transistor and a resistor. To represent this, we need structured data. The technique is to use a compound term, inv(T,R), where T and R are the respective names of the inverter's com- ponent transistor and resistor. Analogously, the name of a nand-gate will be nand(T1,T2,R), where Ti, T2, and R name the two transistors and re- sistor that comprise a nand-gate. Finally, an and-gate can be named in terms of an inverter and a nand-gate. The modified code containing the names appears in Program 2.3.
The query and_gate(G,Inl,1n2,Out)? has solution {G=and(nand(t2, t3,r2) ,inv(ti,rl)) ,In1=n3,In2n5,Out=n1}. Ini, 1n2, and Out have their previous values. The complicated structure for G reflects accurately the functional composition of the and-gate.
Structuring data is important in progranmìing in general and in logic programming in particular. It is used to organize data in a meaningful way. Rules can be written more abstractly, ignoring irrelevant details. More modular programs can be achieved this way, because a change of data representation need not mean a change in the whole program, as shown by the following example.
Consider the following two ways of representing a fact about a lecture course on complexity given on Monday from 9 to 11 by David Harel in the Feinberg building, room A:
course(complexity,monday,9,li,david,harel,feinberg,a).
and
36 Chapter 2
resistor (R,Nodel,Node2) - R is a resistor between Nodel and Node2.
resistor(rl,power,nl).
resistor(r2,power,n2).
transistor ( T,Gate,Source,D rain) * T is a transistor whose gate is Gate, source is Source, and drain is Drain.
transistor(tl,n2,ground,ni).
transistor(t2,n3,n4,n2).
transistor(t3,n5,ground,n4).
inverter(I,Input,Output) - I is an inverter that inverts In put to Output.
inverter(inv(T,R) ,Input,Output) -
transistor(T, Input ,ground,Output),
resistor(R,power,Output).
nand_g ate (Nand,Inputl ,Input2, Output) - Nand is a gate forming the logical nand, Output, of Inputi and Input2.
nand_gate(nand(Tl ,T2 ,R) ,Inputl ,Input2,Output) -
transistor(T1 ,Inputl,X,Output),
transistor (T2 , Input2 ,ground, X),
resistor(R,power,Output).
and_gate(And,Inputl,Input2,Output) And is a gate forming the logical and, Output, of Inputi and Input2.
and_gate(and(N,I),Inputi,Input2,Output) -
nan&gate(N, Input i, Input2 ,X),
inverter(I ,X,Output).
Program 2.3 The circuit database with names
37 Database Programming
course(complexity,time(monday,9,11) ,lecturer(david,harel),
location(feinberg,a)). The first fact represents course as a relation between eight items - a course name, a day, a starting hour, a finishing hour, a lecturer's first name, a lecturer's surname, a building, and a room. The second fact makes course a relation between four items - a name, a time, a lecturer, and a location with further qualification. The time is composed of a day, a starting time, and a finishing time; lecturers have a first name and a surname; and locations are specified by a building and a room. The second fact reflects more elegantly the relations that hold.
The four-argument version of course enables more concise rules to be written by abstracting the details that are irrelevant to the query. Program 2.4 contains examples. The occupied rule assumes a predicate less than or equal, represented as a binary infix operator .
Rules not using the particular values of a structured argument need not "know" how the argument is structured. For example, the rules for duration and teaches represent time explicitly as time (Day »Start, Finish) because the Day or Start or Finish times of the course are de- sired. In contrast, the rule for lecturer does not. This leads to greater modularity, because the representation of time can be changed without affecting the rules that do not inspect it.
We offer no definitive advice on when to use structured data. Not using structured data allows a uniform representation where all the data are simple. The advantages of structured data are compactness of represen- tation, which more accurately reflects our perspective of a situation, and
lecturer (Lecturer ,Course)
course (Course,Time,Lecturer,Location)
duration(Courso ,Length) -
course(Course,time(Day,Start,Finish),Lecturer,Location),
plus (Start, Length ,Finish).
teaches (Lecturer ,Day) -
course(Course,time(Day,Start,Finish) ,Lecturer,Locatïon).
occupied(Room,Day,Time) -
course(Course,time(Day,Start,Finish),Lecturer,Room),
Start Time, Time Finish.
Program 2.4 Course rules
38 Chapter2
modularity. We can relate the discussion to conventional programming languages. Facts are the counterpart of tables, while structured data cor- respond to records with aggregate fields.
We believe that the appearance of a program is important, particularly when attempting difficult problems. A good structuring of data can make a difference when programming complex problems.
Some of the rules in Program 2.4 are recovering relations between two individuals, binai-y relations, from the single, more complicated one. All the course information could have been written in terms of binary relations as follows:
day (complexity, monday).
start_time (complexity, 9).
f inish_time(cornplexity, 11).
lecturer(complexity,harel).
building(complexity,feinberg).
room(complexity, a).
Rules would then be expressed differently, reverting to the previous style of making implicit connections explicit. For example,
teaches(Lecturer,Day) -
lecturer (Course ,Lecturer), day(Course ,Day).
2.2.1 Exercises for Section 2.2
Add rules defining the relations location(Course,Building), busy (Lecturer ,Time), and cannot_meet (Lecturerl ,Lecturer2). Test with your own course facts.
Possibly using relations from Exercise (i), define the relation sched- ule_conflict (Time ,Place ,Coursel ,Course2).
Write a program to check if a student has met the requirements for a college degree. Facts will be used to represent the courses that the student has taken and the grades obtained, and rules will be used to enforce the college requirements.
Design a small database for an application of your own choice. Use a single predicate to express the information, and invent suitable rules.
39 Database Programming
2.3 Recursive Rules
The rules described so far define new relationships in terms of existing ones. An interesting extension is recursive definitions of relationships that define relationships in terms of themselves. One way of viewing recursive rules is as generalization of a set of nonrecursive rules.
Consider a series of rules defining ancestors - grandparents, great- grandparents, etc:
grandparent (Ancestor, Descendant) -
parent (Ancestor,Person), parent(Person,Descendant).
greatgrandparent (Ancestor, Descendant) -
parent (Ancestor,Person), grandparent(Person,Descendant).
greatgreatgrandparent (Ancestor ,Descendant) -
parent (Ancestor,Person), greatgrandparent(Person,
Descendant).
A clear pattern can be seen, which can be expressed in a rule defining the relationship ancestor (Ancestor ,Descendant):
ancestor (Ancestor, Descendant) -
parent (Ancestor,Person), ancestor(Person,Descendant).
This rule is a generalization of the previous rules. A logic program for ancestor also requires a nonrecursive rule, the
choice of which affects the meaning of the program. If the fact ances- tor (X, X) is used, defining the ancestor relationship to be reflexive, peo- ple will be considered to be their own ancestors. This is not the intuitive meaning of ancestor. Program 2.5 is a logic program defining the ances- tor relationship, where parents are considered ancestors.
ancestor(Ancestor,Descendant) Ancestor is an ancestor of Descendant.
ancestor(Ancestor,Descendaxit) - parent (Ancestor , Descendant)
ancestor(Ancestor,Descendant) - parent(Ancestor,Person), ancestor(Person,Descendant).
Program 2.5 The ancestor relationship
40 Chapter 2
The ancestor relationship is the transitive closure of the parent re- lationship. In general, finìding the transitive closure of a relationship is easily done in a logic program by using a recursive rule.
Program 2.5 defining ancestor is an example of a linear recursive pro- gram. A program is linear recursive if there is only one recursive goal in the body of the recursive clause. The linearity can be easily seen from considering the complexity of proof trees solving ancestor queries. A proof tree establishing that two individuals are n generations apart given Program 2.5 and a collection of parent facts has 2 . n nodes.
There are many alternative ways of defining ancestors. The declarative content of the recursive rule in Program 2.5 is that Ancestor is an ances- tor of Descendant if Ancestor is a parent of an ancestor of Descendant. Another way of expressing the recursion is by observing that Ancestor would be an ancestor of Descendant if Ancestor is an ancestor of a par- ent of Descendant. The relevant rule is
ancestor(Ancestor,Descendant) -
ancestor(Ancestor,Person), parent(Person,Descendant).
Another version of defining ancestors is not linear recursive. A pro- gram identical in meaning to Program 2.5 but with two recursive goals in the recursive clause is
ancestor(Ancestor,Descendant) -
parent (Ancestor,Descendant).
ancestor(Ancestor,Descendant) -
ancestor(Ancestor,Person), ancestor(Person,Descendant).
Consider the problem of testing connectivity in a directed graph. A directed graph can be represented as a logic program by a collection of facts. A fact edge(Nodel,Node2) is present in the program if there is an edge from Nodel to Node2 in the graph. Figure 2.4 shows a graph; Program 2.6 is its description as a logic program.
Two nodes are connected if there is a series of edges that can be tra- versed to get from the first node to the second. That is, the relation con- nected(Nodel,Node2), which is true if Nodel and Node2 are connected, is the transitive closure of the edge relation. For example, a and e are connected in the graph in Figure 2.4, but b and f are not. Program 2.7 defines the relation. The meaning of the program is the set of goals con-
41 Database Programming
Figure 2.4 A simple graph
edge(a,b). edge(a,c), edge(b,d).
edge(c,d). edge(d,e). edge(f,g).
Program 2.6 A directed graph
connected(Nodel,Node2) - Nodel is connected to Node2 in the graph defined by the edge/2 relation.
cormected(Node,Node).
connected(Nodel,Node2) - edge(Nodel,Link),
Program 2.7 The transitive closure of the edge relation
nected(X,Y), where X and Y are connected. Note that connected is a transitive reflexive relation because of the choice of base fact.
2.3.1 Exercises for Section 2.3
(i) A stack of blocks can be described by a collection of facts on (Blockl,Block2), which is true if Blocki is on Block2. Define a predicate above(Blockl,Block2) that is true if Blocki is above Block2 in the stack. (Hint: above is the transitive closure of on.)
connected(Link,Node2)
42 Chapter 2
Add recursive rules for left_of and above from Exercise 2.1(iii) on p. 34. Define higher(Objectl,Object2),whichis true if Objecti is on a line higher than Object2 in Figure 2.3. For example, the bicycle is higher than the fish in the figure.
How many nodes are there in the proof tree for connected(a,e) using Programs 2.6 and 2.7? In general, using Program 2.6 and a collection of edge/2 facts, how many nodes are there in a proof tree establishing that two nodes are connected by a path containing n intermediate nodes?
2.4 Logic Programs and the Relational Database Model
Logic programs can be viewed as a powerful extension to the relational database model, the extra power coming from the ability to specify rules. Many of the concepts mtroduced have meaningful analogues in terms of databases. The converse is also true. The basic operations of the rela- tional algebra are easily expressed within logic programming.
Procedures composed solely of facts correspond to relations, the arity of the relation being the arity of the procedure. Five basic operations define the relational algebra: union, set difference, Cartesian product, projection, and selection. We show how each is translated into a logic program.
The union operation creates a relation of arity n from two relations r and s, both of arity n. The new relation, denoted here r_union_s, is the union of r and s. It is defined directly as a logic program by two rules:
r_union_s(X1, . . . ,X) - r(Xi, . . . r_union_s(X1, . . . ,X) - s(Xi, . . . ,X).
Set difference involves negation. We assume a predicate not. Intu- itively, a goal not G is true with respect to a program P if G is not a logical consequence of P. Negation in logic programs is discussed in Chapter 5, where limitations of the intuitive definition are indicated. The definition is correct, however, if we deal only with ground facts, as is the case with relational databases.
The definition of r_diff_s of arity n, where r and s are of arity n, is
43 Database Programming
r_diff_s(Xi, . . ,X) - r(Xi, . . . ,X,), not s(Xi, . . ,X,). Cartesian product can be defined in a single rule. If r is a relation of
arity m, and s is a relation of arity n, then r_x_s is a relation of arity m + n defined by
r_x_s(Xi, . . ,X,Xji, . . . ,Xm) r(Xi, . . . s(Xm+i, . . . ,Xmn).
Projection involves forming a new relation comprising'only some of the attributes of an existing relation. This is straightforward for any particular case. For example, the projection r13 selecting the first and third arguments of a relation r of arity 3 is
r13(Xi,X3) - r(Xì1X2,X3).
Selection is similarly straightforward for any particular case. Consider a relation consisting of tuples whose third components are greater than their second, and a relation where the first component is Smith or Jones. In both cases a relation r of arity 3 is used to illustrate. The first example creates a relation rl:
rl(X1,X2,X3) - r(Xj,X2,X3) ,X3 > X2.
The second example creates a relation r2, which requires a disjunctive relationship, smith_or_j ones:
r2(Xi,X2,X3) - r(Xj,X2,X3), smith_or_jones(Xj). smith_or_j ones (smith).
smith_or_j ones (jones).
Some of the derived operations of the relational algebra are more closely related to the constructs of logic programming We mention two, intersection and the natural join. If r and s are relations of arity n, the intersection, r_meet_s is also of arity n and is defined in a single rule.
r_meet_s(Xi, . . . ,X) - r(X1, . . . ,X), s(Xi, . . . ,X). A natural join is precisely a conjunctive query with shared variables.
44 Chapter 2
2.5 Background
Readers interested in pursuing the connection between logic program- ming and database theory are referred to the many papers that have been written on the subject. A good starting place is the review paper by Gallaire et al. (1984). There are earlier papers on logic and databases in Gallaire and Minker (1978). Another interesting book is about the imple- mentation of a database query language in Prolog (Li, 1984). Our discus- sion of relational databases follows Uliman (1982). Another good account of relational databases can be found in Maier (1983).
In the seven years between the appearance of the first edition and the second edition of this book, the database community has accepted logic programs as extensions of relational databases. The term used for a data- base extended with logical rules is logic database or deductive database. There is now a wealth of material about logic databases. The rewritten version of Ullman's text (1989) discusses logic databases and gives point- ers to the important literature.
Perhaps the major difference between logic databases as taught from a database perspective and the view presented here is the way of evalu- ating queries. Here we implicitly assume that the interpreter from Figure 4.2 will be used, a top-down approach. The database community prefers a bottom-up evaluation mechanism. Various bottom-up strategies for an- swering a query with respect to a logic database are given in Uliman (1989).
In general, an n-ary relation can be replaced by n + i binary relations, as shown by Kowalski (1979a). If one of the arguments forms a key for the relation, as does the course name in the example in Section 2.2, n binary relations suffice.
The addition of an extra argument to each predicate in the circuit, as discussed at the beginning of Section 2.2, is an example of an en- hancement of a logic program. The technique of developing programs by enhancement is of growing importance. More will be said about this in Chapter 13.
3.1 Arithmetic
The simplest recursive data type, natural numbers, arises from the foun- dations of mathematics. Arithmetic is based on the natural numbers. This section gives logic programs for performing arithmetic.
In fact, Prolog programs for performing arithmetic differ considerably from their logical counterparts, as we will see in later chapters. How- ever, it is useful to spend time discussing the logic programs. There are
Recursive Prograniniing
The programs of the previous chapter essentially retrieve information from, and manipulate, finite data structures. In general, mathematical power is gained by considering infinite or potentially infinite structures. Finite instances then follow as special cases. Logic programs harness this power by using recursive data types.
Logical terms can be classified into types. A type is a (possibly infinite) set of terms. Some types are conveniently defined by unary relations. A relation p/i defines the type p to be the set of X's such that p(X).
For example, the male/i and female/i predicates used previously de- fine the male and female types.
More complex types can be defined by recursive logic programs. Such types are called recursive types. Types defined by unary recursive pro- grams are called simple recursive types. A program defining a type is called a type definition.
In this chapter, we show logic programs defining relations over simple recursive types, such as integers, lists, and binary trees, and also pro- grams over more complex types, such as polynomials.
46 Chapter 3
natural_number(X) - X is a natural number.
natural_nuniber(0). natural_number(s(X)) - natural_number(X).
Program 3.1 Defining the natural numbers
two main reasons. First, the operations of arithmetic are usually thought of functionally rather than relationally. Presenting examples for such a familiar area emphasizes the change in thinking necessary for compos- ing logic programs. Second, it is more natural to discuss the underlying mathematical issues, such as correctness and completeness of programs.
The natural numbers are built from two constructs, the constant sym- bol O and the successor function s of arity 1. All the natural numbers are then recursively given as 0, s(0), s(s(0)), s(s(s(0))).....We adopt the convention that s(0) denotes the integer n, that is, n applications of the successor function to O.
As in Chapter 2, we give a relation scheme for each predicate, together with the intended meaning of the predicate. Recall that a program P is correct with respect to an intended meaning M if the meaning of P is a subset of M. It is complete if M is a subset of the meaning of P. It is correct and complete if its meaning is identical to M. Proving correctness establishes that everything deducible from the program is intended. Proving completeness establishes that everything intended is deducible from the program. Two correctness and completeness proofs are given in this section.
The simple type definition of natural numbers is neatly encapsulated in the logic program, shown as Program 3.1. The relation scheme used is natural_number (X), with intended meaning that X is a natural num- ber. The program consists of one unit clause and one iterative clause (a clause with a single goal in the body). Such a program is called minimal recursive.
Proposition Program 3.1 is correct and complete with respect to the set of goals natural_number(s'(0)), for i O.
Proof (1) Completeness. Let n be a natural number. We show that the goal natural_number (n) is deducible from the program by giving an explicit proof tree. Either n is O or of the form sa(0). The proof tree for the goal natural_nuniber(0) is trivial. The proof tree for the goal
47 Recursive Programming
uraI_number(s1 (0)
s(s(0),sm(0) ,sn*m(0))
s (0) ,sm(0) (O
js(s(0) ,sm(0),sm+l (0))
cpIus(0),sm(0),sm(0)
cturaJ_number(sm(O
Figure 3.1 Proof trees establishing completeness of programs
natural_number (s (. .s(0)...)) contains n reductions, using the rule in Program 3.1, to reach the fact riatural_number(0), as shown in the left half of Figure 3.1.
(2) Correctness. Suppose that natural_nuniber(X) is deducible from Program 3.1, in n deductions. We prove that natural_riuniber(X) is in the intended meaning of the program by induction on n. If n = O, then the goal must have been proved using a unit clause, which implies that X = O. If n > o, then the goal must be of the form natural_number(s(X')), since it is deducible from the program, and further, natural_nuxnber(X') is deducible in n - i deductions. By the induction hypothesis, X' is in the intended meaning of the program, i e X'=&c (0) for some k O. u
The natural numbers have a natural order. Program 3.2 is a logic pro- gram defining the relation less than or equal to according to the order. We denote the relation with a binary infix symbol, or operator, , accord- ing to mathematical usage. The goal O < X has predicate symbol < of arity 2, has arguments O and X, and is syntactically identical to '' (O,X).
48 Chapter 3
x Y - X and Y are natural numbers, such that X is less than or equal to Y.
O X - natural_number(X). s(X) < s(Y) - X Y. natural_number(X) - See Program 3.1
Program 3.2 The less than or equal relation
The relation scheme is N1 N2. The intended meaning of Program 3.2 is all ground facts X Y, where X and Y are natural numbers and X is less than or equal to Y. Exercise (ii) at the end of this section is to prove the correctness and completeness of Program 3.2.
The recursive definition of is not computationally efficient. The proof tree establishing that a particular N is less than a particular M has M + 2 nodes. We usually think of testing whether one number is less than another as a unit operation, independent of the size of the numbers. Indeed, Prolog does not define arithmetic according to the axioms pre- sented in this section but uses the underlying arithmetic capabilities of the computer directly.
Addition is a basic operation defining a relation between two natural numbers and their sum. In Section 1.1, a table of the plus relation was assumed for all relevant natural numbers. A recursive program captures the relation elegantly and more compactly, and is given as Program 3.3. The intended meaning of Program 3.3 is the set of facts plus (X,Y,Z), where X, Y, and Z are natural numbers and X+Y=Z.
Proposition Programs 3.1 and 3.3 constitute a correct and complete axiomatization of addition with respect to the standard intended meaning of plus/3.
Proof (1) Completeness. Let X, Y, and Z be natural numbers such that X+Y=Z. We give a proof tree for the goal plus (X,Y,Z). If X equals O, then Y equals Z. Since Program 3.1 is a complete axiomatization of the natural numbers, there is a proof tree for natural_number(Y), which is easily extended to a proof tree for plus (O, Y, Y). Otherwise, X equals 5n(Q) for some n. If Y equals m (0), then Z equals sm (0). The proof tree in the right half of Figure 3.1 establishes completeness.
49 Recursive Programming
plus(X,Y,Z) X, Y , and Z are natural numbers such that Z is the sum of X and Y.
plus(O,X,X) natural_number(X).
plus(s(X),Y,s(Z)) plus(X,Y,Z).
natural_number(X) - See Program 3.1
Program 3.3 Addition
(2) Correctness. Let plus(X,Y,Z) be in the meaning. A simple induc- tive argument on the size of X, similar to the one used in the previous proposition, establishes that X+Y=Z. u
Addition is usually considered to be a function of two arguments rather than a relation of arity 3. Generally, logic programs corresponding to functions of n arguments define relations of arity n + 1. Computing the value of a function is achieved by posing a query with n arguments instantiated and the argument place corresponding to the value of the function uninstantiated. The solution to the query is the value of the function with the given arguments. To make the analogy clearer, we give a functional definition of addition corresponding to the logic program:
o+X = X. s(X)+Y = s(X+Y).
One advantage that relational programs have over functional programs is the multiple uses that can be made of the program. For example, the query plus (s (0) ,s(0) ,s(s(0)))? means checking whether i + i = 2. (We feel free to use the more readable decimal notation when mentioning numbers.) As for , the program for plus is not efficient. The proof tree confirming that the sum of N and M is N + M has N + M + 2 nodes.
Posing the query plus(s(0) ,s(0) ,X)?, an example of the standard use, calculates the sum of 1 and 1. However, the program can just as eas- ily be used for subtraction by posing a query such as plus (s (0) , X, s (s (s(0))))?. The computed value of Xis the difference between 3 and 1, namely, 2. Similarly, asking a query with the first argument uninstanti- ated, and the second and third instantiated, also performs subtraction.
A more novel use exploits the possibility of a query having multiple so- lutions. Consider the query plus(X,Y,s(s(s(0))))?. It reads: "Do there
50 Chapter 3
exist numbers X and Y that add up to 3." In other words, find a partition of the number 3 into the sum of two numbers, X and Y. There are several solutions.
A query with multiple solutions becomes more interesting when the properties of the variables in the query are restricted. There are two forms of restriction: using extra conjuncts in the query, and instanti- ating variables in the query. We saw examples of this when querying a database. Exercise (ii) at the end of this section requires to define a pred- icate even(X), which is true if X is an even number. Assuming such a predicate, the query plus (X,Y,N) ,even(X) ,even(Y)? gives a partition of N into two even numbers. The second type of restriction is exemplified by the query plus(s(s(X)),s(s(Y)),N)?, which insists that each of the numbers adding up to N is strictly greater than 1.
Almost all logic programs have multiple uses. Consider Program 3.2 for , for example. The query s (0) s (s (0))? checks whether i is less than or equal to 2. The query X s(s(0))? finds numbers X less than or equal to 2. The query X Y? computes pairs of numbers less than or equal to each other.
Program 3.3 defining addition is not unique. For example, the logic program
plus(X,O,X) - natural_nuniber(X).
plus(X,s(Y),s(Z)) - plus(X,Y,Z).
has precisely the same meaning as Program 3.3 for plus. Two programs are to be expected because of the symmetry between the first two argu- ments. A proof of correctness and completeness given for Program 3.3 applies to this program by reversing the roles of the symmetric argu- ments.
The meaning of the program for plus would not change even if it consisted of the two programs combined. This composite program is un- desirable, however. There are several different proof trees for the same goal. It is important both for runtime efficiency and for textual concise- ness that axiomatizations of logic programs be minimal.
We define a type condition to be a call to the predicate defining the type. For natural numbers, a type condition is any goal of the form natural_number (X).
In practice, both Programs 3.2 and 3.3 are simplified by omitting the body of the base rule, natural_number (X). Without this test, facts such
51 Recursive Programming
times(X,Y,Z) X, Y, arid Z are natural numbers such that Z is the product of X and Y.
times(O,X,O)
times(s(X) ,Y,Z) es(X,Y,XY), plus(XY,Y,Z).
plus(X,Y,Z) See Program 3.3
Program 3.4 Multiplication as repeated addition
exp(N,X,Y) N, X, and Y are natural numbers such that Y equals X raised to the power N.
exp(s(X) 0,0).
exp(0,s(X) ,s(0))
exp(s(N),X,Y) .- exp(N,X,Z), times(Z,X,Y).
times(X,Y,Z) - See Program 3.4
Program 3.5 Exponentiation as repeated multiplication
as O a and plus(O,a,a), where a is an arbitrary constant, will be in the programs' meanings. Type conditions are necessary for correct programs. However, type conditions distract from the simplicity of the programs arid affect the size of the proof trees. Hence in the following we might omit explicit type conditions from the example programs, Pro- grams 3.4-3.7.
The basic programs shown are the building blocks for more compli- cated relations. A typical example is defining multiplication as repeated addition. Program 3.4 reflects this relation. The relation scheme is times(X,Y,Z), meaning X times Y equals Z.
Exponentiation is defined as repeated multiplication. Program 3.5 for exp(N,X,Y) expresses the relation that XN=Y. It is analogous to Pro- gram 3.4 for times(X,Y,Z), with exp and times replacing times and plus, respectively. The base cases for exponentiation are X0=1 for all pos- itive values of X, and 0N=0 for positive values of N.
A definition of the factorial function uses the definition of multiplica- tion.RecallthatN! =NN i .....2 i.Thepredicatefactorial(N,F) relates a number N to its factorial F. Program 3.6 is its axiomatization.
52 Chapter 3
factoria!(N,F) - F equals N factorial.
Íactorial(O,s(0)) factorial(s(N),F) - factorial(N,F1), times(s(N),F1,F). times(X,Y,Z) - SeeProgram3.4.
Program 3.6 Computing factorials
minimum(N1,N2,Min) - The minimum of the natural numbers Nl and N2 is Min.
minimum(N1,N2,N1) Nl 5 N2.
minimum(Nl,N2,N2) - N2 Nl.
Nl 52 - See Program 3.2
Program 3.7 The minimum of two numbers
Not all relations concerning natural numbers are defined recursively. Relations can also be defined in the style of programs in Chapter 2. An example is Program 3.7 determining the minimum of two numbers via the relation minimum (Nl , N2 , Mm).
Composing a program to determine the remainder after integer divi- sion reveals an interesting phenomenondifferent mathematical defini- tions of the same concept are translated into different logic programs. Programs 3 .8a and 3 .8b give two definitions of the relation mod (X ,Y, Z), which is true if Z is the value of X modulo Y, or in other words, Z is the re- mainder of X divided by Y. The programs assume a relation < as specified in Exercise (i) at the end of this section.
Program 3.8a illustrates the direct translation of a mathematical defi- nition, which is a logical statement, into a logic program. The program corresponds to an existential definition of the integer remainder: "Z is the value of X mod Y if Z is strictly less than Y, and there exists a num- ber Q such that X = Q . Y + Z. In general, mathematical definitions are easily translated to logic programs.
We can relate Program 3.8a to constructive mathematics. Although seemingly an existential definition, it is also constructive, because of the constructive nature of <, plus, and times. The number Q, for example, proposed in the definition will be explicitly computed by times iii any use of mod.
53 Recursive Programming
mod(X,}Z) - Z is the remainder of the integer division of X by Y.
mod(X,Y,Z) - Z < Y, times(Y,Q,QY), plus(QY,Z,X).
Program 3.8a A nonrecursive definition of modulus
mod(X,Y,Z) - Z is the remainder of the integer division of X by Y.
mod(X,Y,X) .- X < Y. mod(X,Y,Z) plus(X1,Y,X), mod(X1,Y,Z).
Program 3.8b A recursive definition of modulus
In contrast to Program 3.8a, Program 3.8b is defined recursively. It con- stitutes an algorithm for finding the integer remainder based on repeated subtraction. The first rule says that X mod Y is X if X is strictly less than Y. The second rule says that the value of X mod Y is the same as X - Y mod Y. The effect of any computation to determine the modulus is to re- peatedly subtract Y from X until it becomes less than Y and hence is the correct value.
The mathematical function X mod Y is not defined when Y is zero. Nei- ther Program 3.8a nor Program 3.8b has goal mod(X,O,Z) in its meaning for any values of X or Z. The test of < guarantees that.
The computational model gives a way of distinguishing between the two programs for mod. Given a particular X, Y, and Z satisfying mod, we can compare the sizes of their proof trees. In general, proof trees produced with Program 3.8b will be smaller than those produced with Program 3.8a. In that sense Program 3.8b is more efficient. We defer more rigorous discussions of efficiency till the discussions on lists, where the insights gained will carry over to Prolog programs.
Another example of translating a mathematical definition directly into a logic program is writing a program that defines Ackermann's function. Ackermann's function is the simplest example of a recursive function that is not primitive recursive. It is a function of two arguments, defined by three cases:
ackermann(O,N) = N + 1.
ackermann(M, O) = ackerrnann(M - 1, 1).
ackermann(M,N) = ackermann(M - 1,ackermann(M,N - 1)).
54 Chapter 3
ackermarìn(X,Y,A) A is the value of Ackermann's function for the natural numbers X and Y.
ackermarm (O ,N , s (N) )
ackermann(s(M) ,O,Val) - ackermann(M,s(0) ,Val).
ackermann(s(M) ,s(N) ,Val)
ackermann(s(M),N,Vall), ackermann(M,Vall,Val).
Program 3.9 Ackermann's function
gcd(X,Y,Z) - Z is the greatest common divisor of the natural numbers X and Y.
gcd(X,Y,Gcd) - mod(X,Y,Z), gcd(Y,Z,Gcd).
gcd(X,O,X) X > O.
Program 3.10 The Euclidean algorithm
Program 3.9 is a translation of the functional definition into a logic pro- gram. The predicate ackermann(M,N,A) denotes that A=ackermann(M,N). The third rule involves two calls to Ackermann's function, one to com- pute the value of the second argument.
The functional definition of Ackermann's function is clearer than the relational one given in Program 3.9. In general, functional notation is more readable for pure functional definitions, such as Ackermann's function and the factorial function (Program 3.6). Expressing constraints can also be awkward with relational logic programs. For example, Pro- gram 3.8a says less directly that X = Q . Y + Z.
The final example in this section is the Euclidean algorithm for finding the greatest common divisor of two natural numbers, recast as a logic program. Like Program 3.8b, it is a recursive program not based on the recursive structure of numbers. The relation scheme is gcd(X,Y,Z), with intended meaning that Z is the greatest common divisor (or gcd) of two natural numbers X and Y. It uses either of the two programs, 3.8a or 3.8b, for mod.
The first rule in Program 3.10 is the logical essence of the Euclidean algorithm. The gcd of X and Y is the same as the gcd of Y and X mod Y. A proof that Program 3.10 is correct depends on the correctness
55 Recursive Programming
of the above mathematical statement about greatest common divisors. The proof that the Euclidean algorithm is correct similarly rests on this result.
The second fact in Program 3.10 is the base fact. It must be specified that Xis greater than Oto preclude gcd(O,O,O) from being in the mean- ing. The gcd of O and O is not well defined.
3.1.1 Exercises for Section 3.1
Modify Program 3.2 for < to axiomatize the relations <, >, and . Discuss multiple uses of these programs.
Prove that Program 3.2 is a correct and complete axiomatization of <
Prove that a proof tree for the query s'(0) < tm(Q) using Pro- gram 3.2 has m + 2 nodes.
Define predicates even(X) and odd(X) for determining if a natural number is even or odd. (Hint: Modify Program 3.1 for natural_ nuniber.)
(y) Write a logic program defining the relation f ib(N,F) to determine the Nth Fibonacci number F.
The predicate times can be used for computing exact quotients with queries such as times(s(s(0)),X,s(s(s(s(0)))))? to find
the result of 4 divided by 2. The query times(s(s(0)),X,s(s(s (0))))? to find 3/2 has no solution. Many applications require the use of integer division that would calculate 3/2 to be 1. Write a program to compute integer quotients. (Hint: Use repeated subtrac- tion.)
Modify Program 3.10 for finding the gcd of two integers so that it performs repeated subtraction directly rather than use the mod function. (Hint: The program repeatedly subtracts the smaller num- ber from the larger number until the two numbers are equal.)
Rewrite the logic programs in Section 3.1 using a different represen- tation of natural numbers, namely as a sum of l's. For example, the modified version of Program 3.1 would be
56 Chapter 3
natural_number (1).
natural_number(1+X) - natural_number(X).
Note that + is used as a binary operator, and O is not defined to be a natural number.
3.2 Lists
The basic structure for arithmetic is the unary successor functor. Al- though complicated recursive functions such as Ackermann's function can be defined, the use of a unary recursive structure is limited. This sec- tion discusses the binary structure, the list.
The first argument of a list holds an element, and the second argument is recursively the rest of the list. Lists are sufficient for most computa- tions - attested to by the success of the programming language Lisp, which has lists as its basic compound data structure. Arbitrarily complex structures can be represented with lists, though it is more convenient to use different structures when appropriate.
For lists, as for numbers, a constant symbol is necessary to terminate recursion. This "empty list," referred to as nil, will be denoted here by the symbol [1. We also need a functor of arity 2. Historically, the usual functor for lists is "" (pronounced dot), which overloads the use of the period. It is convenient to define a separate, special syntax. The term
(X,Y) is denoted [XJY]. Its components have special names: X is called the head and Y is called the tail.
The term [XIY] corresponds to a cons pair in Lisp. The corresponding words for head and tail are, respectively, car and cdr.
Figure 3.2 illustrates the relation between lists written with different syntaxes. The first colunm writes lists with the dot functor, and is the way lists are considered as terms in logic programs. The second colunm gives the square bracket equivalent of the dot syntax. The third column is an improvement upon the syntax of the second colunm, essentially hiding the recursive structure of lists. In this syntax, lists are written as a sequence of elements enclosed in square brackets and separated by commas. The empty list used to terminate the recursive structure is suppressed. Note the use of "cons pair notation" in the third column when the list has a variable tail.
57 Recursive Programming
Formal object Cons pair syntax Element syntax
(a,] 1) Fai F]] Fa]
.(a,(b,F J)) [aIFbIF J]] Fa,b]
.(a,.(b,.(c,F 1))) ]aiFbI]c I] J]]] Fa,b,c]
(aX) Faix] Faix]
Fai]biXl] Fa,biX]
Figure 3.2 Equivalent forms of lists
!ist(Xs) Xs is a list.
list([ ]). list([XiXs]) - list(Xs).
Program 3.11 Defining a list
Terms built with the dot functor are more general than lists. Program 3.11 defines a list precisely. Declaratively it reads: "A list is either the empty list or a cons pair whose tail is a list." The program is analogous to Program 3.1 defining natural numbers, and is the simple type definition of lists.
Figure 3.3 gives a proof tree for the goal list ([a, b, cl). Implicit in the proof tree are ground instances of rules in Program 3.11, for example, list( [a,b , c]) - list( [b, c]). We specify the particular instance here explicitly, as instances of lists in cons pair notation can be confusing. [a, b, c] is an instance of [XIXs] under the substitution {X=a, Xs= [b, c]).
Because lists are richer data structures than numbers, a great variety of interesting relations can be specified with them. Perhaps the most basic operation with lists is determining whether a particular element is in a list. The predicate expressing this relation is member (Element,List). Program 3.12 is a recursive definition of member/2.
Declaratively, the reading of Program 3.12 is straightforward. X is an element of a list if it is the head of the list by the first clause, or if it is a member of the tail of the list by the second clause. The meaning of the program is the set of all ground instances member (X,Xs), where
58 Chapter 3
Figure 3.3 Proof tree verifying a list
member(Element,List) - Element is an element of the list List.
member(X, [XIXs]). member(X,[YIYs]) - member(X,Ys).
Program 3.12 Membership of a list
X is an element of Xs. We omit the type condition in the first clause. Alternatively, it would be written
rnember(X,[XIXs]) - list(Xs).
This program has many interesting applications, to be revealed throughout the book. Its basic uses are checking whether an element is in a list with a query such as member(b, [a,b,c])?, finding an ele- ment of a list with a query such as member (X, [a, b, c])?, and finding a list containing an element with a query such as member(b,X)?. This last query may seem strange, but there are programs that are based on this use of member.
We use the following conventions wherever possible when naming vari- ables in programs involving lists. If X is used to denote the head of a list, then Xs will denote its tail. More generally, plural variable names will denote lists of elements, and singular names will denote individual ele- ments. Numerical suffixes will denote variants of lists. Relation schemes will still contain nmemonic names.
59 Recursive Programming
prefix(Prefix,List) - Prefix is a prefix of List.
prefix([ ],Ys).
prefix([XIXs],[XJYs]) - prefix(Xs,Ys).
suffix(Suffix,List) Suffix is a suffix of List.
suffix(Xs,Xs)
suffix(Xs,[YIYs]) - suffix(Xs,Ys)
Program 3.13 Prefixes and suffixes of a list
Our next example is a predicate sublist(Sub,List) for determining whether Sub is a sublist of List. A sublist needs the elements to be consecutive: [b, c] is a sublist of [a, b, c , d], whereas [a, c] is not.
It is convenient to define two special cases of sublists to make the defi- nition of sublist easier. lt is good style when composing logic programs to define meaningful relations as auxiliary predicates. The two cases con- sidered are initial sublists, or prefixes, of a list, and terminal sublists, or suffixes, of a list. The programs are interesting in their own right.
The predicate prefix(Prefix,List) is true if Prefix is an initial sub- list of List, for example, prefix([a,b] , [a,b,c]) is true. The compan- ion predicate to prefix is suffix(Suffix,List), determining if Suffix is a terminal sublist of List. For example, suffix( [b, c] , [a,b, ci) is true. Both predicates are defined in Program 3.13. A type condition ex- pressing that the variables in the base facts are lists should be added to the base fact in each predicate to give the correct meaning.
An arbitrary sublist can be specified in terms of prefixes and suffixes: namely, as a suffix of a prefix, or as a prefix of a suffix. Program 3.14a expresses the logical rule that Xs is a sublist of Ys if there exists Ps such that Ps is a prefix of Ys and Xs is a suffix of Ps. Program 3.14b is the dual definition of a sublist as a prefix of a suffix.
The predicate prefix can also be used as the basis of a recursive definition of sublist. Thìs is given as Program 3.14c. The base rule reads that a prefix of a list is a sublist of a list. The recursive rule reads that the sublist of a tail of a list is a sublist of the list itself.
The predicate member can be viewed as a special case of sublist de- fined by the rule
member(X,Xs) - sublist([X],Xs).
60 Chapter 3
sublist (Sub,List) - Sub is a sublist of List.
Suffix of a prefix
sublist(Xs,Ys) - prefix(Ps,Ys), suffix(Xs,Ps).
Prefixofasuffix sublist(Xs,Ys) - prefix(Xs,Ss), suffix(Ss,Ys).
C: Recursive definition of a sublist sublist(Xs,Ys) - prefix(Xs,Ys).
sublist(Xs, [YIYs]) - sublist(Xs,Ys).
Prefix of a suffix, using append
sublist (Xs , AsXsBs) -
append(As,XsBs,AsXsBs), append(Xs,Bs,XsBs).
Suffix of a prefix, using append
sublist (Xs , AsXsBs) -
append(AsXs,Bs,AsXsBs), append(As,Xs,AsXs).
Program 3.14 Determining sublists of lists
append (Xs, Ys,XsYs) - XsYs is the result of concatenating the lists Xs and Ys.
append([ ],Ys,Ys).
append([XIXs],Ys,[XIZs]) - append(Xs,Ys,Zs).
Program 3.15 Appending two lists
The basic operation with lists is concatenating two lists to give a third list. This defines a relation, append(Xs,Ys,Zs), between two lists Xs, Ys and the result Zs of joining them together. The code for append, Pro- gram 3.15, is identical in structure to the basic program for combining two numbers, Program 3.3 for plus.
Figure 3.4 gives a proof tree for the goal append([a,b] , [c,d] , [a,b, c , d]). The tree structure suggests that its size is linear in the size of the first list. In general, if Xs is a list of n elements, the proof tree for append(Xs,Ys,Zs) has n + 1 nodes.
There are multiple uses for append similar to the multiple uses for plus. The basic use is to concatenate two lists by posing a query such
61 Recursive Programming
[a,b],[c,d
r cpend([b],[c,d],[b,c,d]
ppend([ ],[c,d],[c,d])
Ftgure 3.4 Proof tree for appending two lists
as append([a,b,c] , [d,e] ,Xs)? with answer Xs=[a,b,c,d,e]. A query such as append(Xs,[c,d],[a,b,c,d])? finds the difference Xs=[a,b] between the lists [c,d] and [a,b,c,d]. Unlike plus, append is not sym- metric in its first two arguments, and thus there are two distinct versions of finding the difference between two lists.
The analogous process to partitioning a number is splitting a list. The query append(As,Bs, [a,b,c,d])?, for example, asks for lists As andBs such that appending Bs to As gives the list [a,b,c,d]. Queries about splitting lists are made more interesting by partially specifying the na- ture of the split lists. The predicates member, sublist, prefix, and suf- fix, introduced previously, can all be defined in terms of append by viewing the process as splitting a list.
The most straightforward definitions are for prefix and suffix, which just specify which of the two split pieces are of interest:
prefix(Xs,Ys) - append(Xs,As,Ys).
suffix(Xs,Ys) - append(As,Xs,Ys).
Sublist can be written using two append goals. There are two distinct variants, given as Programs 3.14d and 314e. These two programs are obtained from Programs 3.14a and 3.14b, respectively, where prefix and suffix are replaced by append goals. Member can be defined using append, as follows:
member(X,Ys) - append(As, [XIXs] ,Ys).
This says that X is a member of Ys if Ys can be split into two lists where X is the head of the second list.
62 Chapter 3
reverse(List, Tsil) - Tsil is the result of reversing the list List.
Naive reverse
reverse(f 1,1 1). reverse([XIXs] ,Zs) reverse(Xs,Ys), append(Ys, [X] ,Zs).
Reverse-accumulate
reverse(Xs,Ys) - reverse(Xs,[ ],Ys).
reverse([XIXs],Acc,Ys) - reverse(Xs,[XIAcc],Ys).
reverse([ ],Ys,Ys).
Program 3.16 Reversing a list
A similar rule can be written to express the relation adj acent (X, Y, Zs) that two elements X and Y are adjacent in a list Zs:
adjacent(X,Y,Zs) - append(As, [X,YIYs] ,Zs).
Another relation easily expressed through append is determining the last element of a list. The desired pattern of the second argument to append, a list with one element, is built into the rule:
last(X,Xs) append(As,[X],Xs).
Repeated applications of append can be used to define a predicate reverse (Li st ,Tsil). The intended meaning of reverse is that Tsil is a list containing the elements in the list List in reverse order to how they appear m List. An example of a goal in the meaning of the program is reverse ([a, b, c] , [c , b, a]). The naive version, given as Program 3.1 6a, is the logical equivalent of the recursive formulation in any language: recursively reverse the tail of the list, and then add the first element at the back of the reversed tail.
There is an alternative way of defining reverse without calling append directly. We define an auxiliary predicate reverse (Xs,Ys,Zs), which is true if Zs is the result of appending Ys to the elements of Xs reversed. It is defined in Program 3.16b. The predicate reverse/3 is related to reverse/2 by the first clause in Program 3.16b.
Program 3.16b is more efficient than Program 3.16a. Consider Fig- ure 3.5, showing proof trees for the goal reverse( [a,b, cl , [c ,b , a]) us- ing both programs. In general, the size of the proof tree of Program 3.16a
63 Recursive Programming
reverse([b,c],[c,b])
reverse([ LE 1) append([ ],[c],[c]) append([ 1,[b],[b])
append([c,b],[a],[c,b,a])
append([b],[a] [ba])
append([ ],[a],[a])
cZTIerse([a,b,c],[c,b,aj
crse([a,b,c],[ ],[c,b,i
erse([c],[b,a],[c,b,a])
reverse([ ],[c,b,a],[c,b,a]
Figure 3.5 Proof trees for reversing a list
64 Chapter 3
!ength(Xs,N) - The list Xs has N elements.
length([ 1,0). length([XIXs],s(N)) - length(Xs,N).
Program 3.17 Determining the length of a list
is quadratic in the number of elements in the list to be reversed, while that of Program 3.16b is linear.
The insight in Program 3.16b is the use of a better data structure for representing the sequence of elements, which we discuss in more detail in Chapters 7 and 15.
The final program in this section, Program 3.17, expresses a rela- tion between numbers and lists, using the recursive structure of each. The predicate length(Xs,N) is true if Xs is a list of length N, that is, contains N elements, where N is a natural number. For example, length([a,b] ,s(s(0))), indicating that [a,b] has two elements, is in the program's meaning.
Let us consider the multiple uses of Program 3.17. The query length ([a,b] ,X)? computes the length, 2, of a list [a,b]. In this way, length is regarded as a function of a list, with the functional definition
length([ 1) = O length([XIXs]) = s(length(Xs)).
The querylength([a,b] ,s(s(0)))? checks whetherthelist [a,b] has length 2. The query length(Xs,s(s(0)))? generates a list of length 2 with variables for elements.
3.2.1 Exercises for Section 3.2
(i) A variant of Program 3.14 for sublist is defined by the following three rules:
subsequence([XIXs] , [XIYs]) - subsequence(Xs,Ys). subsequence(Xs, [Y lYs]) - subsequence(Xs,Ys). subsequence([ ] ,Ys).
Explain why this program has a different meaning from Pro- gram 3.14.
65 Recursive Programming
Write recursive programs for adjacent and last that have the same meaning as the predicates defined in the text in terms of append.
Write a program for double (List,ListList), where every element [ri List appears twice in ListList, e.g., double([1,2,3] [1,1,2, 2,3,3]) is true.
Compute the size of the proof tree as a function of the size of the input list for Programs 3.16a and 3.1Gb defining reverse.
(y) Define the relation suin(Listoflntegers,Sum), which holds if Sum is the sum of the ListOf Integers,
Using plus/3;
Without using any auxiliary predicate.
(Hint: Three axioms are enough.)
3.3 Composing Recursive Programs
No explanation has been given so far about how the example logic pro- grams have been composed. The composition of logic programs is a skill that can be learned by apprenticeship or osmosis, and most definitely by practice. For simple relations, the best axiomatizations have an aesthetic elegance that look obviously correct when written down. Through solv- ing the exercises, the reader may find, however, that there is a difference between recognizing and constructing elegant logic programs
This section gives more example programs involving lists. Their pre- sentation, however, places more emphasis on how the programs might be composed. Two principles are illustrated: how to blend procedural and declarative thinking, and how to develop a program top-down.
We have shown the dual reading of clauses: declarative and procedural. How do they interrelate when composing logic programs? Pragmatically, one thinks procedurally when programming. However, one thinks declar- atively when considering issues of truth and meaning. One way to blend them in logic programming is to compose procedurally and then niter- pret the result as a declarative statement. Construct a program with a
66 Chapter 3
given use in mind; then consider if the alternative uses make declarative sense. We apply this to a program for deleting elements from a list.
The first, and most important, step is to specify the intended meaning of the relation. Clearly, three arguments are involved when deleting ele- ments from a list: an element X to be deleted, a list Li that might have occurrences of X, and a list L2 with all occurrences of X deleted. An ap- propriate relation scheme is delete (Li , X, L2). The natural meaning is all ground instances where L2 is the list Li with all occurrences of X re- moved.
When composing the program, it is easiest to think of one specific use. Consider the query delete([a,b,c,b] ,b,X)?, a typical example of finding the result of deleting an element from a list. The answer here is X= [a, c]. The program will be recursive on the first argument. Let's don our procedural thinking caps.
We begin with the recursive part. The usual form of the recursive ar- gument for lists is [XXs]. There are two possibilities to consider, one where X is the element to be deleted, and one where it is not. In the first case, the result of recursively deleting X from Xs is the desired answer to the query. The appropriate rule is
delete([XIXs],X,Ys) - delete(Xs,X,Ys). Switching hats, the declarative reading of this rule is: "The deletion of
X from [XXs] is Ys if the deletion of X from Xs is Ys." The condition that the head of the list and the element to be deleted are the same is specified by the shared variable in the head of the rule.
The second case where the element to be deleted is different from X, the head of the list, is similar. The result required is a list whose head is X and whose tail is the result of recursively deleting the element. The rule is
delete([XIXs],Z,[XIYs]) - X Z, delete(Xs,Z,Ys).
The rule's declarative reading is: "The deletion of Z from [XIXs] is [XYs] if Z is different from X and the deletion of Z from Xs is Ys." In contrast to the previous rule, the condition that the head of the list and the element to be deleted are different is made explicit in the body of the rule.
The base case is straightforward. No elements can be deleted from the empty list, and the required result is also the empty list. This gives the
67 Recursive Programming
delete(List,X,HasNoXs) - The list HasNoXs is the result of removing all occurrences of X from the list List.
delete([XIXs],X,Ys) - delete(Xs,X,Ys).
delete([XIXs],Z,[XIYs]) XZ, delete(Xs,Z,Ys).
delete([ ] ,X, E 1).
Program 3.18 Deleting all occurrences of an element from a list
select (X,HasXs,OneLessXs) - The list OneLessXs is the result of removing one occurrence of X from the list HasXs.
select (X, [XIXs] ,Xs)
select(X,[YIYs],[YIZs]) select(X,Ys,Zs).
Program 3.19 Selecting an element from a list
fact delete([ ] ,X, E J). The complete program is collected together as Program 3.18.
Let us review the program we have written, and consider alternative formulations. Omitting the condition XZ from the second rule in Pro- gram 3.18 gives a variant of delete. This variant has a less natural mean- ing, since any number of occurrences of an element may be deleted. For example, delete([a,b,c,b],b,[a,c]), delete([a,b,c,b],b,[a,c, b]), delete([a,b,c,b],b,[a,b,c]), and delete([a,b,c,b],b,[a,b, c , b]) are all in the meaning of the variant.
Both Program 3.18 and the variant include in their meaning instances where the element to be deleted does not appear in either list, for ex- ample, delete([a] ,b, [a]) is true. There are applications where this is not desired. Program 3.19 defines select(X,L1,L2), a relation that has a different approach to elements not appearing in the list. The meaning of select (X,L1,L2) is all ground instances where L2 is the list Li where exactly one occurrence of X has been removed. The declarative reading of Program 3.19 is: "X is selected from [XXs] to give Xs; or X is selected from [YYs] to give [YZs] if X is selected from Ys to give Zs,"
A major thrust in programming has been the emphasis on a top-down design methodology, together with stepwise refinement. Loosely, the
68 Chapter 3
methodology is to state the general problem, break it down into subprob- lems, and then solve the pieces. A top-down programming style is one natural way for composing logic programs. Our description of programs throughout the book will be mostly top-down. The rest of this section de- scribes the composition of two programs for sorting a list: permutation sort and quicksort. Their top-down development is stressed.
A logical specification of sorting a list is finding an ordered permuta- tion of a list. This can be written down immediately as a logic program. The basic relation scheme is sort (Xs,Ys), where Ys is a list containing the elements in Xs sorted in ascending order:
sort(Xs,Ys) - permutation(Xs,Ys), ordered(Ys).
The top-level goal of sorting has been decomposed. We must now define permutation and ordered.
Testing whether a list is ordered ascendingly can be expressed in the two clauses that follow. The fact says that a list with a single element is necessarily ordered. The rule says that a list is ordered if the first element is less than or equal to the second, and if the rest of the list, beginning from the second element, is ordered:
ordered([X]).
ordered([X,YIYs]) - X Y, ordered([YJYs]).
A program for permutation is more delicate. One view of the process of permuting a list is selecting an element nondeterrniriistically to be the first element of the permuted list, then recursively permuting the rest of the list. We translate this view into a logic program for permutation, using Program 3.19 for select. The base fact says that the empty list is its own unique permutation:
permutation(Xs, [ZIZs]) - select(Z,Xs,Ys), permutation(Ys,Zs).
permutation([ ],[ 1).
Another procedural view of generating permutations of lists is recur- sively permuting the tail of the list and inserting the head in an arbitrary position. This view also can be encoded immediately. The base part is identical to the previous version:
permutation([XIXsI ,Zs) - permutation(Xs,Ys), insert(X,Ys,Zs).
permutation([ ],[ ]).
69 Recursive Programming
sort(Xs,Ys) - The list Ys is an ordered permutation of the list Xs.
sort(Xs,Ys) - permutation(Xs,Ys), ordered(Ys).
permutation(Xs, [ZIZs]) - select(Z,Xs,Ys), permutation(Ys,Zs).
perrnutation([ ],[ 1). ordered([ ]).
ordered([X])
ordered([X,YIYs]) X Y, ordered([YIYs]).
Program 3.20 Permutation sort
The predicate insert can be defined in terms of Program 3.19 for se- lect:
insert(X,Ys,Zs) '- select(X,Zs,Ys).
Both procedural versions of permutation have clear declarative read- ings.
The "naive" sorting program, which we call permutation sort, is col- lected together as Program 3.20. It is an example of the generate-and-test paradigm, discussed fully in Chapter 14. Note the addition of the extra base case for ordered so that the program behaves correctly for empty lists.
The problem of sorting lists is well studied. Permutation sort is not a good method for sorting lists in practice. Much better algorithms come from applying a "divide and conquer" strategy to the task of sorting. The insight is to sort a list by dividing it into two pieces, recursively sorting the pieces, and then joining the two pieces together to give the sorted list. The methods for dividing and joining the lists must be specified. There are two extreme positions. The first is to make the dividing hard, and the joining easy. This approach is taken by the quicksort algorithm The second position is making the joining hard, but the dividing easy. This is the approach of merge sort, which is posed as Exercise (y) at the end of this section, and insertion sort, shown in Program 3.21.
In insertion sort, one element (typically the first) is removed from the list. The rest of the list is sorted recursively; then the element is inserted, preserving the orderedness of the list.
The insight in quicksort is to divide the list by choosing an arbitrary element in it, and then to split the list into the elements smaller than the
70 Chapter 3
sort(Xs,Ys) - The list Ys is an ordered permutation of the list Xs.
sort([XIXs],Ys) - sort(Xs,Zs), insert(X,Zs,Ys).
sort([ ],[ ]).
insert(X,[ ],[X]).
insert(X,[YIYs],[YIZs]) - X> Y, insert(X,Ys,Zs).
insert(X, [YIYs] , [X,YIYs]) - X y.
Program 3.21 Insertion sort
quicksort (Xs, Ys) - The list Ys is an ordered permutation of the list Xs.
quicksort([XIXs] ,Ys) -
partition(Xs,X,Littles,Bigs),
quicksort(Littles,Ls),
quicksort(Bigs,Bs),
append(Ls, [X lBs] ,Ys).
quicksort([ ],[ 1). partition([XIXs] ,Y, [XILs] ,Bs) X Y, partition(Xs,Y,Ls,Bs).
partition([XIXs] ,Y,Ls, [XIBs]) - X > Y, partition(Xs,Y,Ls,Bs).
partition( E ] ,Y, E J , E i)
Program 3.22 Quicksort
chosen element and the elements larger than the chosen element. The sorted list is composed of the smaller elements, followed by the chosen element, and then the larger elements. The program we describe chooses the first element of the list as the basis of partition.
Program 3.22 defines the quicksort algorithm. The recursive rule for quicksort reads: "Ys is a sorted version of [XIXs] if Littles and Bigs are a result of partitioning Xs according to X; Ls and Bs are the result of sorting Littles and Bigs recursively; and Ys is the result of appending [XBs] to Ls."
Partitioning a list is straightforward, and is similar to the program for deleting elements. There are two cases to consider: when the current head of the list is smaller than the element being used for the parti- tioning, and when the head is larger than the partitioning element. The declarative reading of the first partition clause is: "Partitioning a list whose head is X and whose tail is Xs according to an element Y gives the
71 Recursive Programming
lists [XLitt1es] and Bigs if X is less than or equal to Y, and partitioning Xs according to Y gives the lists Littles and Bigs." The second clause for partition has a similar reading. The base case is that the empty list is partitioned into two empty lists.
3.3.1 Exercises for Section 3.3
Write a program for substitute(X,Y,Li,L2), where L2 is the result of substituting Y for all occurrences of X in Li, e.g., sub- stitute(a,x,[a,b,a,c],[x,b,x,c]) is true, whereas substi- tute (a, x, [a,b, a, cl , [a,b ,x, c] ) is false.
What is the meaning of the variant of select:
select (X, [XIXs] ,Xs). select(X, [YIYs] , [YIZs]) '- X Y, select(X,Ys,Zs).
Write a program for no_doubles(L1 ,L2), where L2 is the result of removing all duplicate elements from Li, e.g., no_doubles ([a, b, c, b], [a,c,b]) is true. (Hint: Use member.)
Write programs for evenpermutation(Xs ,Ys) and oddpermuta- tion(Xs,Ys) that find Ys, the even and odd permutations, respec- tively, of a list Xs. For example, even_permutation([i »2,3] » [2,3, i]) andoddpeimutation([i,2,3],[2,i,3]) aretrue.
(y) Write a program for merge sort.
(vi) Write a logic program for kthlargest (Xs , K) that implements the linear algorithm for finding the kth largest element K of a list Xs. The algorithm has the following steps:
Break the list into groups of five elements. Efficiently find the median of each of the groups, which can be done with a fixed number of comparisons. Recursively find the median of the medians. Partition the original list with respect to the median of medians. Recursively find the kth largest element in the appropriate smaller list.
72 Chapter 3
(vii) Write a program for the relation better_poker_hand(Handl, Hand2,Hand) that succeeds if Hand is the better poker hand be- tween Handi and Hand2. For those unfamiliar with this card game, here are some rules of poker necessary for answering this exercise:
The order of cards is 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, ace.
Each hand consists of five cards.
The rank of hands in ascending order is no pairs < one pair < two pairs < three of a kind < flush < straight < full house < four of a kind < straight flush.
Where two cards have the same rank, the higher denomination wins, for example, a pair of kings beats a pair of 7's.
(Hints: (1) Represent a poker hand by a list of terms of the form card(Suit,Value). For example a hand consisting of the 2 of clubs, the 5 of spades, the queen of hearts, the queen of dia- monds, and the 7 of spades would be represented by the list [card (clubs,2),card(spades,5),card(hearts,queen),card(diamonds,
queen),card(spades,7)1. (2)It maybe helpful to define relations such as has_f lush (Hand), which is true if all the cards in Hand are of the same suit; has_full_house (Hand), which is true if Hand has three cards with the same value but in different suits, and the other two cards have the same different value; and has_straight (Hand), which is true if Hand has cards with consecutive values. (3) The number of cases to consider is reduced if the hand is first sorted.)
3.4 Binary Trees
We next consider binary trees, another recursive data type. These struc- tures have an important place in many algorithms.
Binary trees are represented by the ternary functor tree (Element, Leí t,Right), where Element is the element at the node, and Left and Right are the left and right subtrees respectively. The empty tree is represented by the atom void. For example, the tree
73 Recursive Programming
a
b C
would be represented as
tree(a,tree(b,void,void),tree(c,void,void)).
Logic programs manipulating binary trees are similar to those manip- ulating lists. As with natural numbers and lists, we start with the type definition of binary trees. lt is given as Program 3.23. Note that the pro- gram is doubly recursive; that is, there are two goals in the body of the recursive rule with the same predicate as the head of the rule. This re- suits from the doubly recursive nature of binary trees and will be seen also in the rest of the programs of this section.
Let us write some tree-processing programs. Our first example tests whether an element appears in a tree. The relation scheme is tree_ member(Element,Tree). The relation is true if Element is one of the nodes in the tree. Program 3.24 contains the definition. The declarative reading of the program is: "X is a member of a tree if it is the element at the node (by the fact) or if it is a member of the left or right subtree (by the two recursive rules)."
The two branches of a binary tree are distinguishable, but for many ap- plications the distinction is not relevant. Consequently, a useful concept
binary_tree( Tree) - Tree is a binary tree.
binary_tree (void).
binary_tree (tree(Element ,Lef t ,Right)) -
binary_tree(Left), binarytree(Right).
Program 3.23 Defining binary trees
tree_member(Element,Tree) - Element is an element of the binary tree Tree.
tree_momber(X,tree(X,Left,Right)).
tree_member(X,tree(Y,Left ,Right)) - tree_rnember(X,Left).
tree_membor(X,tree(Y,Left ,Right)) '- tree_member(X ,Rïght).
Program 3.24 Testing tree membership
74 Chapter 3
a b
Figure 3.6 Comparing trees for isomorphism
isotree(Treel,Tree2) - Tree I and Tree2 are isomorphic binary trees.
isotree(void,void).
isotree(tree(X,Leftl,Rightl),tree(X,Left2,Right2))
isotree(Leftl,Left2), isotree(Rightl,Right2)
isotree(tree(X,Leftl,Rightl),tree(X,Left2,Right2))
isotree(Leftl ,Right2), isotree(Rightl ,Left2).
Program 3.25 Determining when trees are isomorphic
is isomorphism, which defines when unordered trees are essentially the same. Two binary trees Ti and T2 are isomorphic if T2 can be obtained by reordering the branches of the subtrees of Ti. Figure 3.6 shows three simple binary trees. The first two are isomorphic; the first and third are not.
Isomorphism is an equivalence relation with a simple recursive defini- tion. Two empty trees are isomorphic. Otherwise, two trees are isomor- phic if they have identical elements at the node and either both the left subtrees and the right subtrees are isomorphic; or the left subtree of one is isomorphic with the right subtree of the other and the two other sub- trees are isomorphic.
Program 3.25 defines a predicate isotree(Treel,Tree2), which is true if Tree i and Tree2 are isomorphic. The predicate is symmetric in its arguments.
Programs related to binary trees involve double recursion, one for each branch of the tree. The double recursion can be manifest in two ways. Programs can have two separate cases to consider, as in Program 3.24 for tree_member. In contrast, Program 3.12 testing membership of a list has only one recursive case. Alternatively, the body of the recursive clause has two recursive calls, as in each of the recursive rules for isotree in Program 3.25.
75 Recursive Programming
substitute (X, Y,TreeX,TreeY) - The binary tree Tree Y is the result of replacing all occurrences of X in the binary tree TreeX by Y.
substïtute(X,Y,void,void)
substitute(X,Y,tree(Node,Left,Right),tree(Nodel,Leftl,Rightl))
replace (X,Y,Node,Nodel)
substitute (X,Y,Left,Leftl)
substitute(X,Y,Rigbt,Rightl).
replace(X,Y,X,Y).
replace(X,Y,Z,Z) - X Z.
Program 3.26 Substituting for a term in a tree
The task in Exercise 3.3(i) is to write a program for substituting for el- ements in lists. An analogous program can be written for substituting elements in binary trees. The predicate substitute(X,Y,OldTree, NewTree) is true if NewTree is the result of replacing all occurrences of X by Y in OldTree. An axiomatization of substitute/4 is given as Program 3.26.
Many applications involving trees require access to the elements ap- pearing as nodes. Central is the idea of a tree traversal, which is a se- quence of the nodes of the tree in some predefined order. There are three possibilities for the linear order of traversal: preorder, where the value of the node is first, then the nodes in the left subtree, followed by the nodes in the right subtree; morder, where the left nodes come first followed by the node itself and then the right nodes; and postorder, where the node comes after the left arid right subtrees.
A definition of each of the three traversals is given in Program 3.27. The recursive structure is identical; the ordy difference between the pro- grams is the order in which the elements are composed by the various append goals.
The final example in this section shows interesting manipulation of trees. A binary tree satisfies the heap property if the value at each node is at least as large as the value at its children (if they exist). Heaps, a class of binary trees that satisfy the heap property, are a useful data structure and can be used to implement priority queues efficiently.
lt is possible to heapify any binary tree containing values for which an ordering exists. That is, the values in the tree are moved around so that
76 Chapter 3
preorder ( Tree,Pre) - Pre is a preorder traversal of the binary tree Tree.
preorder(tree(X,L,R) ,Xs)
preorder(L,Ls), preorder(R,Rs), append([XJLsJ,Rs,Xs).
preorder(void, E 1). morder ( Tree,In) -
In is an morder traversal of the binary tree Tree. inorder(tree(X,L,R) ,Xs) -
inorder(L,Ls), inorder(R,Rs), append(Ls,EXIRs],Xs).
inorder(void, E 1).
postorder ( Tree,Post) - Post is a postorder traversal of the binary tree Tree.
postorder(tree(X,L,R) ,Xs) -
postorder(L,Ls),
postorder(R,Rs),
append(Rs, [X] ,Rsl),
append(Ls,Rsl,Xs).
postorder(void, E 1).
Program 3.27 Traversals of a binary tree
the shape of the tree is preserved and the heap property is satisfied. An example tree and its heapified equivalent are shown in Figure 3.7.
An algorithm for heapifying the elements of a binary tree so that the heap property is satisfied is easily stated recursively. Heapify the left and right subtrees so that they both satisfy the heap property and then ad- just the element at the root appropriately. Program 3.28 embodies this algorithm. The relation heapify/2 lays out the doubly recursive pro- gram structure, and adjust(X,HeapL,HeapR,Heap) produces the final tree Heap satisfying the heap property from the root value X and the left and right subtrees HeapL and HeapR satisfying the heap property.
There are three cases for adj ust/4 depending on the values. If the root value is larger than the root values of the left and right subtrees, then the heap is tree(X,HeapL,HeapR). This is indicated in the first adjust clause in Program 3.28. The second clause handles the case where the root node in the left heap is larger than the root node and the root of the right heap. In that case, the adjustment proceeds recursively on the left heap. The third clause handles the symmetric case where the root node of the right heap is the largest. The code is simplified by relegating the concern whether the subtree is empty to the predicate greater/2.
77 Recursive Programming
Figure 3.7 A binary tree and a heap that preserves the tree's shape
heapify ( Tree,Heap) The elements of the complete binary tree Tree have been adjusted to form the binary tree Heap, which has the same shape as Tree and satisfies the heap property that the value of each parent node is greater than or equal to the values of its children.
heapify(void,void). heapify(tree(X,L,R) Heap)
heapify(L,HeapL), heapify(R,HeapR), adjust(X,HeapL,HeapR,Heap).
adj ust (X , HeapL , HeapR , tree (X , HeapL , HeapR) greater(X,HeapL), greater(X,HeapR).
adj ust(X,tree(X1,L,R),HeapR,tree(X1,HeapL,HeapR)) X < Xl, greater(X1,HeapR), adjust(X,L,R,HeapL).
adjust(X,HeapL,tree(Xl,L,R),tree(Xl,HeapL,HeapR)) -
X < Xl, greater(Xl,HeapL), adjust(X,L,R,HeapR).
greater(X,void).
greater(X,tree(X1,L,R)) - X Xl.
Program 3.28 Adjusting a binary tree to satisfy the heap property
3.4.1 Exercises for Section 3.4
(i) Define a program for subtree(S ,T), where S is a subtree of T.
(II) Define the relation sum_tree(Treeoflntegers,Sum), which holds if Sum is the sum of the integer elements in TreeOflntegers.
(IIi) Define the relation ordered(Treeof Integers), which holds if Tree is an ordered tree of integers, that is, for each node in the tree the elements in the left subtree are smaller than the element in
78 Chapter 3
the node, and the elements in the right subtree are larger than the element in the node. (Hint: Define two auxiliary relations, ordered_left(X,Tree) and ordered_right(X,Tree), which hold if both Tree is ordered and X is larger (respectively, smaller) than the largest (smallest) node of Tree.)
(iv) Define the relation tree_insert(X,Tree,Treel), which holds if Treel is an ordered tree resulting from inserting X into the ordered tree Tree. If X already occurs in Tree, then Tree and Tree i are iden- tical. (Hint: Four axioms suffice.)
(y) Write a logic program for the relation path(X,Tree,Path), where Path is the path from the root of the tree Tree to X.
3.5 Manipulating Symbolic Expressions
The logic programs illustrated so far in this chapter have manipulated natural numbers, lists, and binary trees. The programming style is ap- plicable more generally. This section gives four examples of recursive programming - a program for defining polynomials, a program for sym- bolic differentiation, a program for solving the Towers of Hanoi problem, and a program for testing the satisfiability of Boolean formulae.
The first example is a program for recognizing polynomials in some term X. Polynomials are defined inductively. X itself is a polynomial in X, as is any constant. Sums, differences, and products of polynomials in X are polynomials in X. So too are polynomials raised to the power of a natural number, and the quotient of a polynomial by a constant.
An example of a polynomial in the term x is x2 - 3x + 2. This follows from its being the sum of the polynomials, x2 - 3x and 2, where x2 - 3x is recognized recursively.
A logic program for recognizing polynomials is obtained by expressing the preceding informal rules in the correct form. Program 3.29 defines the relation polynomial (Expression,X), which is true if Expression is a polynomial in X. We give a declarative reading of two rules from the program.
The fact polynomial(X,X) says that a term X is a polynomial in itself. The rule
79 Recursive Programming
polynomial (Expression,X) - Expression is a polynomial ¡n X.
polynomial (X , X)
polynomial (Term, X) -
constant (Term)
polynomial (Termi +Term2 , X) -
polynomial (Terml,X), polynomial(Torm2,X).
polynomial (TermlTorm2 ,X)
polynomial(Terml,X), polynomial(Term2,X).
polynomial (Terml *Term2 , X)
polynomïal(Terml,X), polynomial(Term2,X).
polynomial (Terml/Term2 , X)
polynomial(Terml,X), constant(Term2).
polynomial(TermtN,X)
natural_number(N), polynomial(Term,X).
Program 3.29 Recognizing polynomials
polynomial(Terml+Terin2,X) - polyrioinial(Termi,X), polynomial(Term2,X).
says that the sum Terml+Term2 is a polynomial in X if both Tenni and Term2 are polynomials in X.
Other conventions used in Program 3.29 are the use of the unary pred- icate constant for recognizing constants, and the binary functor T to denote exponentiation. The term XTY denotes x".
The next example is a program for taking derivatives. The relation scheme is derivative(Expression,X,DifferentiatedExpression). The intended meaning of derivative is that DifferentiatedExpres- sion is the derivative of Expression with respect to X.
As for Program 3.29 for recognizing polynomials, a logic program for differentiation is just a collection of the relevant differentiation rules, written in the correct syntax. For example, the fact
derivative(X,X,s(0)).
expresses that the derivative of X with respect to itself is 1. The fact
derivative(sin(X) ,X,cos(X)).
80 Chapter 3
derivative (Expression,X,DifferentiatedEx pression) DifferentiatedEx pression is the derivative of Expression with respect to X.
derivative(X,X,s(0)).
derivative(XIs(N) ,X,s(N)*XIN)
derivative(sin(X) ,X,cos(X))
derivative(cos(X) ,X,-sin(X)).
derivative(elX,X,eX).
derivative(log(X),X,1/X).
derivat ive (F+G ,X ,DF+DG) -
derivative(F,X,DF), derivative(G,X,DG).
derivat ive (F-G ,X ,DF-DG) -
derivative(F,X,DF), derivative(G,X,DG).
derivat ive (F*G ,X ,F*DG+DF*G)
derivative(F,X,DF), derivative(G,X,DG).
derivative(1/F,X,-DF/(F*F)) -
derivative(F,X,DF).
derivative(F/G,X, (G*DF-F*DG)/(G*G)) -
derivative(F,X,DF), derivative(G,X,DG).
Program 3.30 Derivative rules
reads: "The derivative of sin(X) with respect to X is cos(X)." Natural mathematical notation can be used. A representative sample of functions and their derivatives is given in Program 3.30.
Sums and products of terms are differentiated using the sum rule and product rule, respectively. The sum rule states that the derivative of a sum is the sum of derivatives. The appropriate clause is
derivative(F+G,X,DF+DG) -
derivative(F,X,DF), derivative(G,X,DG).
The product rule is a little more complicated, but the logical clause is just the mathematical definition:
derivative(F*G,X,F*DG+DF*G) -
derivative(F,X,DF), derivative(G,X,DG).
Program 3.30 also contains the reciprocal and quotient rules. The chain rule is a little more delicate. It states that the derivative of
f(g(x)) with respect to x is the derivative of f(g(x)) with respect to g(x) times the derivative of g(x) with respect to x. As stated, it involves quan-
81 Recursive Programming
tification over functions, and is outside the scope of the logic programs we have presented.
Nonetheless, a version of the chain rule is possible for each particular function. For example, we give the rule for differentiating XN and sin(X):
derivative(Uls(N) ,X,s(N)*UTN*DU) -
derivative (U,X,DU).
derivative(sin(U),X,cos(U)*DU) - derivative(U,X,DU).
The difficulty of expressing the chain rule for differentiation arises from our choice of representation of terms. Both Programs 3.29 and 3.30 use the 'natural" representation from mathematics where terms represent themselves. A term such as sin(X) is represented using a unary structure sin. If a different representation were used, for example, unary_term(sin,X) where the name of the structure is made accessible, then the problem with the chain rule disappears. The chain rule can then be formulated as
derivative (unary_term(F,U) ,X,DF*DU) -
derivative(unary_term(F,U),U,DF), derivative(IJ,X,DU).
Note that all the rules in Program 3.30 would have to be reformulated in terms of this new representation and would appear less natural.
People take for granted the automatic simplification of expressions when differentiating expressions. Simplification is missing from Program 3.30. The answer to the query derivative (3*x+2,x,D)? is D(3*1+O* x)+O. We would immediately simplify D to 3, but it is not specified in the logic program.
The next example is a solution to the Towers of Hanoi problem, a standard introductory example in the use of recursion. The problem is to move a tower of n disks from one peg to another with the help of an auxiliary peg. There are two rules. Only one disk can be moved at a time, and a larger disk can never be placed on top of a smaller disk.
There is a legend associated with the game. Somewhere hidden in the surroundings of Hanoi, an obscure Far Eastern village when the legend was first told, is a monastery. The monks there are performing a task assigned to them by God when the world was created - solving the preceding problem with three golden pegs and 64 golden disks. At the moment they complete their task, the world will collapse into dust. Since the optimal solution to the problem with n disks takes 2 - i moves, we
82 Chapter 3
hanoi (N,A,B,C,Moves) - Moves is a sequence of moves for solving the Towers of Hanoi puzzle with N disks and three pegs, A, B, and C.
hanoi(s(0),A,B,C,[A to B]). hanoi(s(N),A,B,C,Moves) -
hanoi(N,A,C,B,Msl), hanoi(N,C,B,A,Ms2) append(Msl,[A to BIMs2],Moves).
Program 3.31 Towers of Hanoi
need not lose any sleep over this possibility. The number 264 is comfort- ingly big.
The relation scheme for solving the problem is hanoi(N,A,B,C, Moves). It is true if Moves is the sequence of moves for moving a tower of N disks from peg A to peg B using peg C as the auxiliary peg. This is an extension to usual solutions that do not calculate the sequence of moves but rather perform them. The representation of the moves uses a binary functor to, written as an infix operator. The term X to Y denotes that the top disk on peg X is moved to peg Y. The program for solving the problem is given in Program3.31.
The declarative reading of the heart of the solution, the recursive rule in Program 3.31, is: "Move s is the sequence of moves of s (N) disks from peg A to peg B using peg C as an auxiliary, if Msi is the solution for moving N disks from A to C using B, Ms2 is the solution for moving N disks from C to B using A, and Moves is the result of appending [A to BIMs2] to Msi."
The recursion terminates with moving one disk. A slightly neater, but less intuitive, base for the recursion is moving no disks. The appropriate fact is
hanoi(O,A,B,C,[ ]).
The final example concerns Boolean formulae. A Boolean formula is a term defined as follows: The constants true and
false are Boolean formulae; if X and Y are Boolean formulae, so are Xv Y, XA Y, and -X, where y and A are binary infix operators for disjunction and conjunction, respectively, and is a unary prefix operator for nega- tion.
83 Recursive Programming
saris fiable(Formula) There is a true instance of the Boolean formula Formula.
satisfiabie(true)
satisfiable(XAY) - satisfiable(X), satisfiable(Y).
satisfiabie(XVY) satisfïable(X).
satisfiable(XVY) - satisfiable(Y).
satisfiable(-X) - invalid(X).
invalid(Formula) There is a false instance of the Boolean formula Formula.
invalid (f aise)
invaiid(XVY) - invaiìd(X), invalid(Y).
invaiid(XAY) invaiid(X).
invalid(XAY) - invalid(Y).
invalid(-'-Y) - satisf jable (Y)
Program 3.32 Satisfiability of Boolean formulae
A Boolean formula F is true if
F 'true', F = XAY, and both X and Y are true. F = XvY, and either X or Y (or both) are true. F = X, and X is false.
A Boolean formula F is false if
F = 'false'. F = XAY, and either X or Y (or both) are false. F = XvY, and both X and Y are false. F = =X, and X is true.
Program 3.32 is a logic program for determining the truth or falsity of a Boolean formula. Since it can be applied to Boolean formulae with variables, it is actually more powerful than it seems. A Boolean formula with variables is satisfiable if it has a true instance. It is invalid if it has a false instance. These are the relations computed by the program.
84 Chapter 3
3.5.1 Exercises for Section 3.5
Write a program to recognize if an arithmetic sum is normalized, that is, has the form A + B, where A is a constant and B is a normal- ized sum.
Write a type definition for Boolean formulae.
Write a program for recognizing whether a logical formula is in conjunctive normal form, namely, is a conjunction of disjunctions of literals, where a literal is an atomic formula or its negation.
Write a program for the relation negation_inwards (Fi ,F2), which is true if F2 is the logical formula resulting from moving all nega- tion operators occurring in the formula Fi inside conjunctions and disjunctions.
(y) Write a program for converting a logical formula into conjunctive normal form, that is, a conjunction of disjunctions.
(vi) Consider the following representation of a bag, that is, a list of elements with multiplicities. The function symbol bag(Element, Multiplicity,RestûfBag) should be used. The atom void can be used as an empty bag. For example, the term bag (a, 3, bag (b, 2, void)) represents a list of three copies of an element a, and two copies of an element b. Write logic programs to
Take the union of two bags;
Take the intersection of two bags;
Substitute for an element in a bag;
Convert a list into a bag;
Convert a binary tree into a bag.
3.6 Background
Many of the programs in this chapter have been floating around the logic programming community, and their origins have become obscure. For
85 Recursive Programming
example, several appear in Clocksm and Mellish (1984) and in the uneven collection of short Prolog programs, How to Solve It in Prolog by Coelho et al. (1980).
The latter book has been updated as Coelho and Cotta (1988) and is a source for other simple examples. The exercise on describing poker hands is due to Ken Bowen.
The classic reference for binary trees is Knuth (1968) and for sorting Knuth (1973).
A discussion of the linear algorithm for the kth largest algorithms can be found in most textbooks on algorithms, for example, Horowitz and Sahni (1978). The discussion of the heap property is taken from Horowitz and Sahni (1978).
Many of the basic programs for arithmetic and list processing have a simple structure that allows many correctness theorems to be proved automatically, see, for example, Boyer and Moore (1979) and Sterling and Bundy (1982).
Ackermann's function is discussed by Peter (1967).
4 The Computation Model of Logic Programs
The computation model used in the first three chapters of the book has a severe restriction. All goals appearing in the proof trees are ground. All rule instances used to derive the goals in the proof trees are also ground. The abstract interpreter described assumes that the substitutions giving the desired ground instances can be guessed correctly. In fact, the cor- rect substitutions can be computed rather than guessed.
This chapter presents a general computation model of logic programs. The first section presents a unification algorithm that removes the guess- work in determining instances of terms. The second section presents an appropriately modified abstract interpreter and gives example computa- tions of logic programs.
The computation model of logic programming we present is especially well suited to sequential languages such as Prolog. Our model can be used to describe parallel logic programming languages. However, devel- opers of these languages have often used other models, such as state transitions or dynamic tree creation and destruction (see Section 4.3).
4.1 Unification
The heart of our computation model of logic programs is unification. Unification is the basis of most work in automated deduction and of the use of logical inference in artificial intelligence.
Necessary terminology for describing the algorithm is repeated from Chapter 1, and new definitions are introduced as needed.
88 Chapter 4
Recall that a term t is a common instance of two terms, t1 and t2, if there exist substitutions 9 and 92 such that t equals t191 and t202. A term s is more general than a term t 1f t is an instance of s but s is not an instance of t. A term s is an alphabetic variant of a term t if both s is an instance of t and t is an instance of s. Alphabetic variants are related by the renaming of variables that occur in the terms. For exam- ple, meinber(X,tree(Left,X,Right)) and member(Y,tree(Left,Y,Z)) are alphabetic variants.
A unifier of two terms is a substitution making the terms identical. If two terms have a unifier, we say they unify. There is a close relation be- tween unifiers and common instances. Any unifier determines a common instance, and conversely, any common instance determines a unifier.
For example, append([1,2,3] ,[3,4] ,List) and append([XXs] ,Ys, [XIZs]) unify. A unifying substitution is {X"1,Xs[2,3], Ys=[3,4], List=[1Zs] }. Their common instance, determined by this unifying sub- stitution, is append( [1,2,3] [3,4], [1Zs] ).
A most general unifier, or mgu, of two terms is a unifier such that the associated common instance is most general. It can be shown that if two terms unify, all mgus are equivalent. Making that statement precise is beyond the scope of this book, but we give pointers in Section 4.3. We proceed by giving an algorithm that computes a most general unifier of two terms if one exists.
The algorithm for unification presented here is based on solving equa- tions. The input for the algorithm is two terms, T1 and T2. The output of the algorithm is an mgu of the two terms if they unify, or failure if the terms do not unify. The algorithm uses a pushdown stack for storing the equations that need to be solved and a location, O, for collecting the substitution comprising the output.
The location O is initially empty, and the stack is initialized to contain the equation T1 = T2. The algorithm consists of a loop of popping an equation from the stack and processing it. The loop terminates when the stack becomes empty or if failure occurs in processing an invalid equation.
We consider the possible actions for dealing with a popped equation S = T. The simplest case is if S and T are identical constants or var- iables. This equation is correct, and nothing further needs to be done. The computation continues by popping the next equation from the stack.
89 The Computation Model of Logic Programs
If S is a variable, and T is a term not containing S, the following hap- pens. The stack is searched for all occurrences of S, which are replaced by T. Similarly, all occurrences of S in O are replaced by T. Then the sub- stitution S = T is added to O. It is significant that S does not occur in T. The test embodied by the phrase "not containing" is known as the occurs check.
If T is a variable, and S is a term not containing T, i.e., T satisfies the occurs check with respect to S, the symmetric sequence of actions happens.
Equations are added to the stack if S and T are compound terms with the same principal functor and arity, f(S1,. . .,S,) and f(T1,.. say. For the terms to unify, each of the argument pairs must simultaneously unify. This is achieved by pushing the n equations, S = T, onto the stack.
In any other case, failure is reported, and the algorithm terminates. If the stack is emptied, the terms unify, and the unifier can be found in ¿9. The complete algorithm is given as Figure 4.1. The occurs check is embodied in the phrase "that does not occur in."
We do not prove the correctness of this algorithm, nor analyze its com- plexity. The interested reader is referred to the literature in Section 4.3.
Consider attempting to unify the terms append([a,b] , [c,d] ,Ls) and append([XIXs] ,Ys, [XIZs]). The stack is initialized to the equation
append([a,b] , [c,d] ,Ls) append([XIXs] ,Ys, [XIZs]).
These two terms have the same functor, append, and arity, 3, so we add the three equations relating the subterms of the two terms. These are {a,b]=[XIXs], [c,d]=Ys, and Ls=[XZs].
The next equation, [a, b] = [X Xs], is popped from the stack. These two compound terms have the same functor, ".", and arity, 2, so two equa- tions, aX and [b] =X, are added to the stack. Continuing, the equation a=X is popped. This is covered by the second case in Figure 4.1. X is a variable not occurring in the constant, a. All occurrences of X in the stack are replaced by a. One equation is affected, namely Ls= [XIZs], which becomes Ls= [a Zs]. The equation X=a is added to the initially empty sub- stitution, and the algorithm continues.
The next equation to be popped is Lb] =Xs. Again this is covered by the second case. Xs= [b] is added to the set of substitutions, and the stack is checked for occurrences of Xs. There are none, and the next equation is popped.
90 Chapter 4
Input: Two terms T1 and T2 to be unified
Output: O, the mgu of T1 and T2, or failure
Algorithm: Initialize the substitution O to be empty, the stack to contam the equation T1 = T2, and failure to false.
while stack not empty and no failure do
pop X = Y from the stack
case X is a variable that does not occur in Y:
substitute Y for X in the stack and in O add X = Y to O
Y is a variable that does not occur in X: substitute X for Y in the stack and in O add Y = X to O
X and Y are identical constants or variables: continue
Xisf(X1.....X)andYisf(Y1.....Y) for some functor f and n> O: push X, = Y,i = i n, on the stack
otherwise: failure is true
If failure, then output failure else output O.
Figure 4.1 A unification algorithm
The second case also covers [c,d]=Ys. Another substitution, Ys=[c, dl, is added to the collection, and the final equation, Ls=[aZs], is popped. This is handled by the symmetric first case. Ls does not occur in [aZs], so the equation is added as is to the unifier, and the algorithm terminates successfully. The unifier is {X=a,Xs=[b] Ys= Ec , d], Ls= [aZs] }. The common instance produced by the unifier is append([a,bl , [c,d] , [als]). Note that in this unification, the substi- tutions were not updated.
The occurs check is necessary to prevent the unification of terms such as s (X) and X. There is no finite common instance of these terms. How-
91 The Computation Mode! of Logic Programs
ever, most Prolog implementations omit the occurs check from the unifi- cation algorithm, for pragmatic reasons.
When implementing this unification algorithm for a particular logic programming language, the explicit substitution in both the equations on the stack and the unifier is avoided. Instead, logical variables and other terms are represented by memory cells with different values, and variable binding is implemented by assigning to the memory cell representing a logical variable a reference to the cell containing the representation of the term the variable is bound to. Therefore,
Substitute Y for X in stack and in O. Add X = Y to substitutions.
is replaced by
Make X a reference to Y.
4.1.1 Exercises for Section 4.1
Use the algorithm in Figure 4.1 to compute an mgu of append([b], [c,d] ,L) and append([XIXs] ,Ys, [XIZs]).
Use the algorithm in Figure 4.1 to compute anmguofhnoi(s(N), A,B,C,Ms)andhanoi(s(s(0)),a,b,c,Xs).
4.2 An Abstract Interpreter for Logic Programs
We revise the abstract interpreter of Section 1.8 in the light of the unifi- cation algorithm. The result is our full computation model of logic pro- grams. Ail the concepts introduced previously, such as goal reductions and computation traces, have their analogues in the full model.
A computation of a logic program can be described informally as fol- lows. It starts from some initial (possibly conjunctive) query G and, if it terminates, has one of two results: success or failure. If a computation succeeds, the instance of G proved is conceived of as the output of the computation. A given query can have several successful computations, each resulting in a different output. In addition, it may have nontermi- nating computations, to which we associate no result.
92 Chapter 4
The computation progresses via goal reduction. At each stage, there is some resolvent, a conjunction of goals to be proved. A goal in the resol- vent and clause in the logic program are chosen such that the clause's head unifies with the goal. The computation proceeds with a new resol- vent, obtained by replacing the chosen goal by the body of the chosen clause in the resolvent and then applying the most general unifier of the head of the clause and the goal. The computation terminates when the resolvent is empty. In this case, we say the goal is solved by the program.
To describe computations more formally, we introduce some useful concepts. A computation of a goal Q = Qo by a program P is a (possibly infinite) sequence of triples (Q,G1,C1). Q is a (conjunctive) goal, G is a goal occurring in Q, and C is a clause AB1,. . .,B in P renamed so that it contains new variable symbols not occurring in O, O j i. For all i> O, Q is the result of replacing G by the body of C in Q, and applying the substitution 0, the most general unifier of G and A, the head of C; or the constant true if G is the only goal in Q and the body of C is empty; or the constant fail if G and the head of C do not unify.
The goals B01 are said to be derived from G1 and C. A goal G = Bk0, where Bjk occurs in the body of clause C1, is said to be invoked by G and C. G is the parent of any goal it invokes. Two goals with the same parent goal are sibling goals.
A trace of a computation of a logic program (Q,G1,C1) is the sequence of pairs (G1,G), where 0 is the subset of the mgu 0 computed at the ith reduction, restricted to variables in G.
We present an abstract interpreter for logic programs. It is an adap- tation of the interpreter for ground goals (Figure 1.1). The restriction to using ground instances of clauses to effect reductions is lifted. Instead, the unification algorithm is applied to the chosen goal and head of the chosen clause to find the correct substitution to apply to the new resol- vent.
Care needs to be taken with the variables in rules to avoid name clashes. Variables are local to a clause. Hence variables in different clauses that have the same name are, in fact, different. This is ensured by renaming the variables appearing in a clause each time the clause is chosen to effect a reduction. The new names must not include any of the variable names used previously in the computation.
The revised version of the interpreter is given as Figure 4.2. It solves a query G with respect to a program P. The output of the interpreter is an
93 The Computation Model of Logic Programs
Input:
Output:
Algorithm:
A goal G and a program P
An instance of G that is a logical consequence of P, or no otherwise
lnitjalize the resolvent to G. while the resolvent is not empty do
choose a goal A from the resolvent choose a (renamed) clause A' B1 ..,B,1 from P
such that A and A' unify with mgu E) (if no such goal and clause exist, exit the while loop)
replace A by B1 ,...,B ¡n the resolvent apply O to the resolvent and to G
If the resolvent is empty, then output G, else output rio.
Figure 4.2 An abstract interpreter for logic programs
instance of G if a proof of such an instance is found, or no if a failure has occurred during the computation. Note that the interpreter may also fail to terminate.
An instance of a query for which a proof is found is called a solution to the query.
The policy for adding and removing goals from the resolvent is called the scheduling policy of the interpreter. The abstract interpreter leaves the scheduling policy unspecified.
Consider solving the query append([a,b] , [c,d] ,Ls)? by Program 3.15 for append using the abstract interpreter of Figure 4.2. The resol- vent is initialized to be append([a,bJ , [c,d] ,Ls). lt is chosen as the goal to reduce, being the on'y one. The rule chosen from the program is
append([XIXs],Ys,[XIZs]) - append(Xn,Ys,Zs).
The unifier of the goal and the head of the rule is {X=a,Xs=[b], Ys=[c,d] , Ls[aIZs]}. A detailed calculation of this unifier appeared in the previous section. The new resolvent is the instance of ap- pend(Xn , Ys ,Zn) under the unifier, namely, append( [b] , [c , d] ,Zn). This goal is chosen in the next iteration of the loop. The same clause for append is chosen, but variables must be renamed to avoid a clash of variable names. The version chosen is
append([X1IXs1],Ysl,[X1IZs1]) - apperid(Xsl,Ysl,Zsl).
94 Chapter 4
append([a,b] , [c,d] ,Ls) append([b] , [c,d] ,Zs)
append([ j, [c,d] ,Zsl) true
Output: Ls= [a, b, c , d]
Ls= [aIZs] Zs[bIZsl] Zsl=[c,d]
Figure 4.3 Tracing the appending of two lists
The unifier of the head and goal is {X1=b, Xsl=[ ], Ys1[c,d], Zs=[bZs1]}. The new resolvent is append([ ], [c,d] ,Zsl). This time the fact append( [ ] , Zs2 , Zs2) is chosen; we again rename variables as necessary. The unifier this time is {Zs2=[c,d] , Zsl=[c,d]}. The new resolvent is empty and the computation terminates.
To compute the result of the computation, we apply the relevant part of the mgu's calculated during the computation. The first unification instantiated Ls to [aIZs]. Zs was instantiated to [bZs1] in the second unification, and Zs i further became [c , d]. Putting it together, Ls has the value [a[b[c,d]]],ormore simply, [a,b,c,d].
The computation can be represented by a trace. The trace of the fore- going append computation is presented in Figure 4.3. To make the traces clearer, goals are indented according to the indentation of their parent. A goal has an indentation depth of d+1 if its parent has indentation depth d.
As another example, consider solving the query son(S,haran)? by Program 1.2. It is reduced using the clause son(X,Y) - f ather(Y,X), male (X). A most general unifier is {X=S,Y=haran}. Applying the sub- stitution gives the new resolvent f ather(haran,S), male(S). This is a conjunctive goal. There are two choices for the next goal to reduce. Choosing the goal father (haran , S) leads to the following computation. The goal) unifies with the fact father (haran, lot) in the program, and the computation continues with S instantiated to lot. The new resolvent is male (lot), which is reduced by a fact in the program, and the compu- tation terminates. This is illustrated in the left trace in Figure 4.4.
The other possibility for computing S=haran is choosing to reduce the goal male (S) before father (haran,S). This goal is reduced by the fact male(lot) with S instantiated to lot. The new resolvent is f a- ther (haran, lot), which is reduced to the empty goal by the correspond- ing fact. This is the right trace in Figure 4.4.
95 The Computation Model of Logic Programs
son(S,haran) son(S,haran) father(haran,S) S1ot male(S) S=:]ot male(lot) father(haran,lot)
true true
Figure 4.4 Different traces of the same solution
Solutions to a query obtained using the abstract interpreter may con- tain variables. Consider the query mernber(a,Xs)? with respect to Pro- gram 3.12 for member. This can be interpreted as asking what list Xs has the element a as a member One solution computed by the abstract inter- preter is Xs= [a lYs], namely, a list with a as its head and an unspecified tail. Solutions that contain variables denote an infinity of solutionsall their ground instances.
There are two choices in the interpreter of Figure 4.2: choosing the goal to reduce, and choosing the clause to effect the reduction. These must be resolved in any realization of the computation model. The nature of the choices is fundamentally different.
The choice of goal to reduce is arbitrary; it does not matter which is chosen for the computation to succeed. If there is a successful computa- tion by choosing a given goal, then there is a successful computation by choosing any other goal. The two traces in Figure 4.4 illustrate two suc- cessful computations, where the choice of goal to reduce at the second step of the computation differs.
The choice of the clause to effect the reduction is nondeterministic. Not every choice will lead to a successful computation. For example, in both traces in Figure 4.4, we could have gone wrong. If we had chosen to reduce the goal father (haran,S) with the fact father (haran,yiscah), we would not have been able to reduce the invoked goal male (yiscah). For the second computation, had we chosen to reduce male(S) with male(isaac), the invoked goal f ather(haran,isaac) could not have been reduced.
For some computations, for example, the computation illustrated in Figure 4.3, there is only one clause from the program that can reduce each goal. Such a computation is called deterministic. Deterministic com- putations mean that we do not have to exercise our nondeterministic imagination.
The alternative choices that can be made by the abstract interpreter when trying to prove a goal implicitly define a search tree, as described
96 Chapter 4
more fully in Section 5.4. The interpreter "guesses" a successful path in this search tree, corresponding to a proof of the goal, if one exists. However, dumber interpreters, without guessing abilities, can also be built, with the same power as our abstract interpreter. One possibility is to search this tree breadth-first, that is, to explore all possible choices in parallel. This will guarantee that if there is a finite proof of the goal (i.e., a finite successful path in the search tree), it will be found.
Another possibility would be to explore the abstract search tree depth- first. In contrast to the breadth-first search strategy, the depth-first one does not guarantee finding a proof even if one exists, since the search tree may have infinite paths, corresponding to potentially infinite com- putations of the nondeterministic interpreter. A depth-first search of the tree might get lost in an infinite path, never finding a finite successful path, even if one exists.
In technical terms, the breadth-first search strategy defines a complete proof procedure for logic programs, whereas the depth-first one is in- complete. In spite of its incompleteness, depth-first search is the one incorporated in Prolog, for practical reasons, as explained in Chapter 6.
Let us give a trace of a longer computation, solving the Towers of Hanoi problem with three disks, using Program 3.31. It is a deterministic computation, given as Figure 4.5. The final append goal is given without unifications. It is straightforward to fill them in.
Computations such as that in Figure 4.5 can be compared to compu- tations in more conventional languages. Unification can be seen to sub- sume many of the mechanisms of conventional languages: record alloca- tion, assignment of and access to fields in records, parameter passing, and more. We defer the subject until the computation model for Prolog is introduced in Chapter 6.
A computation of G by P terminates if G = true or fail for some n O. Such a computation is finite and of length n. Successful computations correspond to terminating computations that end in true. Failing com- putations end in fail. All the traces given so far have been of successful computations.
Recursive programs admit the possibility of nonterminating computa- tions. The query append(Xs, [c,d] ,Ys)? with respect to append can be reduced arbitrarily many times using the rule for append. In the process, Xs becomes a list of arbitrary length. This corresponds to solutions of the query appending [c , dl to an arbitrarily long list. The nonterminat- ing computation is illustrated in Figure 4.6.
97 The Computation Mode! of Logic Programs
hanoi(s(s(s(0))),a,b,c,Ms) hanoi(s(s(0)),a,c,b,Msl)
hanoi(s(0),a,b,c,Msl 1) hanoi(s(0),b,c,a,Ms 12) append([a to bija to c,b to ciMsi)
appendU ]ja to c,b to c],Xs) hanoi(s(s(0)),c,b,a,Ms2)
hanoi(s(0),c,a,b,Ms2 1) hanoi(s(0),a,b,c,Ms22) append(]c to aJ,fc to b,a to b],Ms2)
appendU ],[c to ba to b],Ys) append([c to a,c to b,a to bila to b,c to a,
c to b,a to b],Ms) append(Lc to b,a to b],[a to b,c to a,
c to b,a to bJ,Zs) append([a to biLa to b,c to a,
c to b,a to biZsl) appendU ]Ja to b,c to a,
c to b,a to b],Zs2)
Figure 4.5 Solving the Towers of Hanoi
append(Xs,[c,dJ,Ys) append(Xsl ,]c,d],Ysl)
append(Xs2,[c,dI,Ys2) append(Xs 3 ,[c,d],Ys3)
Xs=[XIXs1], Ys=LXYs1l
Xsl=[X1 Xs2], Ysl=[X1 IYs2l Xs2=[X2IXs3], Ys2=LX2JYs3]
Xs3=[X3IXs4I, Ys3=[X3IYs4]
Figure 4.6 A nonterminating computation
Msll=[a to b] Ms12=[b toc] Msl=[a to bIXsJ Xs=[a to c,b to cl
Ms21=Fc to a]
Ms22=[a to bi Ms2=[c to alYs] Ys=[c to b,a to b]
Ms=Lc to aIZsI
Zs=[c to bjZsli
Zsl=[a to bIZs2]
Zs2=[a to b,c to a, c to b,a to b]
98 Chapter 4
All the traces presented so far have an important feature in common. If two goals G and G1 are invoked from the same parent, and G appears before G in the trace, then all goals invoked by G will appear before G in the trace. This scheduling policy makes traces easier to follow, by solving queries depth-first.
The scheduling policy has another important effect: instantiating vari- ables before their values are needed for other parts of the computation. A good ordering can mean the difference between a computation being deterministic or not.
Consider the computation traced in Figure 4.5. The goal
hanoi(s(s(s(0))) ,a,b,c,Ms)
is reduced to the following conjunction
hanoi(s(s(0)) ,a,c,b,Msl),
hanoi(s(s(0)) ,c,b,a,Ms2)
append(Msl, [a to bIMs2] ,Ms).
If the append goal is now chosen, the append fact could be used (incor- rectly) to reduce the goal. By reducing the two hanoi goals first, and all the goals they invoke, the append goal has the correct values for Ms i and Ms2.
4.2.1 Exercises for Section 4.2
Trace the query sort([3,1,2] ,Xs)? using the permutation sort (3.20), insertion sort (3.21), and quicksort (3.22) programs in turn.
Give a trace for the goal derivative(3*sin(x)-4*cos(x),x,D) using Program 3.30 for derivative.
Practice tracing your favorite computations.
4.3 Background
Unification plays a central role in automated deduction and in the use of logical inference in artificial intelligence. It was first described in the landmark paper of Robinson (1965). Algorithms for unification have been
99 The Computation Model of Logic Programs
the subject of much investigation: see, for example, Martelli and Monta- nari (1982), Paterson and Wegman (1978), and Dwork et al. (1984). Typi- cal textbook descriptions appear in Bundy (1983) and Niisson (1980).
The definition of unification presented here is nonstandard. Readers wishing to learn more about unifiers are referred to the definitive dis- cussion on unification in Lassez, Maher, and Marriott (1988). This paper points out inconsistencies of the various definitions of unifiers that have been proposed in the literature, including the version in this book. Es- sentially, we have explained unifiers based on terms to avoid technical issues of composition of substitutions, which are not needed for our de- scription of logic progranmiing computations.
The computation model we have presented has a sequential bias and is influenced by the computation model for Prolog given in Chapter 6. Nonetheless, the model has potential for parallelism by selecting several goals or several rules at a time, and for elaborate control by selecting complicated computation rules. References for reading about different computation models for logic programming are given in Section 6.3.
Another bias of our computation model is the central place of unifi- cation. An exciting development within logic programming has been the realization that unification is just one instance of constraint solving. New computation models have been presented where the solution of equal- fly constraints, i.e., unification, in the abstract interpreter of Figure 4.2 is replaced by solving other constraints. Good starting places to read about the new constraint-based models are Colmerauer (1990), Jaffar and Lassez (1987), and Lassez (1991).
A proof that the choice of goal to reduce from the resolvent is arbitrary can be found in Apt and van Emden (1982) or in the text of Lloyd (1987).
A method for replacing the runtime occurs check with compile-time analysis was suggested by Plaisted (1984).
Attempts have been made to make unification without the occurs check more than a necessary expedient for practical implementations of Prolog. In particular, Colmerauer (1982b) proposes a theoretical model for such unifications that incorporates computing with infinite terms.
A novel use of unification without the occurs check appears in Eggert and Chow (1983), where Escher-like drawings that gracefully tend to in- finity are constructed.
5.1 Semantics
Semantics assigns meanings to programs. Discussing semantics allows us to describe more formally the relation a program computes. Chap- ter 1 informally describes the meaning of a logic program P as the set of ground instances that are deducible from P via a finite number of ap- plications of the rule of universal modus ponens. This section considers more formal approaches.
5 Theory of Logic Programs
A major underlying theme of this book, laid out in the introduction, is that logic programming is attractive as a basis for computation because of its basis in mathematical logic, which has a well-understood, well- developed theory. In this chapter, we sketch some of the growing theory of logic programming, which merges the theory inherited from mathe- matical logic with experience from computer science and engineering. Giving a complete account is way beyond the scope of this book. In this chapter, we present some results to direct the reader in important direc- tions. The first section, on semantics, gives definitions and suggests why the model-theoretic and proof-theoretic semantics give the same result. The main issue in the second section, on program correctness, is termi- nation. Complexity of logic programs is discussed in the third section. The most important section for the rest of the book is Section 4, which discusses search trees. Search trees are vital to understanding Prolog's behavior. Finally, we introduce negation in logic programming.
102 Chapter 5
parent(terach,abraham). parent(abrahani,isaac). parent(isaac ,jacob). parent(jacob,benjainin). ancestor(X,Y) - parent(X,Y). ancestor(X,Z) - parent(X,Y), ancestor(Y,Z).
Program 5.1 Yet another family example
The operational semantics is a way of describing procedurally the meaning of a program. The operational meaning of a logic program P is the set of ground goals that are instances of queries solved by P using the abstract interpreter given in Figure 4.2. This is an alternative for- mulation of the previous semantics, which defined meaning in terms of logical deduction.
The declarative semantics of logic programs is based on the standard model-theoretic semantics of first-order logic. In order to define it, some new terminology is needed.
Definition Let P be a logic program. The Herbrand universe of P, denoted U(P), is the set of all ground terms that can be formed from the constants and function symbols appearing in P. u
In this section, we use two running examplesyet another family data- base example, given as Program 5.1; and Program 3.1 defining the natural numbers, repeated here:
natural_number (0)
natural_number(s(X)) - natural_number(X).
The Herbrand universe of Program 5.1 is the set of all constants appear- ing in the program, namely, {terach, abraham, isaac ,jacob, benj amin}. If there are no function symbols, the Herbrand universe is finite. In Pro- gram 3.1, there is one constant symbol, O, and one unary function sym- bol, s. The Herbrand universe of Program 3.1 is {O, s (0) , s (s (0)), . . . If no constants appear in a program, one is arbitrarily chosen.
Definition The Herbrand base, denoted B(P), is the set of all ground goals that can be formed from the predicates in P and the terms in the Herbrand universe.
103 Theory of Logic Programs
There are two predicates, parent/2 and ancestor/2, in Program 5.1. The Herbrand base of Program 5.1 consists of 25 goals for each predi- cate, where each constant appears as each argument:
{parent(terach,terach), parent(terach,abraham), parerit(terach,isaac), parent(terach,jacob), parent(terach,benjanuin), parent(abraham,terach), parent (abraham, abraham), parent(abraham, isaac), parent(abraham,jacob), parent(abraham,benjanìin), parent(isaac,terach), parent(isaac,abrahain), parent(isaac,isaac), parent(isaac,jacob), parent(isaac,benjaniin), parent(jacob,terach), parent(jacob,abraham), parent(jacob,isaac), parent(jacob,jacob), parent(jacob,berijamin), parent(berijamin,terach), parent(benjainin,abraham), parent (benjamin,isaac), parent(benjamin,jacob), parent(benjamin,benjarniri), ancestor(terach,terach), ancestor(terach,abraham), ancestor(terach,isaac), ancestor(terach,jacob), ancestor(terach,benjamin), ancestor(abrahani,terach), ancestor(abraham,abrahani), ancestor(abraham,isaac), ancestor(abrahain,jacob), ancestor(abraham,benjamin), ancestor(isaac,terach), ancestor(isaac,abrahani), ncestor(isaac,isaac), ancestor(isaac,jacob), ancestor(isaac,benjaxnin), ancestor(jacob,terach), ancestor(jacob,abraham), ancestor(jacob,isaac), ancestor(jacob,jacob), ancestor(jacob,berijainin), ancestor(benjamin,terach), ancestor(benjamin,abrahain), ancestor(benjainin,isaac), ancestor (benjamin, jacob), ancestor (benj amin, benjamin) }.
The Herbrand base is infinite if the Herbrand universe is. For Pro- gram 3.1, there is one predicate, natural_number. The Herbrand base equals tnatural_nuinber(0) ,natural_riumber(s(0)), . . . L
Definition An interpretation for a logic program is a subset of the Herbrand base. u
An interpretation assigns truth and falsity to the elements of the Her- brand base. A goal in the Herbrand base is true with respect to an inter- pretation if it is a member of it, false otherwise.
104 Chapter 5
Definition An interpretation I is a model for a logic program if for each ground instance of a clause in the program AB1,. . .,B, A is in I if B1,. . .,B are inI.
Intuitively, models are interpretations that respect the declarative reading of the clauses of a program.
For Program 3.1, natural_number(0) must be in every model, and natural_number(s(X)) is in the model if natural_number(X) is. Any model of Program 3.1 thus includes the whole Herbrand base.
For Program 5.1, the facts parent(terach,abraham), parent(abra- ham,isaac), parent(isaac,jacob), and parent(jacob,benjamin) must be in every model. A ground instance of the goal ancestor(X,Y) is in the model if the corresponding instance of parent (X, Y) is, by the first clause. So, for example, ancestor(terach,abrahain) is in every model. By the second clause, ancestor(X,Z) is in the model if parent(X,Y) and ancestor(Y,Z) are.
It is easy to see that the intersection of two models for a logic program P is again a model. This property allows the definition of the intersection of all models.
Definition The model obtained as the intersection of all models is known as the minimal model and denoted M(P). The minimal model is the declarative meaning of a logic program. .
The declarative meaning of the program for natural_number, its min- imal model, is the complete Herbrand base {natural_number(0) ,natu- ral_number(s(0)),natural_number(s(s(0))), . ..
The declarative meaning of Program 5.1 is {parent(terach,abrahani), parent(abrahain,isaac), parent(isaac,jacob), parent(jacob,
benjamin), ancestor(terach,abraham), ancestor(abraham,isaac),
ancestor(isaac,jacob), ancestor(jacob,benjamin), ancestor
(terach,isaac), ancestor(terach,jacob), ancestor(terach,
benjamin), ancestor(abraham,jacob), ancestor(abraham,ben- jamin), ancestor(isaac,benjamin)}.
Let us consider the declarative meaning of append, defined as Pro- gram 3.15 and repeated here:
append([XIXs],Ys,[XIZs]) - append(Xs,Ys,Zs).
append([ ] ,Ys,Ys).
105 Theory of Logic Programs
The Herbrand universe is F 1,1F ]],[[ 1,1 ]], . . , namely, all lists that can be built using the constant 1]. The Herbrand base is all combinations of lists with the append predicate. The declarative meaning is all ground in- stances of append([ I ,Xs,Xs), that is, append([ i, [ J 1 i) append( E i , [L i] , [L i]), . . , together with goals such as append ([E 1] , E i , [E ]]), which are logically implied by application(s) of the rule. This is only a subset of the Herbrand base. For example, append( E J , E ] , [E ]]) is not in the meaning of append but is in the Herbrand base.
Denotational semantics assigns meanings to programs based on asso- ciating with the program a function over the domain computed by the program. The meaning of the program is defined as the least fixpoint of the function, if it exists. The domain of computations of logic programs is interpretations.
Definition Given a logic program P, there is a natural mapping Tp from interpreta- tions to interpretations, defined as follows:
Te(I) = [A in B(P):A B1,B2,.. .,B, n 0, is a ground instance of a clause in P, and B1,.. .,B are in I}. i
The mapping is monotonic, since whenever an interpretation I is con- tained in an interpretation J, then Tp(I) is contained in Tp(J).
This mapping gives an alternative way of characterizing models. An interpretation I is a model if and only if Te(I) is contained in I.
Besides being monotonic, the transformation is also continuous, a no- tion that will not be defined here. These two properties ensure that for every logic program P, the transformation Tp has a least fixpoint, which is the meaning assigned to P by its denotational semantics.
Happily, all the different definitions of semantics are actually describ- ing the same object. The operational, denotational, and declarative se- mantics have been demonstrated to be equivalent. This allows us to de- fine the meaning of a logic program as its minimal model.
5.2 Program Correctness
Every logic program has a well-defined meaning, as discussed ¡n Sec- tion 5.1. This meaning is neither correct nor incorrect.
106 Chapter 5
The meaning of the program, however, may or may not be what was intended by the programmer. Discussions of correctness must therefore take mto consideration the intended meaning of the program. Our pre- vious discussion of proving correctness and completeness similarly was with respect to an intended meanrng of a program.
We recall the definitions from Chapter 1. An intended meaning of a program P is a set of ground goals. We use intended meanings to denote the set of goals intended by the programmer for the program to com- pute. A program P is correct with respect to an intended meaning M if M(P) is contained in M. A program P is complete with respect to an in- tended meaning if M is contained in M(P). A program is thus correct and complete with respect to an intended meaning if the two meanings coin- cide exactly.
Another important aspect of a logic program is whether it terminates.
Definition A domain is a set of goals, not necessarily ground, closed under the instance relation. That is, if A is in D and A' is an instance of A, then A' is in D as well.
Definition A termination domain of a program P is a domain D such that every computation of P on every goal in D terminates.
Usually, a useful program should have a termination domain that in- cludes its intended meaning. However, since the computation model of logic programs is liberal in the order in which goals in the resolvent can be reduced, most interesting logic programs will not have interesting ter- mination domains. This situation will improve when we switch to Prolog. The restrictive model of Prolog allows the programmer to compose non- trivial programs that terminate over useful domains.
Consider Program 3.1 defining the natural numbers. This program is terminating over its Herbrand base. However, the program is nonter- minating over the domain {natural_number(X)}. This is caused by the possibility of the nonterminating computation depicted in the trace in Figure 5.1.
For any logic program, it is useful to find domains over which it is terminating. This is usually difficult for recursive logic programs. We
107 Theory of Logic Programs
natural_riuinber(X) Xs(X1) natural_nuinber(X1) X1s(X2)
natural_number(X2) X2s(X3)
Figure 5.1 A nonterminating computation
need to describe recursive data types in a way that allows us to discuss termination.
Recall that a type, introduced in Chapter 3, is a set of terms.
Definition A type is complete if the set is closed under the instance relation. With every complete type T we can associate an incomplete type IT, which is the set of terms that have instances in T and instances not in T. .
We illustrate the use of these definitions to find termination domains for the recursive programs using recursive data types in Chapter 3. Spe- cific instances of the definitions of complete and incomplete types are given for natural numbers and lists. A (complete) natural number is ei- ther the constant O, or a term of the form s(X). An incomplete natural number is either a variable, X, or a term of the form s(0), where X is a variable. Program 3.2 for is terminating for the domain consisting of goals where the first and/or second argument is a complete natural number.
Definition A list is complete if every instance satisfies the definition given in Pro- gram 3.11. A list is incomplete if there are instances that satisfy this definition and instances that do not. .
For example, the list [a,b,c] is complete (proved in Figure 3.3), while the variable X is incomplete. Two more interesting examples: [a, X, c] is a complete list, although not ground, whereas [a,bXs] is incomplete.
A termination domain for append is the set of goals where the first and/or the third argument is a complete list. We discuss domains for other list-processing programs in Section 7.2, on termination of Prolog programs.
108 Chapter 5
5.2.1 Exercises for Section 5.2
Give a domain over which Program 3.3 for plus is terminating.
Define complete and incomplete binary trees by analogy with the definitions for complete and incomplete lists.
5.3 Complexity
We have analyzed informally the complexity of several logic programs, for example, and plus (Programs 3.2 and 3.3) in the section on arith- metic, and append and the two versions of reverse in the section on lists (Programs 3.15 and 3.16). In this section, we briefly describe more formal complexity measures.
The multiple uses of logic programs slightly change the nature of com- plexity measures. Instead of looking at a particular use and specifying complexity in terms of the sizes of the inputs, we look at goals in the meaning and see how they were derived. A natural measure of the com- plexity of a logic program is the length of the proofs it generates for goals in its meaning.
Definition The size of a term is the number of symbols in its textual representation.
. Constants and variables, consisting of a single symbol, have size 1.
The size of a compound term is i more than the sum of the sizes of its arguments. For example, the list [b] has size 3, [a,b] has size 5, and the goal append([a,b] , [c,d] ,Xs) has size 12. In general, a list of n elements has size 2 . n + 1.
Definition A program P is of length complexity L(n) if for any goal G in the meaning of P of size n there is a proof of G with respect to P of length less than equal to L(n).
Length complexity is related to the usual complexity measures in com- puter science. For sequential realizations of the computation model, it corresponds to time complexity. Program 3.15 for append has linear
109 Theory of Logic Programs
length complexity. This is demonstrated in Exercise (i) at the end of this section.
The applicability of this measure to Prolog programs, as opposed to logic programs, depends on using a unification algorithm without an oc- curs check. Consider the runtime of the straightforward program for ap- pending two lists. Appending two lists, as shown in Figure 4.3, involves several unifications of append goals with the head of the append rule append([XXs] ,Ys, [XIZs]). At least three unifications, matching vari- ables against (possibly incomplete) lists, will be necessary. If the occurs check must be performed for each, the argument lists must be searched. This is directly proportional to the size of the input goal. However, if the occurs check is omitted, the unification time will be bounded by a con- stant. The overall complexity of append becomes quadratic in the size of the input lists with the occurs check, but only linear without it.
We introduce other useful measures related to proofs. Let R be a proof. We define the depth of R to be the deepest invocation of a goal in the associated reduction. The goal-size of R is the maximum size of any goal reduced.
Definition A logic program P is of goal-size complexity G(n) if for any goal A in the meaning of P of size n, there is a proof of A with respect to P of goal-size less than or equal to G(n). .
Definition A logic program P is of depth-complexity D(n) if for any goal A in the mearnng of P of size n, there is a proof of G with respect to P of depth D(n).
Goal-size complexity relates to space. Depth-complexity relates to space of what needs to be remembered for sequential realizations, and to space and time complexity for parallel realizations.
5.3.1 Exercises for Section 5.3
(i) Show that the size of a goal in the meaning of append joining a list of length n to one of length ni to give a list of length n + m is 4 . n + 4 m + 4. Show that a proof tree has rn + i nodes. Hence
110 Chapter 5
show that append has linear complexity. Would the complexity be altered if the type condition were added?
Show that Program 3.3 for plus has linear complexity.
Discuss the complexity of other logic programs.
5.4 Search Trees
Computations of logic programs given so far resolve the issue of nonde- terminism by always making the correct choice. For example, the com- plexity measures, based on proof trees, assume that the correct clause can be chosen from the program to effect the reduction. Another way of computationally modeling nondetermrnism is by developing all possible reductions in parallel. In this section, we discuss search trees, a formal- ism for considering all possible computation paths.
Definition A search tree of a goal G with respect to a program P is defined as follows. The root of the tree is G. Nodes of the tree are (possibly con- junctive) goals with one goal selected. There is an edge leading from a node N for each clause in the program whose head unifies with the se- lected goal. Each branch in the tree from the root is a computation of G by P. Leaves of the tree are success nodes, where the empty goal has been reached, or failure nodes, where the selected goal at the node cannot be further reduced. Success nodes correspond to solutions of the root of the tree. u
There are in general many search trees for a given goal with re- spect to a program. Figure 5.2 shows two search trees for the query son(S,haran)? with respect to Program 1.2. The two possibilities cor- respond to the two choices of goal to reduce from the resolvent f a- ther(haran,S) ,male(S). The trees are quite distinct, but both have a single success branch corresponding to the solution of the query S=lot. The respective success branches are given as traces in Figure 4.4.
We adopt some conventions when drawing search trees. The leftmost goal of a node is always the selected one. This implies that the goals in derived goals may be permuted so that the new goal to be selected for
111 Theory of Logic Programs
Figure 5.2 Two search trees
reduction is the first goal. The edges are labeled with substitutions that are applied to the variables in the leftmost goal. These substitutions are computed as part of the unification algorithm.
Search trees correspond closely to traces for deterministic computa- tions. The traces for the append query and hanoi query given, respec- tively, in Figures 4.3 and 4.5 can be easily made into search trees. This is Exercise (i) at the end of this section.
Search trees contain multiple success nodes if the query has mul- tiple solutions. Figure 5.3 contains the search tree for the query ap- pend(As , Bs, [a, b, ci)? with respect to Program 3.15 for append, asking to split the list [a, b, ci into two. The solutions for As and Bs are found by collecting the labels of the edges in the branch leading to the success node. For example, in the figure, following the leftmost branch gives the solution {As=[a,b,c] ,Bs=[ J).
The number of success nodes is the same for any search tree of a given goal with respect to a program.
Search trees can have infinite branches, which correspond to nonter- minating computations. Consider the goal append(Xs, [c,di ,Ys) with respect to the standard program for append. The search tree is given in Figure 5.4. The infinite branch is the nonterminating computation given in Figure 4.6.
112 Chapter 5
Figure 5.3 Search tree with multiple success nodes
Complexity measures can also be defined in terms of search trees. Pro- log programs perform a depth-first traversal of the search tree. There- fore, measures based on the size of the search tree will be a more real- istic measure of the complexity of Prolog programs than those based on the complexity of the proof tree. However, the complexity of the search tree is much harder to analyze.
There is a deeper point lurking. The relation between proof trees and search trees is the relation between nondeterministic computations and deterministic computations. Whether the complexity classes defined via proof trees are equivalent to complexity classes defined via search trees is a reformulation of the classic P=NP question in terms of logic program- ming.
5.4.1 Exercises for Section 5.4
Transform the traces of Figure 4.3 and 4.5 into search trees.
Draw a search tree for the query sort([2,4, 1] ,Xs)? using permu- tation sort.
],Bsa,c]}
{Asl =[ ],Bs=[b,c]}
C {As2=[ },Bs=lcJ}
113 Theory of Logic Programs
5.5 Negation in Logic Programming
Logic programs are collections of rules and facts describing what is true. Untrue facts are not expressed explicitly; they are omitted. When writing rules, it is often natural to include negative conditions. For example, defining a bachelor as an unmarried male could be written as
bachelor(X) - male(X), not married(X).
if negation were allowed. In this section, we describe an extension to the logic programming computation model that allows a limited form of negation.
Researchers have investigated other extensions to logic programming to allow disjunction, and indeed, arbitrary first-order formulae. Dis- cussing them is beyond the scope of this book. The most useful of the extensions is definitely negation.
We define a relation not G and give a semantics. The essence of logic progranmiing is that there is an efficient procedural semantics. There is a natural way to adapt the procedural semantics to negation, namely by negation as failure. A goal G fails, (not G succeeds), if G cannot be derived by the procedural semantics.
{Xs=[X1 jXsl ],Ys=[X1 IYsl]}
aspen
{Xsl =[X2IXs2],Ysl =[X2IYs2J}
aspen
lXs2=[X3IXs3J,Ys2=[X3IYs3]}
Figure 5.4 Search tree with an infinite branch
4
. 4
s
114 Chapter 5
The relation not G is only a partial form of negation from first-order logic. The relation not uses the negation as failure rule. A goal not G will be assumed to be a consequence of a program P if G is not a consequence of P.
Negation as failure can be characterized in terms of search trees.
Definition A search tree of a goal G with respect to a program P is finitely failed if it has no success nodes or infinite branches. The finite failure set of a logic program P is the set of goals G such that G has a finitely failed search tree with respect to P.
A goal not G is implied by a program P by the "negation as failure" rule if G is in the finite failure set of P.
Let us see a simple example. Consider the program consisting of two facts:
likes (abrahain,poinegranates).
likes(isaac,pomegranates).
The goal not likes (sarah, pomegranates) follows from the program by negation as failure. The search tree for the goal likes (sarah, pomegran- ates) has a single failure node.
Using negation as failure allows easy definition of many relations. For example, a declarative definition of the relation disjoint(Xs,Ys) that two lists, Xs and Ys, have no elements in common is possible as follows.
disjoint(Xs,Ys) - not (member(X,Xs), member(X,Ys)).
This reads: "Xs is disjoint from Ys if there is no element X that is a member of both Xs and Ys."
An intuitive understanding of negation as failure is fine for the pro- grams in this book using negation. There are semantic problems, how- ever, especially when integrated with other issues such as completeness and termination. Pointers to the literature are given in Section 5.6, and Prolog's implementation of negation as failure is discussed in Chap- ter 11.
115 Theory of Logic Programs
5.6 Background
The classic paper on the semantics of logic programs is of van Emden and Kowaiski (1976). Important extensions were given by Apt and van Emden (1982). In particular, they showed that the choice of goal to re- duce from the resolvent is arbitrary by showing that the number of suc- cess nodes is an invariant for the search trees. Textbook accounts of the theory of logic programming discussing the equivalence between the declarative and procedural semantics can be found in Apt (1990), Deville (1990), and Lloyd (1987).
In Shapiro (1984), complexity measures for logic programs are com- pared with the complexity of computations of alternating Turing ma- chines. It is shown that goal-size is linearly related to alternating space, the product of length and goal-size is linearly related to alternating tree- size, and the product of depth and goal-size is linearly related to alter- nating time.
The classic name for search trees in the literature is SLD trees. The name SLD was coined by research in automatic theorem proving, which preceded the birth of logic programming. SLD resolution is a particu- lar refinement of the resolution principle introduced in Robinson (1965). Computations of logic programs can be interpreted as a series of reso- lution steps, and in fact, SLD resolution steps, and are still commonly described thus in the literature. The acronym SLD stands for Selecting a literal, using a Linear strategy, restricted to Definite clauses.
The first proof of the correctness and completeness of SLD resolution, albeit under the name LUSH-resolution, was given by Hill (1974).
The subject of negation has received a large amount of attention and interest since the inception of logic programming. The fundamental work on the semantics of negation as failure is by Clark (1978). Clark's results, establishing soundness, were extended by Jaffar et al. (1983), who proved the completeness of the rule.
The concept of negation as failure is a restricted version of the closed world assumption as discussed in the database world. For more infor- mation see Reiter (1978). There has been extensive research on charac- terizing negation in logic programming that has not stabilized at this time. The reader should look up the latest logic programming conference proceedings to find current thinking. A good place to start reading to un- derstand the issue is Kunen (1989).
Leonardo Da Vinci. Portrait of the Florentine poet Bernardo Bellincioni, en- gaged at the Court of Ludovico Sforza. Woodcut, based on a drawing by Leonardo. From Bellincioni's Rime. Milan 1493.
H The Prolog Language
In order to implement a practical programming language based on the computation model of logic programming, three issues need attention. The first concerns resolving the choices remaining in the abstract inter- preter for logic programs, defined in Chapter 4. The second concerns enhancing the expressiveness of the pure computation model of logic programs by adding meta-logical and extra-logical facilities. Finally, ac- cess to some of the capabilities of the underlying computer, such as fast arithmetic and input/output, must be provided. This part discusses how Prolog, the most developed language based on logic programming, han- dles each of these issues.
6 Pure Prolog
A pure Prolog program is a logic program, in which an order is defined both for clauses in the program and for goals in the body of the clause. The abstract interpreter for logic programs is specialized to take advan- tage of this ordering information. This chapter discusses the execution model of Prolog programs in contrast to logic programs, and compares Prolog to more conventional languages.
The relation between logic programming and Prolog is reminiscent of the relation between the lambda-calculus and Lisp. Both are concrete re- alizations of abstract computation models. Logic programs that execute with Prolog's execution mechanism are referred to as pure Prolog. Pure Prolog is an approximate realization of the logic programming compu- tation model on a sequential machine. lt is certainly not the only possi- ble such realization. However, it is a realization with excellent practical choices, which balance preserving the properties of the abstract model with catering for efficient implementation.
6.1 The Execution Model of Prolog
Two major decisions must be taken to convert the abstract interpreter for logic programs into a form suitable for a concrete programming lan- guage. First, the arbitrary choice of which goal in the resolvent to reduce, namely, the scheduling policy, must be specified. Second, the nondeter- ministic choice of the clause from the program to effect the reduction must be implemented.
120 Chapter 6
Several logic programming languages exist, reflecting different choices. Prolog and its extensions (Prolog-II, IC-Prolog, and MU-Prolog, for exam- ple) are based on sequential execution. Other languages, such as PAR- LOG, Concurrent Prolog, GHC, Aurora-Prolog, and Andorra-Prolog, are based on parallel execution. The treatment of nondeternunism distin- guishes between sequential and parallel languages. The distinction be- tween Prolog and its extensions is in the choice of goal to reduce.
Prolog's execution mechanism is obtained from the abstract interpreter by choosing the leftmost goal instead of an arbitrary one and replacing the non- deterministic choice of a clause by sequential search for a unifiable clause and backtracking.
In other words, Prolog adopts a stack scheduling policy. It maintains the resolvent as a stack: pops the top goal for reduction, and pushes the derived goals onto the resolvent stack.
In addition to the stack policy, Prolog simulates the nondeterministic choice of reducing clause by sequential search and backtracking. When attempting to reduce a goal, the first clause whose head unifies with the goal is chosen. If no unifiable clause is found for the popped goal, the computation is unwound to the last choice made, and the next unifiable clause is chosen.
A computation of a goal G with respect to a Prolog program P is the generation of all solutions of G with respect to P. In terms of logic programming concepts, a Prolog computation of a goal G is a complete depth-first traversal of the particular search tree of G obtained by always choosing the leftmost goal.
Many different Prolog implementations exist with differing syntax and programming facilities. Recently, there has been an attempt to reach a Prolog standard based on the Edinburgh dialect of Prolog. At the time of writing, the standard has not been finalized. However a complete draft exists, which we essentially follow. We refer to the Prolog described in that document as Standard Prolog. The syntax of logic programs that we have been using fits within Standard Prolog except that we use some characters not available on a standard keyboard. We give the standard equivalent of our special characters. Thus :- should be used instead of - in Prolog programs to separate the head of a clause from its body. All the programs in this book run (possibly with minor changes) in all Edinburgh-compatible Prologs.
A trace of a Prolog computation is an extension of the trace of a com- putation of a logic program under the abstract interpreter as described
121 Pure Prolog
father(abraharn,isaac). male(isaac). father(haran,lot) male(lot). father (haran,milcah) female (yiscah) father(haran,yiscah). female (milcah). son(X,Y) father(Y,X), male(X). daughter(X,Y) f ather(Y,X), f emale(X).
son (X, haran)? f ather (heran , X) X=lot male ( lot)
true Output: X=lot
father (haran , X) male (milcah) f father (heran, X) male(yiscah) f
no (more) solutions
Xmil cah
X=yi scah
Figure 6.1 Tracing a simple Prolog computation
in Section 4.2. We revise the computations of Chapters 4 and 5, indicat- ing the similarities and differences. Consider the query son(X,haran)? with respect to Program 1.2, biblical family relationships, repeated at the top of Figure 6.1. The computation is given in the bulk of Figure 6.1. lt corresponds to a depth-first traversal of the first of the search trees in Figure 5.2. It is an extension of the first trace in Figure 4.4, since the whole search tree is searched.
The notation previously used for traces must be extended to handle failure and backtracking. An f after a goal denotes that a goal fails, that is there is no clause whose head unifies with the goal. The next goal af- ter a failed goal is where the computation continues on backtracking. It already appears as a previous goal in the trace at the same depth of indentation and can be identified by the variable names. We adopt the Edinburgh Prolog convention that a ";" typed after a solution denotes a continuation of the computation to search for more solutions. Unifica- tions are indicated as previously.
Trace facilities and answers provided by particular Prolog implementa- tions vary from our description. For example, some Prolog implementa- tions always give all solutions, while others wait for a user response after each solution.
122 Chapter 6
append([XIXs] ,Ys, [XIZs]) append(Xs,Ys,Zs).
append([ I ,Ys,Ys).
append(Xs,Ys, [a,b,c])
append(Xsl,Ys, [b,c])
append(Xs2,Ys, [cl)
append(Xs3,Ys, E I) true
Output: (Xs=[a,b,c],Ys=[ ])
append(Xs2,Ys, [cl)
true Output: (Xs=[a,b] ,Ys=[c]
append(Xsl,Ys, [b,c])
true Output: (Xs= [a] , Ys= [b, c])
append(Xs,Ys, [a,b,c])
true Output: (Xs=[ ],Ys=[a,b,c])
no (more) solutions
Figure 6.2 Multiple solutions for splitting a list
Xs=[aIXsl]
Xsl= [bi Xs2]
Xs2= [cl Xs3]
Xs3=[ ],Ys=[ I
Xs2=[ ] ,Ys[c]
Xsl=[ ],Ys=[b,c]
Xs=[ ],Ys=[a,b,c]
The trace of append([a,b], [c,d] ,Ls)? giving the answer Ls=[a,b,c, dl is precisely the trace given in Figure 4.3. Figure 4.5, giving the trace for solving the Towers of Hanoi with three disks, is also a trace of the hanoi program considered as a Prolog program solving the query hanoi(s(s(s(0))) ,a,b,c,Ms)?. The trace of a deterministic computa- tion is the same when considered as a logic program or a Prolog program, provided the order of goals is preserved.
The next example is answering the query append (Xs , Ys, [a, b, c])? with respect to Program 3.15 for append. There are several solutions of the query. The search tree for this goal was given as Figure 5.3. Figure 6.2 gives the Prolog trace.
Tracing computations is a good way to gain understanding of the ex- ecution model of Prolog. We give a slightly larger example, sorting a list with the quicksort program (Program 3.22, reproduced at the top of Figure 6.3). Computations using quicksort are essentially deterministic and show the algorithmic behavior of a Prolog program. Figure 6.3 gives a trace of the query qui cksort ([2, 1,3] , Xs)?. Arithmetic comparisons
123 Pure Prolog
quicksort([XIXS] ,Ys) -
partition(Xs,X,Littles,Bigs),
quicksort(Littles,Ls),
quicksort(Bigs,Bs),
append(Ls, [X lBs] ,Ys).
quïcksort([ ],C ]). partition([XIXs] ,Y, EXILs] ,Bs)
X Y, partition(Xs,Y,Ls,Bs).
partitïon([XIXs] ,Y,Ls, [XIBs])
X > Y, partition(Xs,Y,Ls,Bs).
partition([ ],Y,[ LE J). quicksort([2,1,3] ,qs)
partition([1,3] ,2,Ls,Bs)
12 partition([3] ,2,Lsl,Bs)
32 f partition([3] ,2,Lsl,Bs)
3>2 partition([ I ,2,Lsl,Bsl)
quicksort(E1] ,qsl)
partition([ ] ,1,Ls2,Bs2)
quicksort([ ] ,Qs2)
quicksort([ ] ,Qs3)
append([ I ji] ,Qsi)
quicksort([3] ,Qs4)
partition([ I ,3,Ls3,Bs3)
quicksort([ J ,Qs5)
quicksort([ ] ,Qs6)
appendU ] , [3] ,Qs4) appendUl] ,[2,3] ,Qs)
appendU I ,[2,3] ,Ys) true
Output: (Qsti,2,3])
Figure 6.3 Tracing a quicksort computation
Ls= [1 I Lsi]
Lsi[3 Ls2]
Bs [31 Bsl]
Ls1[ ]Bsl
Ls2=[ ]Bs2
Qs2=[ J
Qs3=[ ]
Qsi=[1]
Ls3=[ ]=Bs3
s5=[ I
Qs6[ ] Qs4 [3]
Qs=EljYs]
Ys[2,3]
124 Chapter 6
are assumed to be unit operations, and the standard program for append is used.
We introduce a distinction between shallow and deep backtracking. Shallow backtracking occurs when the unification of a goal and a clause fails, and an alternative clause is tried. Deep backtracking occurs when the unification of the last clause of a procedure with a goal fails, and control returns to another goal in the computation tree.
It is sometimes convenient to include, for the purpose of this defini- tion, test predicates that occur first in the body of the clause as part of unification, and to classify the backtracking that occurs as a result of their failure as shallow. An example in Figure 6.3 is the choice of a new clause for the goal partition([3] ,2,Lsl,Bs).
6.1.1 Exercises for Section 6.1
Trace the execution of daughter(X,haran)? with respect to Pro- gram 1.2.
Trace the execution of sort([3, 1,2] ,Xs)? with respect to Pro- gram 3.21.
Trace the execution of sort([3, 1,2] ,Xs)? with respect to Pro- gram 3.20.
6.2 Comparison to Conventional Programming Languages
A programming language is characterized by its control and data ma- nipulation mechanisms. Prolog, as a general-purpose progranmiing lan- guage, can be discussed in these terms, as are conventional languages. In this section, we compare the control flow and data manipulation of Prolog to that of Algol-like languages.
The control in Prolog programs is like that in conventional procedural languages as long as the computation progresses forward. Goal invoca- tion corresponds to procedure invocation, and the ordering of goals in the body of clauses corresponds to sequencing of statements. Specifi- cally, the clause A B1,. . .,B can be viewed as the definition of a pro- cedure A as follows:
125 Pure Prolog
procedure A call B1, call B2,
call B, end.
Recursive goal invocation in Prolog is similar in behavior and imple- mentation to that of conventional recursive languages. The differences show when backtracking occurs. In a conventional language, if a compu- tation cannot proceed (e.g., all branches of a case statement are false), a runtime error occurs. In Prolog, the computation is simply undone to the last choice made, and a different computation path is attempted.
The data structures manipulated by logic programs, terms, correspond to general record structures in conventional programming languages. The handling of data structures is very flexible in Prolog. Like Lisp, Prolog is a declaration-free, typeless language.
The major differences between Prolog and conventional languages in the use of data structures arise from the nature of logical variables. Log- ical variables refer to individuals rather than to memory locations. Con- sequently, having once beed specified to refer to a particular individual, a variable cannot be made to refer to another individual. In other words, logic programming does not support destructive assignment where the contents of an initialized variable can change.
Data manipulation in logic programs is achieved entirely via the unifi- cation algorithm. Unification subsumes
Single assignment
Parameter passing Record allocation
Read/write-once field-access in records
We discuss the trace of the quicksort program in Figure 6.3, point- ing out the various uses of unification. The unification of the initial goal quicksort ([2, 1 3] , s) with the head of the procedure definition quicksort ([XI Xs] , Ys) illustrates several features. The unification of [2, 1 3] with the term [X I Xs] achieves record access to the list and also selection of its two fields, the head and tail.
126 Chapter 6
The unification of [1 3] with Xs achieves parameter passing to the partition procedure, because of the sharing of the variables. This gives the first argument of partition. Similarly, the unification of 2 with X passes the value of the second parameter to partition.
Record creation can be seen with the unification of the goal parti- tion( [1,3] .2, Ls ,Bs) with the head of the partition procedure parti- tion([XIXs] ,Z, [XILs1] ,Bsl). As a result, Ls is instantiated to [li Ls 1]. Specifically, Ls is made into a list and its head is assigned the value 1, namely, record creation and field assignment via unifica- tion.
The recursive algorithm embodied by the quicksort program can be easily coded in a conventional programming language using linked lists and pointer manipulation. As discussed, unification is achiev- ing the effect of the necessary pointer manipulations. Indeed, the ma- nipulation of logical variables via unification can be viewed as an abstraction of low-level manipulation of pointers to complex data structures.
These analogies may provide hints on how to implement Prolog effi- ciently on a von Neumann machine. Indeed, the basic idea of compilation of Prolog is to translate special cases of unification to conventional mem- ory manipulation operations, as specified previously.
Conventional languages typically incorporate error-handling or excep- tion-handling mechanisms of various degrees of sophistication. Pure Pro- log does not have an error or exception mechanism built into its defi- nition. The pure Prolog counterparts of nonfatal errors in conventional programs, e.g., a missing case in a case statement, or dividing by zero, cause failure in pure Prolog.
Full Prolog, introduced in the following chapters, includes system predicates, such as arithmetic and I/O, which may cause errors. Current Prolog implementations do not have sophisticated error- handling mechanisms. Typically, on an error condition, a system pred- icate prints an error message and either fails or aborts the computa- tion.
This brief discussion of Prolog's different way of manipulating data does not help with the more interesting question: How does program- ming in Prolog compare with programming in conventional program- ming languages? That is the major underlying topic of the rest of this book.
127 Pure Prolog
6.3 Background
The origins of Prolog are shrouded in mystery. All that is known is that the two founders, Robert Kowaiski, then at Edinburgh, and Alain Colmer- auer at Marseilles worked on similar ideas during the early 1970s, and even worked together one summer The results were the formulation of the logic programming philosophy and computation model by Kowaiski (1974), and the design and implementation of the first logic program- ming language Prolog, by Colmerauer and his colleagues (1973). Three recent articles giving many more details about the beginnings of Prolog and logic programming are Cohen (1988), Kowalski (1988), and Colmer- auer and Roussel (1993).
A major force behind the realization that logic can be the basis of a practical programming language has been the development of efficient implementation techniques, as pioneered by Warren (1977). Warren's compiler identified special cases of unification and translated them into efficient sequences of conventional memory operations. Good accounts of techniques for Prolog implementation, both interpretation and compi- lation, can be found in Maier and Warren (1988) and Ait-Kaci (1991).
Variations of Prolog with extra control features, such as IC-Prolog (Clark and McCabe, 1979), have been developed but have proved too costly in runtirne overhead to be seriously considered as alternatives to Prolog. We will refer to particular interesting variations that have been proposed in the appropriate sections.
Another breed of logic programming languages, which indirectly emerged from IC-Prolog, was concurrent logic languages. The first was the Relational Language (Clark and Gregory, 1981), followed by Concur- rent Prolog (Shapiro, 1983b), PARLOG (Clark and Gregory, 1984), GHC (Ueda, 1985), and a few other proposals.
References for the variations mentioned in the text are, for Prolog- Il (van Caneghem, 1982), IC-Prolog (Clark et al., 1982), and MU-Prolog (Naish, 1986). Aurora-Prolog is described in Disz et al. (1987), while a starting place for reading about AKL, a language emerging from Andorra- Prolog is Janson and Haridi (1991).
The syntax of Prolog stems from the clausal form of logic due to Kowaiski (1974). The original Marseilles interpreter used the terminol- ogy of positive and negative literais from resolution theory. The clause AB1,...,Bwaswritten +AB1.. . .
128 Chapter 6
David H. D. Warren adapted Marseilles Prolog for the DEC-10 at the Uni- versity of Edinburgh, with help from Fernando Pereira. Their decisions have been very influential. Many systems adopted most of the conven- tions of Prolog-10 (Warren et al., 1979), which has become known more generically as Edinburgh Prolog. Its essential features are described in the widespread primer on Prolog (Clocksin and Mellish, 1984). This book follows the description of Standard Prolog existing as Scowen (1991).
A paper by Cohen (1985) delves further into the relation between Pro- log and conventional languages.
7 Prograniming in Pure Prolog
7.1 Rule Order
Two syntactic issues, irrelevant for logic programs, are important to con- sider when composing Prolog programs The rule order, or clause order, of clauses in each procedure must be decided. Also the goal order of goals in the bodies of each clause must be determined The consequences of these decisions can be immense. There can be orders of magnitude of difference in efficiency in the performance of Prolog programs. In ex- treme though quite common cases, correct logic programs will fail to give solutions because of nontermination.
A major aim of logic programming is to enable the programmer to pro- gram at a higher level. Ideally one should write axioms that define the desired relations, maintaining ignorance of the way they are going to be used by the execution mechanism. Current logic programming lan- guages, Prolog in particular, are still far away from allowing this ideal of declarative programming The specific, well-defined choices of how their execution mechanisms approximate the abstract interpreter cannot be ig- nored. Effective logic programming requires knowing and utilizing these choices.
This chapter discusses the consequences of Prolog's execution model for the logic programmer New aspects of the programming task are introduced. Not only must programmers come up with a correct and complete axiomatization of a relation but they must also consider its execution according to the model.
130 Chapter 7
parent(terach,abraham). parent(abraham,isaac). parent(isaac,jacob). parent(jacob,benjaxnin). ancestor(X,Y) - parent(X,Y). ancestor(X,Z) - parent(X,Y), ancestor(Y,Z).
Program 7.1 Yet another family example
The rule order determines the order in which solutions are found.
Changing the order of rules in a procedure permutes the branches in any search tree for a goal using that procedure. The search tree is traversed depth-first. So permuting the branches causes a different order of traversal of the search tree, and a different order of finding solutions. The effect is clearly seen when using facts to answer an existential query. With our biblical database and a query such as father (X, Y)?, changing the order of facts will change the order of solutions found by Prolog. Deciding how to order facts is not very important.
The order of solutions of queries solved by recursive programs is also determined by the clause order. Consider Program 5.1, a simple bibli- cal database together with a program for the relationship ancestor, re- peated here as Program 7.1.
For the query ancestor (terach,X)? with respect to Program 7.1, the solutions will be given in the order, X=abraham, X=isaac, X=j acob, and X=benjamin. If the rules defining ancestor are swapped, the solutions will appear in a different order, namely, X=benj amin, X=j acob, X=isaac, and X=abraham.
The different order of ancestor clauses changes the order of searching the implicit family tree. In one order, Prolog outputs solutions as it goes along. With the other order, Prolog travels to the end of the family tree and gives solutions on the way back. The desired order of solutions is determined by the application, and the rule order of ancestor is chosen accordingly.
Changing the order of clauses for the member predicate (Program 3.12) also changes the order of search. As written, the program searches the list until the desired element is found. If the order of the clauses is reversed, the program always searches to the end of the list. The order of solutions will also be affected, for example, responding to the query member(X, [1,2,3])?. In the standard order, the order of solutions is
131 Programming in Pure Prolog
intuìtive X=1, X=2, X=3. With the rules swapped, the order is X=3, X=2, X=1. The order of Program 3.12 is more intuitive and hence preferable.
When the search tree for a given goal has an infinite branch, the or- der of clauses can determine if any solutions are given at all. Consider the query append(Xs, [c,d] ,Ys) with respect to append. As can be seen from the search tree in Figure 5.4, no solutions would be given. If, how- ever, the append fact appeared before the append rule, an infinite number of pairs Xs , Ys satisfying the query would be given.
There is no consensus as to how to order the clauses of a Prolog pro- cedure. Clearly, the standard dictated in more conventional languages, of testing for the termination condition before proceeding with the iter- ation or recursion is not mandatory in Prolog. This is demonstrated in Program 3.15 for append as well as in other programs in this book. The reason is that the recursive or iterative clause tests its applicability by unification. This test is done explicitly and independently of the other clauses in the procedure.
Clause order is more important for general Prolog programs than it is for pure Prolog programs. Other control features, notably the cut to be discussed in Chapter 11, depend significantly on the clause order. When such constructs are used, clauses lose their independence and modularity, and clause order becomes significant.
In this chapter, for the most part, the convention that the recursive clauses precede the base clauses is adopted.
7.1.1 Exercises for Section 7.1
Verify the order of solutions for the query ancestor(abraham,X)? with respect to Program 7.1, and its variant with different rule order for ancestor, claimed in the text.
What is the order of solutions for the query ancestor(X,benja- miri)? with respect to Program 7.1? What if the rule order for ancestor were swapped?
7.2 Termination
Prolog's depth-first traversal of search trees has a serious problem. If the search tree of a goal with respect to a program contains an infinite
132 Chapter 7
branch, the computation will not terminate. Prolog may fail to find a solution to a goal, even though the goal has a finite computation.
Nontermination arises with recursive rules. Consider adding a relation- ship marri ed (Male ,Female) to our database of family relationships. A sample fact from the biblical situation is married(abraham,sarah). A user querying the married relationship should not care whether males or females are first, as the relationship is commutative. The "obvious" way of overcoming the commutativity is adding a recursive rule mar- ried(X,Y) - married(Y,X). If this is added to the program, no com- putation involving married would ever terminate. For example, the trace of the query married(abraham, sarah)? is given in Figure 7.1.
Recursive rules that have the recursive goal as the first goal in the body are known as left recursive rules. The problematic married axiom is an example. Left recursive rules are inherently troublesome in Prolog. They cause nonterminating computations if called with inappropriate arguments.
The best solution to the problem of left recursion is avoidance. The married relationship used a left recursive rule to express comrnutativìty. Commutative relationships are best handled differently, by defining a new predicate that has a clause for each permutation of the arguments of the relationship. For the relationship married, a new predicate, are_ married (Personi , Person2), say, would be defined using two rules:
are_married(X,Y) - married(X,Y).
are_married(X,Y) - married(Y,X).
Unfortunately, it is not generally possible to remove all occurrences of left recursion. All the elegant minimal recursive logic programs shown in Chapter 3 are left recursive, and can cause nontermination. However,
married(X,Y) - married(Y,X). married(abrahaxn, sarah).
married(abrahain, sarah) married(sarah, abraham)
married(abraham, sarah) married(sarah, abraham)
Figure 7.1 A nonterminating computation
133 Programming in Pure Prolog
the appropriate analysis, using the concepts of domains and complete structures introduced in Section 5.2, can determine which queries will terminate with respect to recursive programs.
Let us consider an example, Program 3.15 for appending two lists. The program for append is everywhere terminating for the set of goals whose first and/or last argument is a complete list. Any append query whose first argument is a complete list will terminate. Similarly, all queries where the third argument is a complete list will terminate. The program will also terminate if the first and/or third argument is a ground term that is not a list. The behavior of append is best summed up by consid- ering the queries that do not terminate, namely, when both the first and third arguments are incomplete lists that are unifiable.
The condition for when a query to Program 3.12 for member terminates is also stated in terms of incomplete lists. A query does not terminate if the second argument is an incomplete list. If the second argument of a query to member is a complete list, the query terminates.
Another guaranteed means of generating nonterminating computa- tions, easy to overlook, is circular definitions. Consider the pair of rules
parent(X,Y) - child(Y,X).
child(X,Y) - parent(Y,X).
Any computation involving parent or child, for example, parent (haran,lot)?, will not terminate. The search tree necessarily contains an infinite branch, because of the circularity.
7.2.1 Exercises for Section 7.2
Discuss the termination behavior of both programs in Program 3.13 determining prefixes and suffixes of lists.
Discuss the termination of Program 3.14c for sublist.
7.3 Goal Order
Goal order is more significant than clause order. lt is the principal means of specifying sequential flow of control in Prolog programs. The pro- grams for sorting lists, e.g., Program 3.22 for quicksort, exploit goal order to indicate the sequence of steps in the sorting algorithms.
134 Chapter 7
We first discuss goal order from the perspective of database program- ming. The order of goals can affect the order of solutions. Consider the query daughter(X,haran)? with respect to a variant of Program 1.2, where the order of the facts f emale(milcah) and f emale(yiscah) is interchanged. The two solutions are given in the order X=milcah, X=yiscah. If the goal order of the daughter rule were changed to be daughter(X,Y) - f emale(X) ,father(Y,X) ., the order of the solutions to the query, given the same database, would be X=yiscah, X=milcah.
The reason that the order of goals in the body of a clause affects the order of solutions to a query is different from the reason that the order of rules in a procedure affects the solution order. Changing rule order does not change the search tree that must be traversed for a given query. The tree is just traversed in a different order. Changing goal order changes the search tree.
Goal order determines the search tree.
Goal order affects the amount of searching the program does in solv- ing a query by deternnning which search tree is traversed. Consider the two search trees for the query son(X,haran)?, given in Figure 5.2. They represent two different ways of finding a solution. In the first case, solu- tions are found by searching for children of haran and checking if they are male. The second case corresponds to the rule for son being written with the order of the goals in its body swapped, namely, son(X,Y) - male (X), parent(Y,X). Now the query is solved by searching through all the males in the program and checking if they are children of ha- ran. If there were many male facts in the program, more search would be involved. For other queries, for example, son(sarah,X)?, the reverse order has advantages. Since sarah is not male, the query would fail more quickly.
The optimal goal order of Prolog programs varies with different uses. Consider the definition of grandparent. There are two possible rules:
grandparent(X,Z) - parent(X,Y), parent(Y,Z).
grandparent(X,Z) - parent(Y,Z), parent(X,Y).
If you wish to find someone's grandson with the grandfather relation- ship with a query such as grandparent (abraham, X)?, the first of the rules searches more directly. If looking for someone's grandparent with
135 Programming in Pure Prolog
a query such as grandparent (X,isaac)?, the second rule finds the solu- tion more directly. If efficiency is important, then it is advisable to have two distinct relationships, grandparent and grandchild, to be used ap- propnately at the user's discretion.
In contrast to rule order, goal order can determine whether computa- tions terminate. Consider the recursive rule for ancestor:
a.ncestor(X,Y) - parent(X,Z), ancestor(Z,Y).
If the goals in the body are swapped, the ancestor program becomes left recursive, and all Prolog computations with ancestor are nontermi- nating.
The goal order is also important in the recursive clause of the quicksort algorithm in Program 322:
quicksort([XIXs] ,Ys) -
partition(Xs,X,Littles,Bigs),
quicksort(Littles,Ls),
quicksort(Bigs,Bs),
append(Ls, [X lBs] ,Ys).
The list should be partitioned into its two smaller pieces before recur- sively sorting the pieces. If, for example, the order of the partition goal and the recursive sorting goal is swapped, no computations terminate.
We next consider Program 3.16a for reversing a list:
reverse([ ],[ ]).
reverse([XIXs] ,Zs) - reverse(Xs,Ys), append(Ys, [X],Zs).
The goal order ìs significant. As written, the program terminates with goals where the first argument is a complete list. Goals where the first argument is an incomplete list give nonterminating computations. If the goals in the recursive rule are swapped, the determining factor of the ter- mination of reverse goals is the second argument. Calls to reverse with the second argument a complete list terminate. They do not terminate if the second argument is an incomplete list.
A subtler example comes from the definition of the predicate sub- list in terms of two append goals, specifying the sublist as a suf- fix of a prefix, as given in Program 3.14e. Consider the query sub- list ([2,3] , [1 2,3,4])? with respect to the program. The query is reduced to append(AsXs,Bs, [1,2,3,4]), append(As, [2,3] ,AsXs)?.
136 Chapter 7
This has a finite search tree, and the initial query succeeds. If Pro- gram 3.14e had its goals reversed, the initial query would be reduced to append(As,[2,31,AsXs),append(AsXs,Bs,[1,2,3,4])?. This leads to a nonterminating computation because of the first goal, as illustrated in Figure 5.4.
A useful heuristic for goal order can be given for recursive programs with tests such as arithmetic comparisons, or determining whether two constants are different. The heuristic is to place the tests as early as possible. An example comes in the program for partition, which is part of Program 3.22. The first recursive rule is
partition([XIXs] ,Y, EXILs] ,Bs) - X Y, partition(Xs,Y,Ls,Bs).
The test X < Y should go before the recursive call. This leads to a smaller search tree.
In Prolog progranmiing (in contrast, perhaps, to life in general) our goal is to fail as quickly as possible. Failing early prunes the search tree and brings us to the right solution sooner.
7.3.1 Exercises for Section 7.3
Consider the goal order for Program 3.14e defining a sublist of a list as a suffix of a prefix. Why is the order of the append goals in Program 3.14e preferable? (Hint: Consider the query sub- list(Xs, [a,b,c])?.) Discuss the clause order, goal order, and termination behavior for substitute, posed as Exercise 3.3(i).
7.4 Redundant Solutions
An important issue when composing Prolog programs, irrelevant for logic programs, is the redundancy of solutions to queries. The mean- ing of a logic program is the set of ground goals deducible from it. No distinction is made between whether a goal in the meaning could be deduced uniquely from the program, or whether it could be deduced in several distinct ways. This distinction is important for Prolog when considering the efficiency of searching for solutions. Each possible de-
137 Programming in Pure Prolog
duction means an extra branch in the search tree. The bigger the search tree, the longer a computation will take. It is desirable in general to keep the size of the search tree as small as possible.
Having a redundant program may cause, in an extreme case, exponen- tial increase in runtime, in the event of backtracking. If a conjunction of n goals is solved, and each goal has one redundant solution, then in the event of backtracking, the conjunction may generate 2" solutions, thus possibly changing a polynomial-time program (or even a linear one) to be exponential.
One way for redundancy to occur in Prolog programs is by covering the same case with several rules. Consider the following two clauses defining the relation minimum.
minimum(X,Y,X) - X < Y.
minimum(X,Y,Y) - Y X.
The query minimuni(2,2,M)? with respect to these two clauses has a unique solution M=2, which is given twice; one is redundant.
Careful specification of the cases can avoid the problem. The second clause can be changed to
minimuni(X,Y,Y) - Y < X.
Now only the first rule covers the case when the two numbers have equal values.
Similar care is necessary with the definition of partition as part of Program 3.22 for quicksort. The programmer must ensure that only one of the recursive clauses for partition covers the case when the number being compared is the same as the number being used to split the list.
Another way redundancy appears in programs is by having too many special cases. Some of these can be motivated by efficiency. An extra fact can be added to Program 3.15 for append, namely, append(Xs, E J ,Xs), to save recursive computations when the second argument is an empty list. In order to remove redundancy, each of the other clauses for append would have to cover only lists with at least one element as their second argument.
We illustrate these points when composing Program 7.2 for the relation merge (Xs , Ys , Ze), which is true if Xs and Ys are lists of integers sorted in ascending order and Zs is the ordered list resulting from merging them.
138 Chapter 7
merge(Xs,Ys,Zs) - Zs is an ordered list of integers obtained from merging the ordered lists of integers Xs and Ys.
merge([XIXs],[YIYs],[XIZs]) -
X < Y, merge(Xs,[YIYs],Zs).
merge([XIXs] ,[YIYs] , [X,XIZs]) -
X Y, merge(Xs,Ys,Zs).
merge([XIXs],[YIYs],[YIZs]) -
X > Y, merge([XIXs],Ys,Zs).
merge([ I, [XIXs] , [XIXs]). merge(Xs, E I ,Xs).
Program 7.2 Merging ordered lists
There are three separate recursive clauses. They cover the three pos- sible cases: when the head of the first list is less than, equal to, or greater than the head of the second list. We discuss the predicates <, =:=, and> in Chapter 8. Two cases are needed when the elements iii ei- ther list have been exhausted. Note that we have been careful that the goal merge ( [ I , [ I , [ ]) is covered by only one fact, the bottom one.
Redundant computations occur when using member to find whether a particular element occurs in a particular list, and there are multiple occurrences of the particular element being checked for in the list. For example, the search tree for the query member (a, [a, b, a, c]) would have two success nodes.
The redundancy of previous programs was removed by a careful con- sideration of the logic. In this case, the member program is correct. If we want a different behavior, the solution is to compose a modified version of member.
Program 7.3 defines the relation member_check(X,Xs) which checks whether an element X is a member of a list Xs. The program is a vari- ant of Program 3.12 for member that adds a test to the recursive clause. It has the same meaning but, as a Prolog program, it behaves differ- ently. Figure 7.2 shows the difference between the search trees for the identical query to the two programs. The left tree is for the goal mem- ber(a, [a,b,a,c]) with respect to Program 3.12. Note there are two suc- cess nodes. The right tree is for the goal member_check(a, [a,b,a,c]) with respect to Program 7.3. It has only one success node.
139 Programming in Pure Prolog
member_check (X,Xs) X is a member of the list Xs.
rnember_check(X, [XIXs]).
member_check(X,[YIYs]) - X Y, member_check(X,Ys).
Program 7.3 Checking for list membership
Figure 7.2 Variant search trees
We restrict use of Program 7.3 to queries where both arguments are ground. This is because of the way is implemented in Prolog, discussed in Section 11.3.
7.5 Recursive Programming in Pure Prolog
Lists are a very useful data structure for many applications written in Prolog. In this section, we revise several logic programs of Sections 3.2 and 3.3 concerned with list processing. The chosen clause and goal or- ders are explained, and their termination behavior presented. The section also discusses some new examples. Their properties are analyzed, and a reconstruction offered of how they are composed.
member check(aja,b,a,c
aa, member_check(a,[b,a,c
140 Chapter 7
select_first (X,Xs, Ys) Ys is the list obtained by removing the first occurrence of X from the list Xs.
selectfirst(X, [XIXs] ,Xs)
select_first(X, [YIYs] , [YIZs])
X Y, select_first(X,Ys,Zs).
Program 7.4 Selecting the first occurrence of an element from a list
Programs 3.12 and 3.15 for member and append, respectively, are cor- rect Prolog programs as written. They are both minimal recursive pro- grams, so there is no issue of goal order. They are in their preferred clause order, the reasons for which have been discussed earlier in this chapter. The termination of the programs was discussed in Section 7.2.
Program 3.19 for select is analogous to the program for member:
select (X, [XIXs] ,Xs). select(X,[YIYs],[YIZs]) - select(X,Ys,Zs).
The analysis of select is similar to the analysis of member. There is no issue of goal order because the program is minimal recursive. The clause order is chosen to reflect the intuitive order of solutions to queries such as select(X,[a,b,c],Xs), namely, {X=a,Xs=[b,c]},{X=b,Xs=[a,c]}, {X=c,Xs=[a,b]}. The first solution is the result of choosing the first element, and so forth. The program terminates unless both the second and third arguments are incomplete lists.
A variant of select is obtained by adding the test X Y in the recur- sive clause. As before, we assume that is only defined for ground argu- ments. The variant is given as Program 7.4 defining the relation select_ f irst(X,Xs,Ys). Programs 3.12 and 7.3 defining member and mnember_ check have the same meaning. Program 7.4, in contrast, has a different meaningfromProgram3.19.Thegoalselect(a,[a,b,a,c],[a,b,c]) is in the meaning of select, whereas select_f irst (a, [a,b,a,c] , [a,b, c]) is not in the meaning of select_first.
The next program considered is Program 3.20 for permutation. The order of clauses, analogously to the clause order for append, reflects the more likely mode of use:
permutation(XsJXÌYs]) - select(X,Xs,Zs), permnutation(Zs,Ys). permutation( E J , E ]).
141 Progromming in Pure Prolog
nonmember(X,Xs) - X is not a member of the list Xs.
nonmember(X,[YIYs]) - X Y, nonmernber(X,Ys).
nonmember(X,[ I).
Program 7.5 Nonmembership of a list
The goal order and the termination behavior of permutation are closely related. Computations of permutation goals where the first argument is a complete list will terminate. The query calls select with its sec- ond argument a complete list, which terminates generating a complete list as its third argument. Thus there is a complete list for the recur- sive permutation goal. If the first argument is an incomplete list, the permutation query will not terminate, because it calls a select goal that will not terminate. 1f the order of the goals in the recursive rule for permutation is swapped, the second argument of a permutation query becomes the significant one for determining termination. If it is an incomplete list, the computation will not terminate; otherwise it will.
A useful predicate using is nonmember(X,Ys) which is true if Xis not a member of a list Ys. Declaratively the definition is straightforward: An element is a norimember of a list if it is not the head and is a nonmember of the tail. The base case is that any element is a nonmember of the empty list. This program is given as Program 7.5.
Because of the use of , nonmember is restricted to ground instances. This is sensible intuitively. There are arbitrarily many elements that are not elements of a given list, and also arbitrarily many lists not containing a given element. Thus the behavior of Program 7.5 with respect to these queries is largely irrelevant.
The clause order of nonmember follows the convention of the recursive clause preceding the fact. The goal order uses the heuristic of putting the test before the recursive goal.
We reconstruct the composition of two programs concerned with the subset relation. Program 7.6 defines a relation based on Program 3.12 for member, and Program 7.7 defines a relation based on Program 3.19 for select. Both consider the occurrences of the elements of one list in a second list.
142 Chapter 7
members(Xs,Ys) Each element of the list Xs is a member of the list Ys.
members([XIXs] ,Ys) - member(X,Ys), members(Xs,Ys).
members([ ],Ys).
Program 7.6 Testing for a subset
selects(Xs,Ys) - The list Xs is a subset of the list Ys.
selects([XIXs],Ys) - select(X,Ys,Ysl),
selects([ I ,Ys)
select(X,Ys,Zs) - See Program 3.19.
Program 7.7 Testing for a subset
selects(Xs,Ysl)
Program 7.6 defining members (Xs,Ys) ignores the multiplicity of ele- ments in the lists. For example, members ([b, b] , [a, b, cl) is in the mean- ing of the program. There are two occurrences of b in the first list, but only one in the second.
Program 7.6 is also restrictive with respect to termination. If either the first or the second argument of a members query is an incomplete list, the program will not terminate. The second argument must be a complete list because of the call to member, while the first argument must also be complete, since that is providing the recursive control. The query members (Xs, [1 ,2,3])? asking for subsets of a given set does not terminate. Since multiple copies of elements are allowed in Xs, there are an infinite number of solutions, and hence the query should not terminate.
Both these limitations are avoided by Program 7.7. The revised relation is selects (Xs,Ys). Goals in the meaning of Program 7.7 have at most as many copies of an element in the first list as appear in the second. Related to this property, Program 7.7 terminates whenever the second argument is a complete list. A query such as selects(Xs, [a,b,c]) has as solution all the subsets of a given set.
We now consider a different example: translating a list of English words, word for word, into a list of French words. The relation is trans- late(Words,Mots), where Words is a list of English words and Mots the corresponding list of French words. Program 7.8 performs the trans-
143 Programming in Pure Prolog
translate(Words,Mots) - Mots is a list of French words that is the translation of the list of English words Words.
tx-anslate([WordlWords] [Mot Mots]) -
dict(Word,Mot), translate(Words,Mots)
translate([ ],[ J). dïct(the,le) dict(dog,chien)
dïct(chases,chasso) dict(cat,chat)
Program 7.8 Translating word for word
lation. It assumes a dictionary of pairs of corresponding English and French words, the relation scheme being dict(Word,Mot). The trans- lation is very naive, ignoring issues of number, gender, subject-verb agreement, and so on. Its range is solving a query such as trans- late([the,dog,chases,the,cat]),X)? with solution X=[le,chien, chasse , le , chat]. This program can be used in multiple ways. English sentences can be translated to French, French ones to English, or two sentences can be checked to see if they are correct mutual translations.
Program 7.8 is a typical program performing mapping, that is, convert- ing one list to another by applying some function to each element of the list. The clause order has the recursive rule(s) first, and the goal order calls dict first, so as not to be left recursive.
We conclude this section with a discussion of the use of data structures in Prolog programs. Data structures are handled somewhat differently in Prolog than in conventional programming languages. Rather than having a global structure, all parts of which are accessible, the programmer specifies logical relations between various substructures of the data.
Taking a more procedural view, in order to build and modify struc- tures, the Prolog programmer must pass the necessary fields of the struc- ture to subprocedures. These fields are used and/or acquire values dur- ing the computation. Assignment of values to the structures happens via unification.
Let us look more closely at a generic example - producing a single output from some given input. Examples are the standard use of ap- pend, joining two lists together to get a third, and using Program 7.8 to translate a list of English words into French. The computation proceeds recursively. The initial call instantiates the output to be an incomplete
144 Chapter 7
list [X I Xs]. The head X is instantiated by the call to the procedure, often in unification with the head of the clause. The tail Xs is progressively in- stantiated while solving the recursive call. The structure becomes fully instantiated with the solution of the base case and the termination of the computation.
Consider appending the list Ic, d] to the list [a, b], as illustrated in Figure 4.3. The output Ls=[a,b,c,d] is constructed in stages, as Ls=[alZs], Zs=[bZs1], and finally Zsl=[c,d], when the base fact of append is used. Each recursive call partially instantiates the originally in- complete list. Note that the recursive calls to append do not have access to the list being computed. This is a top-down construction of recursive structures and is typical of programming in Prolog.
The top-down construction of recursive data structures has one limi- tation. Pieces of the global data structure cannot be referred to deeper in the computation. This is illustrated in a program for the relation no_ doubles (XXs,Xs), which is true if Xs is a list of all the elements appear- ing in the list XXs with all duplicates removed.
Consider trying to compose no_doubles top-down. The head of the recursive clause will be
no_doubles([XIXs], . . .) -
where we need to fill iii the blank. The blank is filled by calling no_ doubles recursively on Xs with output Ys and integrating Ys with X. If X has not appeared in the output so far, then it should be added, and the blank will be [XYs]. If X has appeared, then it should not be added and the blank is Ys. This cannot be easily said. There is no way of knowing what the output is so far.
A program for no_doubles can be composed by thinking differently about the problem. Instead of deternrniing whether an element has al- ready appeared in the output, we can determine whether it will appear. Each element X is checked to see if it appears again in the tail of the list Xs. If X appears, then the result is Ys, the output of the recursive call to no_doubles. If X does not appear, then it is added to the recursive result. This version of no_doubles is given as Program 7.9. It uses Program 7.5 for nonmember.
A problem with Program 7.9 is that the list without duplicates may not have the elements in the desired order. For example, no_doubles ([a, b, c , b] , Xs)? has the solution Xs= [a, c , b], where the solution Xs= [a, b, c]
145 Programming in Pure Prolog
no_doubles (Xs, Ys) Ys is the list obtained by removing duplicate elements from the list Xs.
no_doubles([XIXs] ,Ys) -
mernber(X,Xs), no_doubles(Xs,Ys).
no_doubles([XIXs] ,[XIYs]) -
nomnember(X,Xs), rio_doubles(Xs,Ys)
no_doubles([ 1,1 J). nonmember(X,Xs) - See Program 7.5.
Program 7.9 Removing duplicates from a list
may be preferred. This latter result is possible if the program is rewrit- ten. Each element is deleted from the remainder of the list as it is found. In terms of Program 7.9, this is done by replacing the two recursive calls by a rule
no_doubles([XIXs] , [XIYs]) -
delete(X,Xs,Xsl), no_doubles(Xsl,Ys).
The new program builds the output top-down. However, it is inefficient for large lists, as will be discussed in Chapter 13. Briefly, each call to delete rebuilds the whole structure of the list.
The alternative to building structures top-down is building them bottom-up. A simple example of bottom-up construction of data struc- tures is Program 3.16b for reversing a list:
reverse(Xs,Ys) '- reverse(Xs,[ ],Ys).
reverse([XIXs] ,Revs,Ys) - reverse(Xs, [XIRevs] ,Ys).
reverse([ I ,Ys,Ys).
An extra argument is added to reverse/2 and used to accumulate the values of the reversed list as the computation proceeds. This procedure for reverse builds the output list bottom-up rather than top-down. In the trace in Figure 7.3 solving the goal reverse([a,b,c] ,Xs), the suc- cessive values of the middle argument of the calls to reverse/3 E ], [a], [b, a], and [c , b, a] represent the structure being built.
A bottom-up construction of structures allows access to the partial results of the structure during the computation. Consider a relation nd_ reverse(Xs ,Ys) combining the effects of no.doubles and reverse. The
146 Chapter 7
reverse([a,b,c] ,Xs)
reverse([a,b,c] , E I ,Xs)
reverse([b,c] , [a] ,Xs)
reverse([c] , [b,a] ,Xs)
reverse([ I, [c,b,a] ,Xs) true
Xs[c,b,a]
Figure 7.3 Tracing a reverse computation
nd_reverse(Xs, Ys) Ys is the reversal of the list obtained by removing duplicate elements from the list Xs.
nd_reverse(Xs ,Ys) - nd_reverse(Xs, [ I ,Ys).
nd_reverse([XIXs] ,Revs,Ys) -
member(X,Revs), nd_reverse(Xs,Revs,Ys).
nd_reverse([XIXs] ,Revs,Ys) -
nonmember(X,Revs), nd_reverse(Xs, [XlRevs] ,Ys).
n&reverse([ I ,Ys,Ys).
nonmember(X,Xs) - See Program 7.5.
Program 7.10 Reversing with no duplicates
meaning of nd_reverse is that Ys is a list of elements in Xs in reverse or- der and with duplicates removed. Analogously to reverse, nd_reverse calls nd_reverse/3 with an extra argument that builds the result bottom- up. This argument is checked to see whether a particular element ap- pears, rather than checking the tail of the list as in Program 7.9 for no_doubles. The program is given as Program 7.10.
We emphasize the characteristics of bottom-up construction illus- trated here. One argument behaves as an accumulator of the final data structure. It is augmented in the recursive call, so that the more complex version is in the body of the clause rather than in its head. This contrasts with top-down construction, where the more complex version of the data structure being built is in the head of the clause. Another argument is used solely for returning the output, namely, the final value of the ac- cumulator. It is instantiated with the satisfaction of the base fact. The argument is explicitly carried unchanged in the recursive call.
The tecimique of adding an accumulator to a program can be general- ized. It is used in Chapter 8 discussing Prolog programs for arithmetic.
147 Programming in Pure Prolog
Accumulators cari also be viewed as a special case of incomplete data structures, as is discussed in Chapter 15.
7.5.1 Exercise for Section 7.5
(i) Write Program 7.9 for no_doubles, building the structure bottom- up
7.6 Background
Prolog was envisaged as a first approximation to logic programming, which would be superseded by further research. Its control has always been acknowledged as being limited and naive. An oft-cited slogan, cred- ited to Kowaiski (1979b), is "Algorithm Logic + Control." The particular control provided in pure Prolog was intended as just one solution on the path to declarative prograniming and intelligent control. Time has shown otherwise. The control of Prolog has proven adequate for a large range of applications, and the language has not only endured but has blossomed.
Nonetheless, logic programming researchers have investigated other forms of control. For example, LOGLISP (Robinson and Sibert, 1982) has breadth-first traversal of the search tree, and IC-Prolog (Clark and Mc- Cabe, 1979) has co-routining. MU-Prolog (Naish, 1986) allows suspension to provide a correct implementation of negation and to prevent the com- putation from searching infinite branches in certain cases. Wait declara- tions are generated (Naish, 1985b) that are related to the conditions on termination of Prolog programs given in Section 7.2.
A methodology for systematically constructing simple Prolog pro- grams is given in Deville (1990). Essential to Deville's methods are specifications, a subject touched upon in Section 13.3.
Analysis of Prolog programs, and logic programs more generally, has become a hot topic of research. Most analyses are based on some form of abstract interpretation, a topic beyond the scope of this book. The initial work in Prolog can be found in Mellish (1985), and a view of leading research groups can be found in a special issue of the Journal of Logic Programming (1993).
Extensive work has also appeared recently on analyzing termination of Prolog programs. A starting place for this topic is PlUmer (1990).
8 Arithmetic
The logic programs for performing arithmetic presented in Section 3.1 are very elegant, but they are not practical. Any reasonable computer provides very efficient arithmetic operations directly in hardware, and practical logic programming languages cannot afford to ignore this fea- ture. Computations such as addition take unit time on most computers independent of the size of the addends (as long as they are smaller than some large constant). The recursive logic program for plus (Program 3.3) takes time proportional to the first of the numbers being added. This could be improved by switching to binary or decimal notation but still won't compete with direct execution by dedicated hardware.
Every Prolog implementation reserves some predicate names for system-related procedures. Queries to these predicates, called system predicates, are handled by special code in the implementation in contrast to calls to predicates defined by pure Prolog programs. A Prolog imple- mentor should build system predicates that complement pure Prolog naturally and elegantly. Other names for system predicates are evaluable predicates, builtiri predicates, or bips, the latter two being referred to in the draft for Standard Prolog.
8.1 System Predicates for Aiitkmetic
The role of the system predicates for arithmetic introduced in Prolog is to provide an interface to the underlying arithmetic capabilities of the computer in a straightforward way. The price paid for this efficiency is
150 Chapter 8
that some of the machine-oriented arithmetic operations are not as gen- eral as their logical counterparts. The interface provided is an arithmetic evaluator, which uses the underlying arithmetic facilities of the com- puter. Standard Prolog has a system predicate is (Value ,Expression) for arithmetic evaluation. Goals with the predicate is are usually written in binary infix form, taking advantage of the operator facility of Prolog, about which we now digress.
Operators are used in order to make programs more readable. People are very flexible and learn to adjust to strange surroundingsthey can become accustomed to reading Lisp and Fortran programs, for example. We believe nonetheless that syntax is important; the power of a good notation is well known from mathematics. Ari integral part of a good syntax for Prolog is the ability to specify and use operators.
Operators, for example and <, have already been used in earlier chapters. Standard Prolog provides several operators, which we intro- duce as they arise. Programmers can also define their own operators using the built-in predicate op/3. An explanation of the mechanism for operator declarations, together with a list of pre-defined operators and their precedences is given in Appendix B.
Queries using the arithmetic evaluator provided by Prolog have the form Value is Expression?. Queries to the evaluator are interpreted as follows. The arithmetic expression Expression is evaluated and the result is unified with Value. Once arithmetic evaluation succeeds, the query succeeds or fails depending on whether unification succeeds or fails.
Here are some examples of simple addition, illustrating the use and behavior of the evaluator. The query (X is 3+5)? has the solution X=8. This is the standard use of the evaluator, instantiating a variable to the value of an arithmetic expression. The query (8 is 3+5)? succeeds. Hav- ing both arguments to is instantiated allows checking the value of an arithmetic expression. (3+5 is 3+5)? fails because the left-hand argu- ment, 3+5, does not unify with 8, the result of evaluating the expression.
Standard Prolog specifies a range of arithmetic operations that should be supported by Prolog for both integers and reals represented as floating-point numbers. In particular, the evaluator provides for addi- tion, subtraction, multiplication, and division (+, -, *, I) with their usual mathematical precedences. In this book, we restrict ourselves to integer arithmetic.
151 Arithmetic
What happens if the term to be evaluated is not a valid arithmetic ex- pression? An expression can be invalid for one of two reasons, which should be treated differently, at least conceptually. A term such as 3+x for a constant x cannot be evaluated. In contrast, a term 3+Y for a variable Y may or may not be evaluable, depending on the value of Y.
The semantics of any logic program is completely defined, and, in this sense, logic programs cannot have runtime "errors." For example, the goal X is 3+Y has solutions (X=3,Y=O}. However, when interfacing logic programs to a computer, the limitations of the machine should be taken into account. A runtime error occurs when the machine cannot determine the result of the computation because of insufficient information, that is, uninstantiated variables. This is distinct from goals that simply fail. Extensions to Prolog and other logic languages handle such "errors" by suspending until the values of the concerned variables are known. The execution model of Prolog as introduced does not permit suspension. Instead of simply failure, we say an error condition occurs.
The query (X is 3+x)? fails because the right-hand argument cannot be evaluated as an arithmetic expression. The query (X is 3Y)? is an example of a query that would succeed if Y were instantiated to an arith- metic expression. Here an error condition should be reported.
A common misconception of beginning Prolog programmers is to re- gard is as taking the place of assignment as in conventional program- ming languages. It is tempting to write a goal such as (N is N+1). This is meaningless. The goal fails if N is instantiated, or causes an error if N is a variable.
Further system predicates for arithmetic are the comparison operators. Instead of the logically defined <, (written =<), >, (written >), Prolog directly calls the underlying arithmetic operations. We describe the behavior of <; the others are virtually identical. To answer the query (A < B)?, A and B are evaluated as arithmetic expressions. The two resultant numbers are compared, and the goal succeeds if the result of evaluating A is less than the result of evaluating B. Again, if A or B is not an arithmetic expression, the goal will fail, and an error condition should result if A or B are not ground.
Here are some simple examples. The query (1 < 2)? succeeds, as does the query (3-2 < 2*3+1)?. On the other hand, (2 < 1)? fails, and (N < 1)? generates an error when N is a variable.
152 Chapter 8
Tests for equality and inequality of values of arithmetic expressions are implemented via the builtin predicates =: = and =1=, which evaluate both of their arguments and compare the resulting values.
8.2 Arithmetic Logic Programs Revisited
Performing arithmetic via evaluation rather than logic demands a recon- sideration of the logic programs for arithmetic presented in Section 3.1. Calculations can certainly be done more efficiently. For example, finding the minimum of two numbers can use the underlying arithmetic com- parison. The program syntactically need not change from Program 3.7. Similarly, the greatest common divisor of two integers can be computed efficiently using the usual Euclidean algorithm, given as Program 8.1. Note that the explicit condition .3 > O is necessary to avoid multiple solutions when J equals O and errors from calling mod with a zero ar- gument.
Two features of logic programs for arithmetic are missing from their Prolog counterparts. First, multiple uses of programs are restricted. Sup- pose we wanted a predicate plus (X,Y,Z) that performed as before, built using is. The obvious definition is
plus(X,Y,Z) - Z is X+Y.
This works correctly if X and Y are instantiated to integers. However, we cannot use the same program for subtraction with a goal such as plus(3,X,8)?, which raises an error condition. Meta-logical tests are needed if the same program is to be used for both addition and sub- traction. We defer this until meta-logical predicates are introduced in Chapter 10.
Programs effectively become specialized for a single use, and it is tricky to understand what happens when the program is used differently.
greatesLconimon_divisor (X, Y,Z) - Z is the greatest common divisor of the integers X and Y.
greatest_common_divisor(I,O,I).
greatest_common_divisor(I , .J,Gcd) -
J > O, B. is I mod J, greatest_common_divisor(J,R,Gcd).
Program 8.1 Computing the greatest common divisor of two integers
153 Arithmetic
factorial(N,F) - F is the integer N factorial.
factorial(N,F) -
N > 0, Nl is N-1, factorial(N1,F1), F is N*F1.
factorial (0,1).
Program 8.2 Computing the factorial of a number
Program 3.7 for minimum, for example, can be used reliably only for find- ing the minimum of two integers.
The other feature missing from Prolog programs for arithmetic is the recursive structure of numbers. In logic programs, the structure is used to determine which rule applies, and to guarantee termination of compu- tations. Program 8.2 is a Prolog program for computing factorials closely corresponding to Program 3.6. The recursive rule is more clumsy than before. The first argument in the recursive call of factorial must be cal- culated explicitly rather than emerging as a result of unification. Further- more, the explicit condition determining the applicability of the recursive rule, N > O, must be given. This is to prevent nonterminating computa- tions with goals such as factorial(-1,N)? or even factorial(3,F)?. Previously, in the logic program, unification with the recursive structure prevented nonterminating computations.
Program 8.2 corresponds to the standard recursive definition of the factorial function. Unlike Program 3.6, the program can be used only to calculate the factorial of a given number. A factorial query where the first argument is a variable will cause an error condition.
We must modify the concept of correctness of a Prolog program to accommodate behavior with respect to arithmetic tests. Other system predicates that generate runtime "errors" are handled similarly. A Prolog program is totally correct over a domani D of goals if for all goals in D the computation terminates, does not produce a runtime error, and has the correct meaning. Program 8.2 is totally correct over the domain of goals where the first argument is an integer.
8.2.1 Exercises for Section 8.2
(i) The Nth triangular number is the sum of the numbers up to and in- cluding N. Write a program for the relation triangle(N,T), where T is the Nth triangular number. (Hint: Adapt Program 8.2.)
154 Chapter 8
(ii) Write a Prolog program for power(X,N,V), where V equals XN. Which way can it be used? (Hint: Model it on Program 3.5 for exp.)
(ill) Write Prolog programs for other logic programs for arithmetic given in the text and exercises in Section 3.1.
(iv) Write a Prolog program to generate a Huifman encoding tree from a list of symbols and their relative frequencies.
8.3 Transforming Recursion into Iteration
In Prolog there are no iterative constructs as such, and a more general concept, namely recursion, is used to specify both recursive and iterative algorithms. The main advantage of iteration over recursion is efficiency, mostly space efficiency. In the implementation of recursion, a data struc- ture (called a stack frame) has to be maintained for every recursive call that has not terminated yet. A recursive computation involving n recur- sive procedure calls would require, therefore, space linear in n. On the other hand, an iterative program typically uses only a constant amount of memory, independent of the number of iterations.
Nevertheless, there is a restricted class of recursive programs that cor- responds quite closely to conventional iterative programs. Under some conditions, explained further in Section 11.2 on tail recursion optimiza- tion, such Prolog programs can be implemented with almost the same efficiency as iterative programs in conventional languages. For this rea- son, it is preferable to express a relation using an iterative program, if possible. In this section, we show how recursive programs can be made iterative using accumulators.
Recall that a pure Prolog clause is iterative if it has one recursive call in the body. We extend this notion to full Prolog, and allow zero or more calls to Prolog system predicates before the recursive call. A Prolog procedure is iterative if it contains only unit clauses and iterative clauses.
Most simple arithmetic calculations can be implemented by iterative programs.
Factorials can be computed, for example, in a loop where the numbers up to the desired factorial are multiplied together. A procedure in a
155 Arithmetic
factorial(N);
I is O; T is 1;
while I < N do I is I + 1; T is T * I end;
return T.
Figure 8.1 Computing factorials iteratively
factorial(N,F) F is the integer N factorial.
factorïal(N,F) - factorial(0,N,i,F).
factorial(I,N,T,F)
I < N, Ii is 1+1, Ti is T*I1, factorial(I1,N,T1,F).
factorial(N,N,F,F).
Program 8.3 An iterative factorial
Pascal-like language using a while loop is given in Figure 8.1. Its iterative behavior can be encoded directly in Prolog with an iterative program.
Prolog does not have storage variables, which can hold intermediate results of the computation and be modified as the computation pro- gresses. Therefore, to implement iterative algorithms, which require the storage of intermediate results, Prolog procedures are augmented with additional arguments, called accumWators. Typically, one of the interme- diate values constitutes the result of the computation upon termination of the iteration. This value is unified with the result variable using the unit clause of the procedure.
This technique is demonstrated by Program 8.3, which is a Prolog del- initìon of factorial that mirrors the behavior of the while ioop in Fig- ure 8.1. It uses factorial(I ,N,T,F), which is true if F is the value of N factorial, and I and T are the values of the corresponding loop variables before the (i+1)th iteration of the loop.
The basic iterative loop is performed by the iterative procedure f acto- rial/4. Each reduction of a goal using Íactorial/4 corresponds to an iteration of the while loop. The call of f actorial/4 by factorìal/2 cor- responds to the initialization stage. The first argument of factorial/4, the loop counter, is set to O.
156 Chapter 8
factorial(N,F) - F is the integer N factorial.
factorial(N,F) - factorial(N,1,F).
factorial(N,T,F) -
N > O, Ti is T*N, Nl is N-1, factorial(Nl,Ti,F).
factorial(O,F,F).
Program 8.4 Another iterative factorial
The third argument of f actorial/4 is used as an accumulator of the running value of the product. It is initialized to 1 in the call to f ac- torial/4 by factorial/2. The handling of both accumulators in Pro- gram 8.3 is a typical programming technique in Prolog. It is closely re- lated to the use of accumulators in Programs 3.16b and 7.10 for collect- ing elements in a list.
Accumulators are logical variables rather than locations in memory. The value is passed between iterations, not an address. Since logical variables are "write-once," the updated value, a new logical variable, is passed each time. Stylistically, we use variable names with the suffix 1, for example, Ti and Il, to indicate updated values.
The computation terminates when the counter I equals N. The rule for f actorial/4 in Program 8.3 no longer applies, and the fact succeeds. With this successful reduction, the value of the factorial is returned. This happens as a result of the unification with the accumulator in the base clause. Note that the logical variable representing the solution, the final argument of f actorial/4, had to be carried throughout the whole computation to be set on the final call of factorial. This passing of values in arguments is characteristic of Prolog programs and might seem strange to the newcomer.
Program 8.3 exactly mirrors the while loop for factorial given in Fig- ure 8.1. Another iterative version of factorial can be written by count- ing down from N to 0, rather than up from O to N. The basic program structure remains the same and is given as Program 8.4. There is an ini- tialization call that sets the value of the accumulator, and recursive and base clauses implementing the while loop.
Program 8.4 is marginally more efficient than Program 8.3. In general, the fewer arguments a procedure has, the more readable it becomes, and the faster it runs.
157 Arithmetic
between(I,J,K) - K is an integer between the integers I and J inclusive.
betwoen(I,J,I) - I < J. between(I,J,K) I < J, Il is 1+1, between(I1,J,K).
Program 8.5 Generating a range of integers
sumlist(Is,Sum) Sum is the sum of the list of integers Is.
sunilist([IIIs],Sum) sun1list(Is,IsSum), Sum is I+IsSuni. sumlist([ ],O).
Program 8.6a Summing a list of integers
sumlist(Is,Sum) - Sum is the sum of the list of integers Is.
sumlist(Is,Sum) - sumlist(Is,O,Suin). sujnlist([IIIs] ,Temp,Suin) -
Tempi is Temp+I, sumlist(Is,Templ,Sum). suinlist([ ] ,Suxn,Suni).
Program 8.6b Iterative version of summing a list of integers using an accu- mulator
A useful iterative predicate is between (I , .3 , K), which is true if K is an integer between I and J inclusive. lt can be used to generate nondeter- ministically integer values within a range (see Program 8.5). This is useful in generate-and-test programs, explained in Section 14.1, and in failure- driven loops, explained in Section 12.5.
Iterative programs can be written for calculations over lists of integers as well. Consider the relation sumlist(IntegerList,Sum), where Sum is the sum of the integers in the list IntegerList. We present two pro- grams for the relation. Program 8.6a is a recursive formulation. To sum a list of integers, sum the tail, and then add the head. Program 8.6b uses an accumulator to compute the progressive sum precisely as Program 8.3 for factorial uses an accumulator to compute a progressive product. An auxiliary predicate, sunilist/3, is introduced with an extra argument for the accumulator, whose starting value, O, is set in the initial call to
158 Chapter 8
inner_product (Xs, Ys, Value) - Value is the inner product of the vectors represented by the lists of integers Xs and Ys.
inner_product ([XI Xs] , [Y I Ys] ,IP) -
inner_product(Xs,Ys,IP1), IP is X*Y+IP1.
inner_product([ ],[ ],O).
Program 8.7a Computing inner products of vectors
inner_product (Xs, Ys,Value) - Value is the limer product of the vectors represented by the lists of integers Xs and Ys.
inner_product (XS,YS, IP) - inner_product(Xs ,Ys,O, IP)
inner_product([XIXs] , [Y lYs] ,Temp, IP) -
Tempi is X*Y+Temp, inner_product(Xs,Ys,Templ ,IP).
inner_product([ ],[ ],IP,IP).
Program 8.7b Computing inner products of vectors iteratively
sumlist/3. The sum is passed out in the final call by unification with the base fact. The only difference between Program 8.6b and the iterative ver- sions of factorial is that the recursive structure of the list rather than a counter is used to control the iteration.
Let us consider another example. The inner product of two vec- tors is the sum X1 Y1 + + X Y. If we represent vectors as lists, it is straightforward to write a program for the relation inner_ product (Xs,Ys,IP), where IP is the inner product of Xs and Ys. Pro- grams 8.7a and 8.7b are recursive and iterative versions, respectively. The iterative version of inner_product bears the same relation to the recursive inner_product that Program 8.6b for suinlist bears to Pro- gram 8.6a.
Both Programs 8.7a and 8.7b are correct for goals inner_product (Xs, Ys , Zs), where Xs and Ys are lists of integers of the same length. There is a built-in check that the vectors are of the same length. The programs fail if Xs and Ys are of different lengths.
The similarity of the relations between Programs 8.6a and 8.6b, and Programs 8.7a and 8.7b, suggests that one may be automatically trans- formed to the other. The transformation of recursive programs to equiv-
159 Arithmetic
area(Chain,A rea) Area is the area of the polygon enclosed by the list of points Chain, where the coordinates of each point are represented by a pair (X,Y) of integers.
area( [Tuple] 0).
area([(Xl,Yl),(X2,Y2)IXYs],Area)
area([(X2,Y2)IXYs] Areal),
Area is (Xl*Y2-Yl*X2)/2 + Areal.
Program 8.8 Computing the area of polygons
aient iterative programs is an interesting research question. Certainly it can be done for the simple examples shown here.
The sophistication of a Prolog program depends on the underlying logical relation it axiomatizes. Here is a very elegant example of a simple Prolog program solving a complicated problem.
Consider the following problem: Given a closed planar polygon chain {P1,P2.....P}, compute the area of the enclosed polygon and the orienta- tion of the chain. The area is computed by the line integral
1/2 Jxdy-ydx,
where the integral is over the polygon chain. The solution is given in Program 8.8, which defines the relation
area(Chain,Area). Chain is given as a list of tuples, for example, [(4,6),(4,2), (0,8), (4,6)]. The magnitude of Area is the area of the poly- gon bounded by the chain. The sign of Area is positive if the orientation of the polygon is counterclockwise, and negative if it is clockwise.
The query area([(4,6),(4,2),(O,8),(4,6)],Area)? has the solu- tion Area = -8. The polygon gains opposite orientation by reversing the order of the tuples. The solution of the query area([(4,6),(0,8), (4,2) ,(4,6)] ,Area)? is Area = 8.
The program shown is not iterative. Converting it to be iterative is the subject of Exercise (y) at the end of the section.
An iterative program can be written to find the maximum of a list of integers. The relation scheme is maxlist(Xs,Max), and the program is given as Program 8.9. An auxiliary predicate maxlist(Xs,X,Max) is used for the relation that Max is the maximum of X and the elements in the list Xs. The second argument of maxlist/3 is initialized to be the first
160 Chapter 8
maxlist(Xs,N) - N is the maximum of the list of integers Xs.
maxlist([XIXs3,M) - maxlist(Xs,X,M).
rnaxlist([XIXs],Y,M) - maximuin(X,Y,Y1), rnaxlist(Xs,Y1,M).
maxlist([ ],M,M).
maximuni(X,Y,Y) - X Y.
maximum(X,Y,X) - X > Y.
Program 8.9 Finding the maximum of a list of integers
length(Xs,N) - Xs is a list of length N.
length([XIXs],N) - N > 0, Ni is N-1, length(Xs,N1). length([ 1,0).
Program 8.10 Checking the length of a list
element of the list. Note that the maximum of an empty list is not defined by this program.
The standard recursive program for finding the maximum of a list of integers constitutes a slightly different algorithm. The recursive formula- tion finds the maximum of the tail of the list and compares it to the head of the list to find the maximum element. In contrast, Program 8.9 keeps track of the rurming maximum as the list is traversed.
Program 3.17 for finding the length of a list is interesting, affording several ways of translating a logic program into Prolog, each of which has its separate features. One possibility is Program 8.10, which is iterative. Queries length (Xs , N)? are handled correctly if N is a natural number, testing if the length of a list is N, generating a list of N uninstantiated elements, or failing. The program is unsuitable, however, for finding the length of a list with a call such as length([1,2,3 ,N)?. This query generates an error.
The length of a list can be found using Program 8.11. This program cannot be used, however, to generate a list of N elements. In contrast to Program 8.10, the computation does not terminate if the first argument is an incomplete list. Different programs for length are needed for the different uses.
161 Arithmetic
iength(Xs,N) N is the length of the list Xs.
length([XIXs],N) - length(Xs,N1), N is Ni-4-i.
length([ 1,0)
Program 8.11 Finding the length of a list
range(M,N,Ns) - Ns is the list of integers between M and N inclusive.
range(M,N,[MINs]) M < N, Ml is M+1, range(M1,N,Ns).
range(N,N, [N]).
Program 8.12 Generating a list of integers in a given range
Similar considerations about the intended use of a program occur when trying to define the relation range(M,N,Ns), where Ns is the list of integers between N and N inclusive. Program 8.12 has a specific use: generating a list of numbers in a desired range. The program is totally correct over all goals rarige(M,N,Ns) where M and N are instantiated. The program cannot be used, however, to find the upper and lower limits of a range of integers, because of the test M < N. Removing this test would allow the program to answer a query range(M,N, [1,2,3])?, but then it would not terminate for the intended use, solving queries such as range (1,3, Ns) ?.
8.3.1 Exercises for Section 8.3
Write an iterative version for triangle(N,T), posed as Exer- cise 8.2(i).
Write an iterative version for power(X,N,V), posed as Exercise 8.2(u).
Rewrite Program 8.5 so that the successive integers are generated in descending order.
Write an iterative program for the relation timeslist (Integer- List,Product) computing the product of a list of integers, anal- ogous to Program 8.6b for sumlist.
162 Chapter 8
(y) Rewrite Program 8.8 for finding the area enclosed by a polygon so that it is iterative.
Write a program to find the minimum of a list of integers.
Rewrite Program 8.11 for finding the length of a list so that it is iterative. (Hint: Use a counter, as in Program 8.3.)
Rewrite Program 8.12 so that the range of integers is built bottom- up rather than top-down.
8.4 Background
The examples given in this chapter are small and do not especially ex- ploit Prolog's features. Algorithms that are fundamentally recursive are more interesting in Prolog. A good example of such a program is the Fast Fourier Transform, for which efficient versions have been written in Pro- log.
A good place for reading about Huifman encoding trees for Exercise 8.2(iv) is Abelson and Sussman (1985).
A program for transforming recursive programs to iterative ones, which handles the examples in the text, is described in Bloch (1984).
Program 8.8, computing the area of a polygon, was shown to us by Martin Nils son.
9 Structure Inspection
Standard Prolog has several predicates related to the structure of terms. These predicates are used to recognize the different types of terms, to decompose terms into their functor and arguments, and to create new terms. This chapter discusses the use of predicates related to term struc- ture.
9.1 Type Predicates
Type predicates are unary relations that distinguish between the different types of terms. System predicates exist that test whether a given term is a structure or a constant, and further, whether a constant is an atom, an integer or floating-point. Figure 9.1 gives the four basic type predicates in Standard Prolog, together with their intended meanings.
Each of the basic predicates in Figure 9.1 can be regarded as an infi- nite table of facts. The predicate integer/i would consist of a table of integers:
integer(0). integer(i). integer(-1).
The predicate atom/i would consist of a table of atoms in the program:
atoin(foo). atom (bar). .
The predicate compound/i would consist of a table of the function sym- bols in the program with variable arguments, etc.
compound(father(X,Y)). compound(son(X,Y)).
164 Chapter 9
integer(X) - X is an integer.
atom(X) - X is an atom.
real(X) - X is a floating-point number.
compound(X) - X is a compound term.
Figure 9.1 Basic system type predicates
Other type predicates can be built from the basic type predicates. For example, that a number is either an integer or floating-point can be rep- resented by two clauses:
number(X) - integer(X).
number(X) - real(X).
Standard Prolog includes a predicate number/i effectively defined in this way. It also includes a predicate atomic (X), which is true if X is an atom or a number. In this book, we prefer to call the predicate con- stant/i. To run under Standard Prolog, the following clause may be necessary:
constant(X) - atomic(X).
To illustrate the use of type predicates, the query integer (3)? would succeed, but the query atom(3)? would fail. One might expect that a call to a type predicate with a variable argument, such as integer (X) ?, would generate different integers on backtracking. This is not practical for implementation, however, and we would prefer that such a call re- port an error condition. In fact, Standard Prolog specifies that the call integer (X)? should fail.
The only terms not covered by the predicates in Figure 9.1 are vari- ables. Prolog does provide system predicates relating to variables. The use of such predicates, however, is conceptually very different from the use of structure inspection predicates described in this chapter. Meta- logical predicates (theìr technical name) are the subject of Chapter 10.
We give an example of the use of a type predicate as part of a pro- gram for flattening a list of lists. The relation flatten(Xs,Ys) is true if Ys is the list of elements occurring in the list of lists Xs. The elements of Xs can themselves be lists or elements, so elements can be arbitrarily deeply nested. An example of a goal in the meaning of flatten is f lat- ten([[a] , [b, [c,d]] ,e] , [a,b,c,d,e]).
165 Structure Inspection
flatten(Xs,Ys) Ys is a list of the elements of Xs.
flatten([XIXs] ,Ys)
flatten(X,Ysl), flatten(Xs,Ys2), append(Ysl,Ys2,Ys).
flatten(X,[X])
constant(X), X[ 1. flatten([ ],[ 1).
Program 9.la Flattening a list with double recursion
The simplest program for flattening uses double recursion. To flatten an arbitrary list [XIXs], where X can itself be a list, flatten the head of the list X, flatten the tail of the list Xs, and concatenate the results:
flatten([XIXs] ,Ys)
flatten(X,Ysl), flatten(Xs,Ys2), append(Ysl,Ys2,Ys).
What are the base cases? The empty list is flattened to itself: A type predicate is necessary for the remaining case. The result of flattening a constant is a list containing the constant:
flatten(X, [X]) - constant(X), Xl E ]
The condition constant (X) is necessary to prevent the rule being used when X is a list. The complete program for f latten is given as Pro- gram 9.la.
Program 9.la, although very clear declaratively, is not the most effi- cient way of flattening a list. In the worst case, which is a left-linear tree, the program would require a number of reductions whose order is qua- dratic in the number of elements in the flattened list.
A program for flatten that constructs the flattened list top-down is a little more involved than the doubly recursive version. lt uses an auxiliary predicate flatten(Xs,Stack,Ys), where Ys is a flattened list containing the elements in Xs and a stack Stack to keep track of what needs to be flattened. The stack is represented as a list.
The call of flatten/3 by f latten/2 initializes the stack to the empty list. We discuss the cases covered by f latten/3. The general case is flattening a list [XXs], where X is itself a list. ¡ri this case Xs is pushed onto the stack, and X is recursively flattened. The predicate list(X) is used to recognize a list. It is defined by the fact list ( [XIXs]):
166 Chapter 9
flatten(Xs,Ys) - Ys is a list of the elements of Xs.
flatten(Xs,Ys) - flatten(Xs,[ ],Ys).
flatten([XIXs] ,S,Ys) -
list (X), flatten(X, [XsIS] ,Ys)
flatten([XIXs],S,[XIYs]) -
constant(X), X E I , flatten(Xs,S,Ys).
flatten([ ],[XIS],Ys) -
flatten(X,S,Ys).
flatten([ 1,1 ],E I). list([XIXs]).
Program 9.lb Flattening a list using a stack
flatten([XIXs] ,S,Ys) - list(X), flatten(X, [XsIS] ,Ys).
When the head of the list is a constant other then the empty list, it is added to the output, and the tail of the list is flattened recursively:
flatten([XIXs],S,[XIYs]) -
constant(X),X[ 1, flatten(Xs,S,Ys).
When the end of the list is reached, there are two possibilities, depending on the state of the stack. If the stack is nonempty, the top element is popped, and the flattening continues:
flatten([ ],[XIS],Ys) - flatten(X,S,Ys).
If the stack is empty, the computation terminates:
flatten([ ],[ ],[ 1).
The complete program is given as Program 9.lb. A general technique of using a stack is demonstrated in Program 9.lb.
The stack is managed by unification. Items are pushed onto the stack by recursive calls to a consed list. Items are popped by unifying with the head of the list and recursive calls to the tail. Another application of stacks appears in Programs 17.3 and 17.4 simulating pushdown au- tomata.
Note that the stack parameter is an example of an accumulator. The reader can verify that the revised program requires a number of
reductions linear in the size of the flattened list.
167 Structure Inspection
9.1.1 Exercise for Section 9.1
(i) Rewrite Program 9.la for f latten(Xs,Ys) to use an accumulator instead of the call to append, keeping it doubly recursive.
9.2 Accessing Compound Terms
Recognizing a term as compound is one aspect of structure inspection. Another aspect is providing access to the furictor name, arity, and argu- ments of a compound term. One system predicate for delving into com- pound terms is functor (Term, F, Arity). This predicate is true if Term is a term whose principal functor has name F and arity Arity. For example, functor(father(haran,lot),father,2)? succeeds.
The functor predicate can be defined, analogously to the type pred- icates, by a table of facts of the form functor(f(Xl,...,XN),f,N) for each functor f of arity N, for example, functor(father(X,Y) father, 2), functor(son(X,Y) ,son,2).....Standard Prolog considers constants to be functors of arity O, with the appropriate extension to the functor table.
Calls to functor can fail for various reasons. A goal such as f unc- tor(father(X,Y) ,son,2) does not unify with an appropriate fact in the table. Also, there are type restrictions on the arguments of f uric- tor goals. For example, the third argument of functor, the arity of the term, cannot be an atom or a compound term. If these restrictions are violated, the goal fails. A distinction can be made between calls that fail and calls that should give an error because there are infinitely many so- lutions, such as functor(X,Y,2)?.
The predicate functor is commonly used in two ways, term decompo- sition and creation. The first use finds the functor name and arity of a given term. For example, the query functor(father(haran,lot) ,X,Y)? has the solution {X=father,Y"2}. The second use builds a term with a particular functor name and arity. A sample query is functor (T ,father, 2)? with solution T=father(X,Y).
The companion system predicate to functor is arg(N,Term,Arg), which accesses the arguments of a term rather than the functor name.
168 Chapter 9
subterm (Sub, Term) Sub is a subterm of the ground term Term.
subterm(Term,Term).
subterm(Sub,Term) -
compound(Term), functor(Term,F,N), subterm(N,Sub,Term).
subterm(N ,Sub,Term) -
N > 1, Nl is N-1, subterm(N1,Sub,Term).
subterm(N,Sub,Term) -
arg(N,Term,Arg), subterm(Sub,Arg).
Program 9.2 Fmdmg subterms of a term
The goal arg (N ,Term, Arg) is true if Arg is the Nth argument of Term. For example, arg(1,father(haran,lot) ,haran) is true.
Like functor/3, arg/3 is commonly used in two ways. The term de- composition use finds a particular argument of a compound term. A query exemplifying this use is arg(2,father(haran,lot) ,X)? with so- lution X=lot. The term creation use instantiates a variable argument of a term. For example, the query arg(1,father(X,lot) ,haran)? succeeds, instantiating X to haran.
The predicate arg is also defined as if there is an infinite table of facts. A fragment of the table is
arg(i,father(X,Y) ,X). arg(2,father(X,Y) ,Y).
arg(i,son(X,Y) ,X).
Calls to arg fail if the goal does not unify with the appropriate fact in the table, for example, arg(1,father(haran,lot) abraham). They also fail if the type restrictions are violated, for example, if the first argument is an atom. An error is reported with a goal such as arg (1 , X, Y).
Let us consider an example of using functor and arg to inspect terms. Program 9.2 axiomatizes a relation subterm(T1 T2), which is true if Ti is a subterm of T2. For reasons that will become apparent later, we restrict Ti and T2 to be ground.
The first clause of Program 9.2 defining subterm/2 states that any term is a subterm of itself. The second clause states that Sub is a subterm of a compound term Term if it is a subterm of one of the arguments. The number of arguments, i.e., the arity of the principal functor of the term,
169 Structure Inspection
is found and used as a ioop counter by the auxiliary subterm/3, which iteratively tests all the arguments.
The first clause of subterm/3 decrements the counter and recursively calls subterm. The second clause covers the case when Sub is a subterm of the Nth argument of the term.
The subteim procedure can be used in two ways: to test whether the first argument is indeed a subterm of the second; and to generate sub- terms of a given term. Note that the clause order determines the order in which subterms are generated. The order in Program 9.2 gives sub- terms of the first argument before subterms of the second argument, and so on. Swapping the order of the clauses changes the order of solu- tions.
Consider the query subterm (a, f (X , Y))?, where the second argument is not ground. Eventually the subgoal subterrn(a,X) is reached. This suc- ceeds by the first subterm rule, instantiating X to a. The subgoal also matches the second subterm rule, invoking the goal compound(X), which generates an error. This is undesirable behavior.
We defer the issues arising when performing structure inspection on nonground terms to Chapter 10, where meta-logical predicates with suit- able expressive power are introduced. For the rest of this chapter, all programs are assumed to take only ground arguments unless otherwise stated.
Program 9.2 is typical code for programs that perform structure in- spection. We look at another example, substituting for a subterm in a term.
The relation scheme for a general program for substituting subterms is substitute (Old,New,OldTerm,Newlerm), where NewTerm is the result of replacing all occurrences of Old in OldTeim by New. Program 9.3 imple- menting the relation generalizes substituting for elements in a list, posed as Exercise 3.3(i) and the logic program (Program 3.26) substituting for elements in binary trees.
Program 9.3 is a little more complicated than Program 9.2 for sub- term but conforms to the same basic pattern. The clauses for substi- tute/4 cover three different cases. The last, handling compound terms, calls an auxiliary predicate substitute/5, which iteratively substitutes in the subterms. The arity of the principal functor of the term is used as the initial value of a loop counter that is successively decremented to control the iteration. We present a particular example to illustrate
170 Chapter 9
substitute( Old,New,OldTerm,NewTerrn) - NewTenn is the result of replacing all occurrences of Oid in OldTerm by New.
substitute(Old,New, Dld,New).
substitute(Old,New,Term,Term) -
constant(Term), Term Old.
substitute(Old,New,Term,Termi) -
compound(Term),
f unctor (Term , F , N)
functor(Terml ,F,N),
substitute(N,Old,New,Term,Terml).
substitute(N,Old,New,Term,Terml) -
N > O,
arg (N , Term , Arg)
substitute(Old,New ,Arg,Argl),
arg(N,Terml ,Argl),
Nl is N-i,
substitute(Ni3Old,New,Term,Termi).
substitute(O,Old,New,Term,Termi).
Program 9.3 A program for substituting in a term
the interesting points lurking in the code. A trace of the query substi- tute (cat ,dog, owns(j arie ,cat) , X)? is given in Figure 9.2.
The query fails to unify with the fact in Program 9.3. The second rule is also not applicable because owns(jane,cat) is not a constant.
The third substitute rule is applicable to the query. The second call of functor is interesting. Name and Arity have been instantiated to owns and 2, respectively, in the previous call of furictor, so this call builds a term that serves as the answer template to be filled in as the computation progresses. This explicit term building has been achieved by implicit uni- fication in previous Prolog programs. The call to substitute/5 succes- sively instantiates the arguments of Termi. In our example, the second argument of owns (Xi,X2) is instantiated to dog, and then Xl is mstanti- ated to jane.
The two calls to arg serve different tasks in substitute/5. The first call selects an argument, while the second call of arg instantiates an argument.
Substitution in a term is typically done by destructive assignment in conventional languages. Destructive assignment is not possible directly
171 Structure Inspection
substitute(cat,dog,owns(jane,cat) ,X)
constant(owns(jane,cat)) f
substitute (cat ,dog, owns(jane, cat) ,X)
coinpound(owns(jane ,cat))
functor(owns(jane,cat) ,F,N)
functor(X owns, 2)
substitute(2,cat,dog,owns(jane,cat),owns(X1,X2))
2>0 arg(2,owns(jane,cat) Arg)
substitute(cat ,dog, cat ,Argl)
arg (2 , owns (Xl, X2) , dog)
Nl is 2-1
substitute(1,cat,dog,owns(jarie,cat), owns(X1
1>0 arg(l,owns(jane,cat) ,Arg2)
substitute(cat ,dog,jane,Arg3)
constant (j ane)
jane cat
arg(1,owns(X1,dog) jane)
N2 is 1-1
substitute(0,cat,dog,owns(jane,cat),owns
0>0 f substitute(0,cat,dog,owns(jane,cat) ,owns
true Output: (Xowns (j ane ,dog))
gure 9.2 Tracing the substitute predicate
in Prolog. Program 9.3 typifies how Prolog handles changing data struc- tures. The new term is recursively built as the old term is being traversed, by logically relating the corresponding subterms of the terms.
Note that the order of the second arg goal and the recursive call to substitute/5 can be swapped. The modified clause for substitute/5 is logically equivalent to the previous one and gives the same result in the context of Program 9.3. Procedurally, however, they are radically different.
Another system predicate for structure inspection is a binary operator called, for historical reasons, univ. The goal Term =.. List succeeds
if List is a list whose head is the functor name of the term Term and whose tail is the list of arguments of Term. For example, the query (f a- ther(haran,lot) =. . [f ather,haran,lot])? succeeds.
Xowns (j ane,
cat)
F=owns , N2
Xowns(X1 ,X2)
Argcat Argl=dog
X2dog N1=l
,dog))
Arg2j ane
Arg3=j ane
Xljane N2=0
(jane,dog))
(jane,dog))
172 Chapter 9
subterm(Sub,Term) - Sub is a subterm of the ground term Term.
subterm(Term,Term).
subterm(Sub,Term) -
compound(Term), Term =.. [FIArgs], subterm_list(Sub,Args).
subterm_list (Sub, [Arg I Args])
subterm (Sub Arg).
subterm_list (Sub, [Arg I Args]) - subterm_list (Sub Args).
Program 9.4 Subterm defined using univ
Like functor and arg, Univ has two uses. Either it builds a term given a list, for example, (X =. . [f ather,haran,lot])? with solution X=father(haran,lot), or it builds a list given a term, for example, (f a- ther(haran, lot) =.. Xs)? with solution Xs= [f ather,haran, lot].
In general, programs written using functor and arg can also be written with Univ. Program 9.4 is an alternative definition of subterin, equivalent to Program 9.2. As in Program 9.2, an auxiliary predicate investigates the arguments; here it is subterm_list. Univ is used to access the list of arguments, Args, of which subterms are recursively found by subterm_ list.
Programs using univ to inspect structures are usually simpler. How- ever, programs written with functor and arg are in general more effi- cient than those using univ, since they avoid building intermediate struc- tures.
A neat use of univ is formulating the chain rule for symbolic differ- entiation. The chain rule states that d/dx{f(g(x)} = d/dg(x){f(g(x)J x d/dx{g(x)}. In Section 3.5, we noted that this rule could not be expressed as a single clause of a logic program as part of Program 3.30. A Prolog rule encapsulating the chain rule is
derivative (F_G_X,X ,DF*DG) -
F_G_X =.. [F,G_X],
derivative(F_G_X,G_X,DF),
derivative(G_X,X,DG).
The function F_G_X is split up by univ into its function F and argument G_X, checking that F is a function of arity i at the same time. The deriva-
173 Structure Inspection
Term =.. List - List is a list containing the ftmctor of Term followed by the arguments of Term.
Term .. [FIArgsI
functor(Term,F,N), args(O,N,Term,Args).
args(I,N,Term,[krglArgs])
I < N, Il is 1+1, arg(I1,Term,Arg), args(I1,N,Term,Args).
args(N,N,Term,[ J).
Program 9.Sa Constructing a list corresponding to a term
tive of F with respect to its argument is recursively calculated, as is the derivative of G_X. These are combined to give the solution. Univ can be defined m terms of functor and arg. Two different def-
initions are necessary, however, to cover both building lists from terms and building terms from lists. One definition does not suffice, because of errors caused by uninstantiated variables. Other system predicates are similarly precluded from flexible use.
Program 9.5a behaves correctly for building a list from a term. The functor F is found by the call to functor, and the arguments are re- cursively found by the predicate args. The first argument of args is a counter that counts up, so that the arguments will appear in order in the final list. If Program 9.5a is called with Term uninstantiated, an error will be generated because of an incorrect call of furictor.
Program 9.5b behaves correctly for constructing a term from a list. The length of the list is used to determine the number of arguments The term template is built by the call to functor, and a different variant of args is used to fill in the arguments. Program 9.5b results in an error if used to build a list, because of the goal length(Args ,N) being called with uninstantiated arguments.
9.2.1 Exercises for Section 9.2
Define a predicate occurrences (Sub,Term,N), true if N is the num- ber of occurrences of subterm Sub in Term, Assume that Term is ground.
Define a predicate position(Subterm,Term,Position), where Po- sition is a list of argument positions identifying Subterm within Term. For example, the position of X in 2'sin(X) is [2,1], since
174 Chapter 9
Term =..List - The functor of Term is the first element of the list List, and its arguments are the rest of List's elements.
Term =.. EFIArgs] -
length(Args,N), Î unctor(Term,F,N), args(Args,Term,i).
args([ArglArgs] ,Term,N) -
arg(N,Term,Arg), Nl is N+1, args(Args,Terin,Nl).
args([ ],Term,N).
length(Xs,N) - See Program 8.11.
Program 9.Sb Constructing a term corresponding to a list
sin(X) is the second argument of the binary operator "", and X is the first argument of sin(X). (Hint: Add an extra argument for Program 9.2 for subterni, and build the position list top-down.)
Rewrite Program 9.5a so that it counts down. (Hint: Use an accumu- lator.)
Define functor and arg in terms of univ. How can the programs be used?
(y) Rewrite Program 9.3 for substitute so that it uses univ.
9.3 Background
Prolog does not distinguish between object-level and meta-level type predicates. We have taken a different approach, by defining the type test predicates to work only on instantiated terms and by treating the meta- logical test predicates (e.g., var/i, discussed in Section 10.1) separately. The predicates for accessing and constructing terms, functor, arg, and
originate from the Edinburgh family The origin of =.. is in the old Prolog-10 syntax for lists, which used the operator ,. . instead of the current in lists, e.g., [a,b,c,. .Xs] instead of [a,b,cXs]. The . . on the right-hand side suggested or reminded that the right-hand side of the equality is a list.
Several of the examples in this section were adapted from O'Keefe (1983).
Exercises 9.2(1) and 9.2(u) are used in the equation solver in Chapter 23.
10 Meta-Logical Predicates
A useful extension to the expressive power of logic programs is provided by the meta-logical predicates. These predicates are outside the scope of first-order logic, because they query the state of the proof, treat variables (rather than the terms they denote) as objects of the language, and allow the conversion of data structures to goals.
Meta-logical predicates allow us to overcome two difficulties involving the use of variables encountered in previous chapters. The first difficulty is the behavior of variables in system predicates. For example, evaluating an arithmetic expression with variables gives an error. So does calling type predicates with variable arguments. A consequence of this behavior is to restrict Prolog programs to have a single use in contrast to the multiple uses of the equivalent logic programs.
The second difficulty is the accidental instantiation of variables during structure inspection. Variables need to be considered as specific objects rather than standing for an arbitrary unspecified term. In Chapter 9 we handled the difficulty by restricting inspection to ground terms only.
This chapter has four sections, each for a different class of meta-logical predicates. The first section discusses type predicates that determine whether a term is a variable. The second section discusses term com- parison. The next sections describe predicates enabling variables to be manipulated as objects. Finally, a facility is described for converting data into executable goals.
176 Chapter 10
10.1 Meta-Logical Type Predicates
The basic meta-logical type predicate is var (Term), which tests whether a given term is at present an uninstantiated variable. Its behavior is simi- lar to the type predicates discussed in Section 9.1. The query var (Term)? succeeds if Term is a variable and fails if Term is not a variable. For exam- ple, var(X)? succeeds, whereas both var(a)? and var([XXs])? fail.
The predicate var is an extension to pure Prolog programs. A table carmot be used to give all the variable names. A fact var (X) means that all instances of X are variables rather than that the letter X denotes a variable. Being able to refer to a variable name is outside the scope of first-order logic in general or pure Prolog in particular.
The predicate nonvar (Term) has the opposite behavior to var. The query nonvar (Term)? succeeds if Term is not a variable and fails if Term is a variable.
The meta-logical type predicates can be used to restore some flexibility to programs using system predicates and also to control goal order. We demonstrate this by revising some programs from earlier chapters.
Consider the relation plus (X,Y,Z). Program 10.1 is a version of plus that can be used for subtraction as well as addition. The idea is to check which arguments are instantiated before calling the arithmetic evaluator. For example, the second rule says that if the first and third arguments, X and Z, are not variables, the second argument, Y, can be determined as their difference. Note that if the arguments are not integers, the evalua- tion will fail, the desired behavior.
The behavior of Program 10.1 resembles that of Program 3.3, the logic program for plus. Further, it does not generate any errors. Nonetheless, it does not have the full flexibility of the recursive logic program: it cannot be used to partition a number mto two smaller numbers, for
plus(X,Y,Z) - The sum of the numbers X and Y is Z.
plus(X,Y,Z) -
plus(X,Y,Z) - plus(X,Y,Z) -
Program 10.1
nonvar(X),
nonvar(X),
nonvar(Y),
nonvar(Y), Z
nonvar(Z), Y
nonvar(Z), X
Multiple uses for plus
is X+Y.
is Z-X.
is Z-Y.
177 Meta-Logical Predicates
length(Xs,N) - The list Xs has length N.
length(Xs,N) nonvar(Xs), lengthl(Xs,N). length(Xs,N) - var(Xs), nonvar(N), lerigth2(Xs,N). lengthl(Xs,N) See Program 8.11. length2(Xs,N) .- See Program 8.10.
Program 10.2 A multipurpose length program
example. To partition a number involves generating numbers, for which a different program is needed. This is posed as Exercise (ii) at the end of this section.
Meta-logical goals placed initially in the body of a clause to decide which clause in a procedure should be used are called meta-logical tests. Program 10.1 for plus is controlled by meta-logical tests. These tests re- fer to the current state of the computation. Knowledge of the operational semantics of Prolog is required to understand them.
Standard Prolog in fact endows the type predicates with a meta-logical ability. For example, if X is a variable the goal integer(X) fails, rather than giving an error. This enables the rules from Program 10.1 to be writ- ten using the system predicate integer rather than nonvar, for example,
plus(X,Y,Z) - integer(X), integer(Y), Z is X+Y.
We feel it is preferable to separate type checking, which is a perfectly le- gitimate first-order operation, from meta-logical tests, which are a much stronger tool.
Another relation that can have multiple uses restored is length (Xs , N) determining the length N of a list Xs. Separate Prolog programs (8.10 and 8.11) are needed to find the length of a given list and to generate an arbitrary list of a given length, despite the fact that one logic program (3.17) performs both functions. Program 10.2 uses meta-logical tests to define a single length relation. The program has an added virtue over Programs 8.10 and 8.11. lt avoids the non-terminating behavior present in both, when both arguments are uninstantiated.
Meta-logical tests can also be used to make the best choice of the goal order of clauses in a program. Section 7.3 discusses the definition of grandparent:
178 Chapter lO
grandparent(X,Z) - X is the grandparent of Z.
grandparent(X,Z) - nonvar(X), parent(X,Y), parent(Y,Z).
grandparent(X,Z) - nonvar(Z), parent(Y,Z), parent(X,Y).
Program 10.3 A more efficient version of grandparent
ground(Term) - Term is a ground term.
ground(Term) -
nonvar (Term), constant(Term).
ground(Term) -
nonvar (Term),
compound (Term),
f unctor (Term , F , N)
ground (N , Term)
ground(N,Term) -
N > O,
arg (N , Term , Arg)
ground (Arg)
Nl is N-1,
ground(Nl ,Term).
ground(O,Term).
Program 10.4 Testing if a term is ground
grandparent(X,Z) - parent(X,Y), parent(Y,Z).
The optimum goal order changes depending on whether you are search- ing for the grandchildren of a given grandparent or the grandparents of a given grandchild. Program 10.3 is a version of grandparent that will search more efficiently.
The basic meta-logical type predicates can be used to define more in- volved meta-logical procedures. Consider a relation ground (Term), which is true if Term is ground. Program 10.4 gives a definition.
The program is in the style of the programs for structure inspection given in Section 9.2, in particular Program 9.3 for substitute. The two clauses for ground/i are straightforward. In both cases, a meta-logical test is used to ensure that no error is generated. The first clause says that constant terms are ground. The second clause deals with structures.
179 Me ta-Logical Predicates
It calls an auxiliary predicate ground/2, which iteratively checks that all the arguments of the structure are ground.
We look at a more elaborate example of using meta-logical type predi- cates; writing a unification algorithm The necessity of Prolog to support unification for matching goals with clause heads means that explicit uni- fication is readily available. Prolog's underlying unification can be used to give a trivial definition
unify (X , X)
which is the definition of the system predicate =12, namely, X=X. Note that this definition depends on Prolog's underlying mechanism
for unification, and hence does not enforce the occurs check. A more explicit definition of Prolog's unification is possible using mcta-
logical type predicates. Although more cumbersome and less efficient, this definition is useful as a basis for more elaborate unification algo- rithms One example is unification with the occurs check, described in Section 10.2. Another example is unification ni other logic programming languages that can be embedded in Prolog, such as read-only unification of Concurrent Prolog.
Program 10.5 is an explicit definition of unification. The relation unify(Ternil,Term2) is true if Termi unifies with Term2. The clauses of unify outline the possible cases. The first clause of the program says that two variables unify The next clause is an encapsulation of the rule for unification that if X is a variable, then X unifies with Y.
The other case bearing discussion in Program 10.5 is unifying two com- pound terms, as given in the predicate term_unify(X,Y). This predicate checks that the two terms X and Y have the same principal functor and arity, and then checks that all the arguments unify, using uriify_args, in a way similar to the structure inspection programs shown before.
10.1.1 Exercises for Section 10.1
Write a version of Program 8.12 for range that can be used in mul- tiple ways.
Write a version of Program 10.1 for plus that partitìons a number as well as performing addition and subtraction. (Hint: Use between to generate numbers.)
180 Chapter 10
unify(Terml,Term2) - Termi arLd Term2 are unified, ignoring the occurs check.
unify(X,Y) -
var(X), var(Y), XY.
unify(X,Y) -
var(X), nonvar(Y), X=Y.
unify(X,Y) -
var(Y), nonvar(X), YX.
unify(X,Y) -
nonvar(X), nonvar(Y), constant(X), constant(Y), X=Y.
unify(X,Y) -
nonvar(X), nonvar(Y), compound(X), compound(Y), term_unify(X,Y).
term_unify(X,Y) -
functor(X,F,N), functor(Y,F,N), unify_args(N,X,Y).
unify_args(N,X,Y) -
N > O, unify_arg(N,X,Y), Nl is N-1, unify_args(N1,X,Y).
unify_args(O,X,Y).
unify_arg(N,X,Y) -
arg(N,X,ArgX), arg(N,Y,ArgY), unify(ArgX,ArgY).
Program 10.5 Unification algorithm
10.2 Comparing Nonground Terms
Consider the problem of extending the explicit unification program, Pro- gram 10.5, to handle the occurs check. Recall that the occurs check is part of the formal definition of unification, which requires that a variable not be unified with a term containing this variable. In order to implement it in Prolog, we need to check whether two variables are identical (not just unifiable, as any two variables are). This is a meta-logical test.
Standard Prolog provides a system predicate, =/2, for this purpose. The query X == Y? succeeds if X and Y are identical constants, identical variables, or both structures whose principal functors have the same name and arity, and recursively X == Y1? succeeds for all corresponding arguments X and Y. of X and Y. The goal fails otherwise. For example, X == 5? fails (in contrast to X = 5?).
There is also a system predicate that has the opposite behavior to ==. The query X \== Y? succeeds unless X and Y are identical terms.
181 Meta-Logical Predicates
unify(Terml,Term2) - Term 1 and Term2 are unified with the occurs check.
uriify(X,Y)
var(X), var(Y), XY.
unify(X,Y)
var(X), nonvar(Y), not_occurs_in(X,Y), X=Y.
i.mify(X,Y) -
var(Y), nonvar(X), not_occurs_in(Y,X), YX.
unify(X,Y) -
nonvar(X), nonvar(Y), constant(X), constant(Y), X=Y.
unify(X,Y)
nonvar(X), nonvar(Y), compound(X), compound(Y), term_unify(X,Y).
noLoccursjn(X,Term) - The variable X does not occur in Term.
not_occurs_in(X,Y) -
var(Y), X \== Y.
not_occurs_in(X ,Y)
nonvar(Y), constant(Y).
not_occurs_in(X,Y)
nonvar(Y), compound(Y), functor(Y,F,N), not_occurs_in(N,X,Y).
not_occurs_in(N,X,Y)
N>O, arg(N,Y,Arg), not_occurs_in(X,Arg), Nl is N-1,
uot_occurs_in(N1 ,X ,Y).
not_occurs_in(O,X,Y).
term_unify(X,Y) .- See Program 10.5.
Program 10.6 Unification with the occurs check
The predicate \== can be used to define a predicate not_occurs_ in(Sub,Term), which is true if Sub does not occur in leIm, the relation that is needed in the unification algorithm with the occurs check. not_ occurs_in (Sub, Term) is a meta-logical structure inspection predicate. lt is used in Program 10.6, a variant of Program 10.5, to implement unifica- tion with the occurs check.
Note that the definition of not_occurs_in is not restricted to ground terms. Lifting the restriction on Program 9.2 for subterm is not as easy. Consider the query subterm(X,Y)?. This would succeed using Pro- gram 9.2, instantiating X to Y.
We define a meta-logical predicate occurs_in(Sub,Term) that has the desired behavior.
182 Chapter 10
occurs_in(Sub,Terrn) - Sub is a subterm of the (possibly nonground) term Term.
Using ==
occurs_in(X,Term) -
subterm(Sub,Term), X == Sub.
Using freeze
occurs_in(X,Terni) -
freeze(X,Xf), freeze(Term,Termf), subterm(Xf,Termf).
subterm(X,Term) - See Program 9.2.
Program 10.7 Occurs in
The predicate == allows a definition of occurs_in based on Pro- gram 9.2 for subterm. All the subterms of the given term are generated on backtracking and tested to see if they are identical to the variable. The code is given in Program 10.7a.
As defined, subterm works properly only for ground terms. However, by adding meta-logical type tests, as in the definition of not_occurs_in in Program 10.6, this problem is easily rectified.
10.3 Variables as Objects
The delicate handling of variables needed to define occurs_in in Sec- tion 10.2 highlights a deficiency in the expressive power of Prolog. Vari- ables are not easily manipulated. When trying to inspect, create, and reason about terms, variables can be unwittingly instantiated.
A similar concern occurs with Program 9.3 for substitute. Consider the goal substitute(a,b,X,Y), substituting a for b in a variable X to give Y. There are two plausible behaviors for substitute in this case. Logically there is a solution when X is a and Y is b. This is the solution actually given by Program 9.3, achieved by unification with the base fact substitute (OLd ,New, Old ,New).
In practice, another behavior is usually preferred. The two terms X and a should be considered different, and Y should be instantiated to X. The other base case from Program 9.3,
substitute(Old,New,Term,Term) - constant(Term), Term Old.
183 Meta-Logical Predicates
covers this behavior. However, the goal would fail because a variable is not a constant.
We can prevent the first (logical) solution by using a meta-logical test to ensure that the term being substituted in is ground. The unification implicit in the head of the clause is then only performed if the test succeeds, and so must be made explicit. The base fact becomes the rule
substitute(Old,New,Terni,New) - ground(Terni), Old = Term.
Treating a variable as different from a constant is handled by a special rule, again relying on a meta-logical test:
substitute(Old,New,Var,Var) - var(Var).
Adding the two preceding clauses to Program 9.3 for substitute and adding other meta-logical tests allows the program to handle nonground terms. However, the resultant program is inelegant. It is a mixture of procedural and declarative styles, and it demands of the reader an under- standing of Prolog's control flow. To make a medical analogy, the symp- toms have been treated (undesirable instantiation of variables), but not the disease (inability to refer to variables as objects). Additional meta- logical primitives are necessary to cure the problem.
The difficulty of mixing object-level and meta-level manipulation of terms stems from a theoretical problem. Strictly speaking, meta-level programs should view object-level variables as constants and be able to refer to them by name.
We suggest two system predicates, f reeze(Term,Frozen) and melt (Frozen, Thawed), to allow explicit manipulation of variables. Freezing a term Term makes a copy of the term, Frozen, where all the uninstantiated variables in the term become unique constants. A frozen term looks like, and can be manipulated as, a ground term.
Frozen variables are regarded as ground atoms during unification. Two frozen variables unify if and only if they are identical Similarly, if a frozen term and an uninstantiated variable are unified, they become an identical frozen term. The behavior of frozen variables in system predi- cates is the behavior of the constants. For example, arithmetic evaluation involving a frozen variable will fail.
The predicate freeze is meta-logical in a similar sense to var. It en- ables the state of a term during the computation to be manipulated di- rectly.
184 Chapter 10
The predicate freeze allows an alternative definition of occurs_in from the one given m Section 10.2. The idea is to freeze the term so that variables become ground objects. This makes Program 9.2 for subterm, which works correctly for ground terms, applicable. The definition is given as Program 10.7b.
Freezing gives the ability to tell whether two terms are identical. Two frozen terms, X and Y, unify if and only if their unfrozen versions are identical, that is, X == Y. This property is essential to the correct behav- ior of Program 10.7b.
The difference between a frozen term and a ground term is that the frozen term can be "melted back" into a nonground term. The compan- ion predicate to freeze is melt(Frozen,Thawed). The goal melt(X,Y) produces a copy Y of the term X where frozen variables become regular Prolog variables. Any instantiations to the variables in X during the time when X has been frozen are taken into account when melting Y.
The combination of freeze and melt allows us to write a variant of substitute, non_ground_substitute, where variables are not acciden- tally instantiated. The procedural view of non_ground_substitute is as follows. The term is frozen before substitution; the substitution is per- formed on the frozen term using the version of substitute, which works correctly on ground terms; and then the new term is melted:
non_ground_substitute(X,Y,Old,New) -
freeze(Dld,Oldl), substitute(X,Y,Oldl,01d2),
melt(01d2,New).
The frozen term can also be used as a template for making copies. The system predicate melt_new (Frozen,Term) makes a copy Term of the term Frozen, where frozen variables are replaced by new variables.
One use of melt_new is to copy a term. The predicate copy(Term,Copy) produces a new copy of a term. It can be defined in a single rule:
copy(Term,Copy) - f reeze(Term,Frozen), melt_new(Frozen, Copy).
Standard Prolog provides the predicate copy_term(Terml ,Term2) for copying terms. It is true if and only if Term2 unifies with a term T that is a copy of Termi except that all the variables of Termi have been replaced by fresh variables.
Unfortunately, the predicates freeze/2, melt/2, and melt_new/2 as described here are not present in existing Prolog implementations. They
185 Meta-Logical Predicates
numbervars(Term,N1,N2) The variables in Term are numbered from Nl to N2 - 1.
nuinbervars('$VAR' (N) ,N,N1) - Nl is N+1.
nunibervars (Term, N, N) - norivar(Term), constant(Temm).
nunibervars(Term,N1 ,N2) - norivar(Term), compound(Temm), f unctor (Term , Name , N) nuxnbervars (O ,N , Term , Nl, N2)
nuinbervars (N ,N Term, Nl, Nl) nuinbervars(I,N,Term,N1,N3)
I <N Il is 1+1, arg(Il,Term,Arg), nuinbervars(Arg,N1 ,N2), nuinbervars(I1,N,Term,N2,N3).
Program 10.8 Numbering the variables in a term
will be useful nonetheless in expressing and explaining the behavior of extra-logical predicates, discussed in Chapter 12.
A useful approximation to freeze is the predicate nunbervars (Term, Nl ,N2), which is provided in many Edinburgh Prolog libraries. A call to the predicate is true if the variables appearing in Term can be numbered from Nl to N2-1. The effect of the call is to replace each variable in the term by a term of the foriii '$VAR' (N) where N lies between Nl and N2. For example, the goal numbervars(append([XIXs] ,Ys, [XIZs] ,1,N) suc- ceeds with the substitution {X='$VAR(l)', Xs'$VAR' (2), Ys='$VAR' (3), Zs='$VAR' (4), N=5}. Code implementing numbervars is given as Program 10.8. lt is in the same style as the structure inspection utilities given in Chapter 9.
10.4 The Meta-Variable Facility
A feature of Prolog is the equivalence of programs and data - both can be represented as logical terms. In order for this to be exploited, programs need to be treated as data, and data must be transformed into programs. In this section, we mention a facility that allows a term to be
186 Chapter 10
X;Y - X or Y.
X Y - X. X Y - Y.
Program 10.9 Logical disjunction
converted into a goal. The predicate call(X) calls the goal X for Prolog to solve.
In practice, most Prolog implementations relax the restriction we have imposed on logic programs, that the goals in the body of a clause must be nonvariable terms. The me ta-variable facility allows a variable to ap- pear as a goal in a conjunctive goal or in the body of the clause. During the computation, by the time it is called, the variable must be instan- tiated to a term. It will then be treated as usual. If the variable is not instantiated when it comes to be called, an error is reported. The meta- variable facility is a syntactic convenience for the system predicate call.
The meta-variable facility greatly facilitates meta-prograrnrning, in par- ticular the construction of meta-interpreters and shells. Two important examples to be discussed in later chapters are Program 12.6, a simple shell, and Program 17.5, a meta-interpreter. It is also essential for defin- ing negation (Program 11.6) and allowing the definition of higher-order predicates to be described in Section 16.3.
We give an example of using the meta-variable facility with a definition of logical disjunction, denoted by the binary infix operator "; ". The goal (X;Y) is true if X or Y is true. The definition is given as Program 10.9.
10.5 Background
An excellent discussion of meta-logical system predicates in DEC-10 Pro- log, and how they are used, can be found in O'Keefe (1983).
The unification procedure for Concurrent Prolog, written in Prolog, is in Shapiro (1983b).
The difficulty in correctly manipulating object-level variables in Prolog at the meta-level has been raised by several people. The discussion first extensive discussion is in Nakashima et al. (1984), where the predicates freeze, melt, and melt_new are introduced. The name freeze was a little
187 Meta-Logica! Predicates
unfortunate, as it has been suggested for other additions to pure Prolog. Most notable is Colmerauer's geler (Colmerauer, 1982a), which allows the suspension of a goal and gives the programmer more control over goal order. This predicate is provided by Sicstus Prolog as freeze. The discussion of Nakashima and colleagues, although publicized in the first editon of this book, was largely ignored, to be revived by Barklund (1989) musing over "What is a variable in Prolog?" and by attempts to do meta- programming in constraint logic programming languages, for example, Heintze et aI. (1989) and [im and Stuckey (1990).
The Gödel project (Hill and Uoyd, 1993) has advocated replacing Pro- log by a language that facilitates explicit manipulation of variables at a meta-level. In Lloyd and Hill (1989), the terms ground and nonground representation are used. Prolog uses a nonground representation, and adding freeze and nuinbervars allows a ground representation.
11.1. Green Cuts: Expressing Determinism
Consider the program merge(Xs,Ys,Zs) (Program 11.1), which merges two sorted lists of numbers Xs and Ys into the combined sorted list Zs.
Merging two lists of sorted numbers is a deterministic operation. Only one of the five merge clauses applies for each nontrivial goal in a given computation. To be more specific, when comparing two numbers X and Y, for example, only one of the three tests X < Y, X = : = Y, and X > Y can be true. Once a test succeeds, there is no possibility that any other test will succeed.
11 Cuts and Negation
Prolog provides a single system predicate, called cut, for affecting the procedural behavior of programs. Its main function is to reduce the search space of Prolog computations by dynamically pruning the search tree. The cut can be used to prevent Prolog from following fruitless com- putation paths that the programmer knows could not produce solutions.
The cut can also be used, inadvertently or purposefully, to pi-une com- putation paths that do contain solutions. By doing so, a weak form of negation can be effected.
The use of cut is controversial. Many of its uses can only be inter- preted procedurally, in contrast to the declarative style of programming we encourage. Used sparingly, however, it can improve the efficiency of programs without compromising their clarity.
190 Chapter 11
merge(Xs,Ys,Zs) - Zs is an ordered list of integers obtained from merging the ordered lists of integers Xs and Ys.
merge([XJXs], [YIYs] , [XIZs]) - X < Y, merge(Xs, [YIYs] ,Zs).
merge([XIXs], [YIYs] , [X,YZs]) - X=:=Y, merge(Xs,Ys,Zs).
merge([XXs], [YIYs] , [YIZs]) - X > Y, merge([XIXs] ,Ys,Zs).
merge(Xs, E ] ,Xs).
merge([ ],Ys,Ys).
Program 11.1 Merging ordered lists
The cut, denoted !, can be used to express the mutually exclusive nature of the tests. It is placed after the arithmetic tests. For example, the first merge clause is written
merge([XIXs],[YIYs],[XIZs]) x < Y, !, merge(Xs,[Y IYs],Zs).
Operationally, the cut is handled as follows. The goal succeeds and commits Prolog to all the choices made since the
parent goal was unified with the head of the clause the cut occurs in. Although this definition is complete and precise, its ramifications and
implications are not always intuitively clear or apparent. Misunderstandings concerning the effects of a cut are a major source
for bugs for experienced and inexperienced Prolog programmers alike. The misunderstandings fall into two categories: assuming that the cut prunes computation paths it does not, and assuming that it does not prune solutions where it actually does.
The following implications may help clarify the foregoing terse defini- tion:
First, a cut prunes all clauses below it. A goal p unified with a clause containing a cut that succeeded would not be able to produce solutions using clauses that occur below that clause. Second, a cut prunes all alternative solutions to the conjunction of goals that appear to its left in the clause. For example, a conjunctive goal followed by a cut will produce at most one solution. On the other hand, the cut does not affect the goals to its right in the clause. They can produce more than one solution in the event of backtracking. However, once this conjunction fails, the search proceeds
191 Cuts and Negation
i =:2,Imerge([3,5],[3],Xsi
Figure 11.1 The effect of cut
from the last alternative prior to the choice of the clause containing the cut.
Let us consider a fragment of the search tree of the query merge ([1 3, 5] , [2,3] ,Xs)? with respect to Program 112, a version of merge with cuts added. The fragment is given as Figure 11.1. The query is first re- ducedtotheconjunctivequeryl 2,!,merge([3,5],[2,3],Xsl)?;the goal i < 2 is successfully solved, reaching the node marked (*) in the search tree. The effect of executing the cut is to prune the branches marked (a) and (b).
Continuing discussion of Program 11.2, the placement of the cuts in the three recursive clauses of merge is after the test.' The two base cases of merge are also deterministic. The correct clause is chosen by unifica- tion, and thus a cut is placed as the first goal (and in fact the only goal) in the body of the rule. Note that the cuts eliminate the redundant solution to the goal merge ( [ J , L J , Xs). Previously, this was accomplished more awkwardly, by specifying that Xs (or Ys) had at least one element.
1. The cut after the third merge clause is unliecessary in any practica! sense. Proce- durally, it will not cause any reduction of search. But it makes the program more symmetric, and like the old joke says about chicken soup, it doesn't hurt.
192 Chapter 11
merge(Xs,Ys,Zs) - Zs is an ordered list of integers obtained from merging the ordered lists of integers Xs and Ys.
merge([XIXs],[YIYs],[XIZs]) - X < Y, !, merge(Xs,[YIYs],Zs).
merge([XXs] ,[YIYs] , [X,YIZs]) -
X=:=Y, !, merge(Xs,Ys,Zs).
merge([XIXs] , [Y lYs], [YIZs]) - X > Y, !, merge([XIXs],Ys,Zs).
merge(Xs, E ] ,Xs) -
merge([ ],Ys,Ys) - L
Program 11.2 Merging with cuts
We restate the effect of a cut in a general clause C = A - B1,. . . ,Bk,!, Bk+2, . . . ,B in a procedure defining A. If the current goal G unifies with the head of C, and B1,. . .,B further succeed, the cut has the following effect. The program is committed to the choice of C for reducing G; any alternative clauses for A that might unify with G are ignored. Further, should B fail for i> k + 1, backtracking goes back only as far as the L Other choices remaining in the computation of B, i k, are pruned from the search tree. If backtracking actually reaches the cut, then the cut fails, and the search proceeds from the last choice made before the choice of G to reduce C.
The cuts used in the merge program express that merge is determinis- tic. That is, only one of the clauses can be used successfully for proving an applicable goal. The cut commits the computation to a single clause, once the computation has progressed enough to determine that this is the only clause to be used.
The information conveyed by the cut prunes the search tree, and hence shortens the path traversed by Prolog, which reduces the computation time. In practice, using cuts in a program is even more important for saving space. Intuitively, knowing that a computation is deterministic means that less information needs to be kept for use in the event of backtracking. This can be exploited by Prolog implementations with tail recursion optimization, discussed in Section 11.2.
Let us consider some other examples. Cuts can be added to the pro- gram for computing the minimum of two numbers (Program 3.7) in pre- cisely the same way as for merge. Once an arithmetic test succeeds, there
193 Cuts and Negation
minimum(X,Y,Min) - Min is the minimum of the numbers X and Y.
minimuin(X,Y,X) - XY, L minimuin(X,Y,Y) - X > Y, L
Program 11.3 minimum with cuts
polynomial ( Term,X) - Term is a poiynornial in X.
polynomial(X,X) - L
polynomial(Term,X) -
constant (Term),
polynomial (Terml+Term2 , X)
polynomial (Terni, X), polynomial(Term2,X).
polynomial (Terml-Term2 , X)
!, polynomial(Terml,X), polynomial(Term2,X).
polynomial (Terml*Term2 ,X)
polynomial(Termi,X), polynomial (Term2,X).
polynomial (Terml/Term2 , X)
polynomial (Terml,X) constant (Term2)
polynomial(TermIN,X) -
!, integer(N), N > O, polynomial (Term,X).
Program 11.4 Recog zing polynomials
is no possibility for the other test succeeding. Program 11.3 is the appro- priately modified version of minimum.
A more substantial example where cuts can be added to indicate that a program is deterministic is provided by Program 3.29. The program defines the relation polynomial(Term,X) for recognizing if Term is a polynomial in X. A typical rule is
polynomial (Ternil+Term2 , X) -
polynomial(Terml,X), polynomial(Term2,X).
Once the term being tested has been recognized as a sum (by unifying with the head of the rule), it is known that none of the other polynomial rules will be applicable. Program 11.4 gives the complete polynomial program with cuts added. The result is a deterministic program that has a mixture of cuts after conditions and cuts after unification.
194 Chapter 11
When discussing the Prolog programs for arithmetic, which use the un- derlying arithmetic capabilities of the computer rather than a recursive logic program, we argued that the increased efficiency is often achieved at the price of flexibility. The logic programs lost their multiple uses when expressed as Prolog programs. Prolog programs with cuts also have less flexibility than their cut-free equivalents. This is not a problem if the intended use of a program is one-way to begin with, as is often the case.
The examples so far have demonstrated pruning useless alternatives for the parent goal. We give an example where cuts greatly aid efficiency by removing redundant computations of sibling goals. Consider the re- cursive clause of an interchange sort program:
sort(Xs,Ys) -
append(As, [X,YIBs] ,Xs),
X > Y,
append(As, [Y,XIBs] ,Xsl),
sort (Xsl,Ys).
The program searches for a pair of adjacent elements that are out of order, swaps them, and continues until the list is ordered. The base clause is
sort(Xs,Xs) - ordered(Xs).
Consider a goal sort([3,2, 1] ,Xs). This is sorted by swapping 3 and 2, then 3 and 1, and finally 2 and i to produce the ordered list [1 ,2,3]. It could also be sorted by first swapping 2 and 1, then swapping 3 and 1, and finally swapping 3 and 2, to arrive at the same solution. We know there is only one sorted list. Consequently there is no point in searching for another alternative once an interchange is made. This can be indi- cated by placing the cut after the test X > Y. This is the earliest it is known that an interchange is necessary. The interchange sort program with cut is given as Program 11.5.
The addition of cuts to the programs described in this section does not alter their declarative meaning; all solutions to a given query are found. Conversely, removing the cuts should similarly not affect the meaning of the program. Unfortunately, this is not always the case. A distinction has been made in the literature between green cuts and red cuts. Green cuts have been considered in this section. The addition and removal of green cuts from a program do not affect the program's meaning. Green cuts
195 Cuts and Negation
sort(Xs,Ys) Ys is an ordered permutation of the list of integers Xs.
sort(Xs,Ys)
append(As, [X,YIBs] ,Xs),
X > Y,
append(As, [Y,XIBs] ,Xsl),
sort (Xsl,Ys)
sort(Xs,Xs) -
ordered(Xs)
ordered(Xs) See Program 3.20.
Program 11.5 Interchange sort
prune on'y computation paths that do not lead to new solutions. Cuts that are not green are red.
The cut interacts with system predicates such as call and ;, intro- duced in Chapter 10, and with predicates such as riot, introduced later in this chapter. The question is what scope should cut have, that is, which choice points should be affected. Since such tricky uses of cut are not presented or advocated in this book, we defer discussion of the scope of cut until Chapter 17 on interpreters,
Exercises for Section 11.1
Add cuts to the partition program from quicksort, Program 3.22.
Add cuts to the differentiation program, Program 3.30.
(il) Add cuts to the insertion sort program, Program 3.21.
11.2 Tail Recursion Optimization
As noted in Section 8.3, the main difference from a performance point of view between recursion and iteration is that recursion requires, in general, space linear in the number of recursive calls to execute, whereas
196 Chapter 11
iteration can be executed in constant space, independent of the number of iterations performed.
Recursive programs defined free of side effects might be considered more elegant and pleasing than their iterative counterparts defined in terms of iteration and local variables. However, an order of magnitude in space complexity seems an unacceptable price for such aesthetic plea- sures. Fortunately, there is a class of recursive programs, precisely those that can be translated directly into iterative ones, that can be executed in constant space.
The implementation technique that achieves this space saving is called tail recursion optimization, or more precisely, last call optimization. Intu- itively, the idea of tail recursion optimization is to execute a recursive program as if it were an iterative one.
Consider the reduction of a goal A using the clause
A' - B1,B2, . . . ,B.
with most general unifier 9. The optimization is potentially applicable to the last call in the body of a clause, B. It reuses the area allocated for the parent goal A for the new goal B.
The key precondition for this optimization to apply is that there be no choice points left from the time the parent goal A reduced to this clause to the time the last goal B is reduced. In other words, A has no alternative clauses for reduction left, and there are no choice points left in the computation of goals to the left of B, namely, the computation of the conjunctive goal (B1,B2,. . .,B1)O, was deterministic.
Most implementations of tail recursion optimization can recognize to a limited extent at runtime whether this condition occurs, by comparing backtracking-related information associated with the goals B and A. An- other implementation technique, clause indexing, also interacts closely with tail recursion optimization and enhances the ability of the imple- mentation to detect that this precondition occurs. Indexing performs some analysis of the goal, to detect which clauses are applicable for reduction, before actually attempting to do the unifications. Typically, indexing is done on the type and value of the first argument of the goal.
Consider the append program:
append([XIXs] ,Ys, [XIZs]) - append(Xs,Ys,Zs).
append( [ J ,Ys ,Ys).
197 Cuts and Negation
If it is used to append two complete lists, then by the time the recursive append goal is executed, the preconditions for tail recursion optimiza- tion hold. No other clause is applicable to the parent goal (if the first argument unifies with [XXs], it certainly won't unify with L], since we assumed that the first argument is a complete list). There are no other goals in the body besides append, so the second precondition holds vac- uously.
However, for the implementation to know that the optimization ap- plies, it needs to know that the second clause, although not tried yet, is not applicable. Here indexing comes into play. By analyzing the first argument of append, it is possible to know that the second clause would fail even before trying it, and to apply the optimization in the recursive call to append.
Not all implementations provide indexing, and not all cases of deter- minism can be detected by the indexing mechanisms available. Therefore it is in the interest of the programmer to help an implementation that supports tail recursion optimization to recognize that the preconditions for applying it hold.
There is a sledgehammer technique for doing so: Add a cut before the last goal of a clause, in which tail recursion optimization should always apply, as in
A1 - B1,B2, . . . ,!,B.
This cut prunes both alternative clauses left for the parent goal A, and any alternatives left for the computation of (B1,B2,. .
In general, it is not possible to answer if such a cut is green or red, and the programmer's judgment should be applied.
It should be noted that the effect of tail recursion optimization is en- hanced greatly when accompanied with a good garbage collector. Stated negatively, the optimization is not very significant without garbage col- lection. The reason is that most tail recursive programs generate some data structures on each iteration. Most of these structures are tempo- rary and can be reclaimed (see, for instance, the editor in Program 12.5). Together with a garbage collector, such programs can run, in principle, forever. Without it, although the stack space they consume would remain constant, the space allocated to the uncollected temporary data struc- tures would overflow.
198 Chapter 11
notX - X is not provable.
not X - X, !, fail. not X.
Program 11.6 Negation as failure
11.3 Negation
The cut can be used to implement a version of negation as failure. Pro- gram 11.6 defines a predicate not (Goal), which succeeds if Goal fails. As well as using cut, the program uses the meta-variable facility described in Chapter 10, and a system predicate fail that always fails.
Standard Prolog provides a predicate fail_if (Goal), which has the same behavior as not/i. Other Prologs provide the same predicate under the name \+/i. The rationale for not calling the system predicate not is that the predicate does not implement true logical negation, and it is misleading to label it as such. We believe that the user easily learns how the predicate differs from true negation, as we will explain, and programmers are helped rather than misled by the name.
Let us consider the behavior of Program 11.6 in answering the query not G? The first rule applies, and G is called using the meta-variable facility. If G succeeds, the cut is encountered. The computation is then committed to the first rule, and not G fails. If the call to G fails, then the second rule of Program 11.6 is used, which succeeds. Thus not G fails if G succeeds and succeeds if G fails.
The rule order is essential for Program 11.6 to behave as intended. This introduces a new, not entirely desirable, dimension to Prolog programs. Previously, changing the rule order only changed the order of solutions. Now the meaning of the program can change. Procedures where the rule order is critical in this sense must be considered as a single unit rather than as a collection of individual clauses.
The termination of a goal not G depends on the termination of G. If G terminates, so does not G. If G does not terminate, then not G may or may not terminate depending on whether a success node is found in the search tree before an infinite branch. Consider the following nontermi- nating program:
199 Cuts and Negation
married(abraham, sarah).
married(X,Y) married(Y,X).
The query not married(abrahaxn,sarah)? terminates (with failure) even though married (abraham, sarah)? does not termrnate.
Program 11.6 is incomplete as an implementation of negation by fail- ure. The incompleteness arises from Prolog's incompleteness in realizing the computation model of logic programs. The definition of negation as failure for logic programs is in terms of a finitely failed search tree. A Prolog computation is not guaranteed to find one, even if it exists. There are goals that could fail by negation as failure, that do not terminate un- der Prolog's computation rule. For example, the query not (p (X) ,q(X))? does not terminate with respect to the program
p(s(X)) p(X). q(a).
The query would succeed if the q(X) goal were selected first, since that gives a finitely failed search tree.
The incorrectness of Program 11.6 stems from the order of traver- sal of the search tree and arises when not is used in conjunction with other goals. Consider using not to define a relationship unniarried_ student(X) for someone who is both not married and a student, as in the following program:
unmarried_student(X) not married(X), student(X).
student (bill).
married(joe).
The query unmarried_student (X)? fails with respect to the preceding data, ignoring that X=bill is a solution logically implied by the rule and two facts. The failure occurs in the goal not married(X), since there is a solution X=j 0e. The problem can be avoided here by swapping the order of the goals in the body of the rule.
A similar example is the query not (X=1), X=2?, which fails although there is a solution X=2.
The implementation of negation as failure is not guaranteed to work correctly for nonground goals, as the foregoing examples demonstrate. In most implementations of Prolog, it is the responsibility of the pro- grammer to ensure that negated goals are ground before they are solved.
200 Chapter ii
variants ( Termi , Term2) - Tenni and Term2 are variants.
variants(Terxnl ,Term2) -
verify ( (numbervars (Termi 0, N),
numbervars (Term2 0, N),
Term 1=Term2)).
verify(Goal) - Goal has a true instance. Verifying this is not done constructively, so variables are not instantiated in the process.
verify(Goal) - not(not Goal).
numbervars(Term,N,N1) - See Program 10.8.
Program 11.7 Testing if terms are variants
This can be done either by a static analysis of the program or by a run- time check, using the predicate ground defined in Program 10.4.
The predicate not is very useful. It allows us to define interesting con- cepts. For example, consider a predicate disjoint(Xs,Ys), true if two lists Xs and Ys have no elements in common. It can be defined as
disjoint(Xs,Ys) not (meniber(Z,Xs), member(Z,Ys)).
Many other examples of using not will appear in the programs through- out this book.
An interesting property of not (Goal) is that it never instantiates the arguments in Goal. This is because of the explicit failure after the call to Goal succeeds, which undoes any bindings made. This property can be exploited to define a procedure verify(Goal), given as part of Pro- gram 11.7, which determines whether a goal is true without affecting the current state of the variable bindings. Double negation provides the means.
We note in passing that negation as implemented in Prolog shares a feature with negation in natural language. A doubly negated statement is not the same as the equivalent affirmative statement.
The program for verify can be used in conjunction with Program 10.8 for numbervars to define a notion of equality intermediate between uniti- ability provided by =/2 and syntactic equality provided by ==/2. The predicate variants(X,Y) defined in Program 11.7 is true if two terms X and Y are variants. Recall from Chapter 4 that two terms are variants
201 Cuts and Negation
x y - X and Y are not unifiable.
X X - ', fail.
X Y.
Program 11.8 Implementing
if they are instances of each other. This can be achieved with the follow- ing trick, implemented in Program 11.7. Instantiate the variables using numbervars, test whether the terms unify, and undo the instantiation.
The three forms of comparison =12, variant/2, arid ==/2 are pro- gressively stronger, with unifiability being the weakest and most general. Identical terms are variants, and variant terms are unifiable The distinc- tion between the different comparisons vanishes for ground terms; for ground terms all three comparisons return the same results.
The conjunction of cut and fail used in the first clause of not in Pro- gram 11.6 is known as the cut-fail combination. The cut-fail combination is a technique that can be used more generally. It allows early failure. A clause with a cut-fail combination says that the search need not (and will not) proceed.
Some cuts in a cut-fail combination are green cuts. That is, the program has the same meaning if the clause containing the cut-fail combination is removed. For example, consider Program 10.4 defining the predicate ground. An extra clause can be added, which can reduce the search with- out affecting the meaning:
ground(Term) - var(Term), !, fail.
The use of cut in Program 11.6 implementing not is riot green, but red. The program does not behave as intended if the cut is removed.
The cut-fail combination is used to implement other system predi- cates involving negation. For example, the predicate (written as \= in Standard Prolog) can be simply implemented via unification and cut-fail, rather than via an infinite table, with Program 11.8. This program is also only guaranteed to work correctly for ground goals.
With ingenuity, and a good understanding of unification and the ex- ecution mechanism of Prolog, interesting definitions can be found for many meta-logical predicates. A sense of the necessary contortions can
202 Chapter 11
be found in the program for same_var(X,Y), which succeeds if X and Y are the same variable and otherwise fails:
same_var(foo,Y) var(Y), , fail.
same_var(X,Y) var(X), var(Y).
The argument for its correctness follows: "If the arguments to samevar are the same variable, binding X to f oo will bind the second argument as well, so the first clause will fail, and the second clause will succeed. If either of the arguments is not a variable, both clauses will fail. If the arguments are different variables, the first clause will fail, but the cut stops the second clause from being considered."
Exercises for Section 1L3
Define the system predicate \== using == and the cut-fail combina- tion.
Define nonvar using var and the cut-fail combination.
11.4 Red Cuts: Omitting Explicit Conditions
Prolog's sequential choice of rules and its behavior in executmg cut are the key features necessary to compose the program for not. The pro- grammer can take into account that Prolog will only execute a part of the procedure if certain conditions hold. This suggests a new, and mis- guided, style of programming in Prolog, where the explicit conditions governing the use of a rule are omitted.
The prototypical (bad) example in the literature is a modified version of Program 11.3 for minimum. The comparison in the second clause of the program can be discarded to give the program
minimum(X,Y,X) XY, minimum(X,Y,Y).
The reasoning offered to justify the program is as follows: "If X is less than or equal to Y, then the minimum is X. Otherwise the minimum is Y, and another comparison between X and Y is unnecessary." Such a comparison is performed, however, by Program 11.3.
203 Cuts and Negation
There is a severe flaw with this reasoning. The modified program has a different meaning from the standard program for minimum. It succeeds on the goal miriimuxn(2,5,5). The modified program is a false logic pro- gram.
The incorrect minimum goal implied by the modified program can be avoided. It is necessary to make explicit the unification between the first and third arguments, which is implicit in the first rule. The modified rule is
minimum(X,Y,Z) - XY, !, Z=X.
This technique of using the cut to cornniit to a clause after part of the unification has been done is quite general. But for minimum the resultant code is contrived. lt is far better to simply write the correct logic pro- gram, adding cuts if efficiency is important, as done in Program 11.3.
Using cut with the operational behavior of Prolog in mind is problem- atic. lt allows the writing of Prolog programs that are false when read as logic programs, that is, have false conclusions but behave correctly because Prolog is unable to prove the false conclusions. For example, if minimum goals are of the form minimum (X, Y, Z), where X and Y are instan- tiated, but Z is not, the modified program behaves correctly.
The only effect of the green cuts presented in Section 11.1 is to prune from the search tree branches that are known to be useless. Cuts whose presence in a program changes the meaning of that program are called red cuts. The removal of a red cut from a program changes its meaning, i.e., the set of goals it can prove.
A standard Prolog programming technique using red cuts is the omis- sion of explicit conditions. Knowledge of the behavior of Prolog, specifi- cally the order in which rules are used in a program, is relied on to omit conditions that could be inferred to be true. This is sometimes essen- tial in practical Prolog programming, since explicit conditions, especially negative ones, are cumbersome to specify and inefficient to run. But mak- ing such omissions is error-prone.
Omitting an explicit condition is possible if the failure of the previous clauses implies the condition. For example, the failure of the comparison XY in the minimum code implies that X is greater than Y. Thus the test X > Y can be omitted. In general, the explicit condition is effectively the negation of the previous conditions. By using red cuts to omit conditions, negation is being expressed implicitly.
204 Chapter 11
delete(Xs,X,Ys) - Ys is the result of deleting all occurrences of X from the list Xs.
delete([XIXs],X,Ys) - !, delete(Xs,X,Ys).
delete([XIXs],Z,[XIYs]) - X Z, !, delete(Xs,Z,Ys).
delete([ ],X,[ 3).
Program 11.9a Deleting elements from a list
delete(Xs,X,Ys) - Ys is the result of deleting all occurrences of X from the list Xs.
delete([XIXs],X,Ys) , delete(Xs,X,Ys).
delete([XIXs],Z,[XIYs]) - delete(Xs,Z,Ys).
delete([ ],X,[ 3).
Program 11.9b Deleting elements from a list
Consider Program 11.5 for interchange sort. The first (recursive) rule applies whenever there is an adjacent pair of elements in the list that are out of order. When the second sort rule is used, there are no such pairs and the list must be sorted. Thus the condition ordered(Xs) can be omitted, leaving the second rule as the fact sort(Xs,Xs). As with minimum, this is an incorrect logical statement.
Once the ordered condition is removed from the program, the cut changes from green to red. Removing the cut from the variant without the ordered condition leaves a program that gives false solutions.
Let us consider another example of omitting an explicit condition. Con- sider Program 3.18 for deleting elements in a list. The two recursive clauses cover distinct cases, corresponding to whether or not the head of the list is the element to be deleted. The distinct nature of the cases can be indicated with cuts, as shown in Program 11.9a.
By reasoning that the failure of the first clause implies that the head of the list is not the same as the element to be deleted, the explicit inequality test can be omitted from the second clause. The modified program is given as Program 11.9b. The cuts in Program 11.9a are green in comparison to the red cut in the first clause of Program 11 .9b.
In general, omitting simple tests as in Program 11.9b is inadvisable. The efficiency gain by their omission is minimal compared to the loss of readability and modifiability of the code.
205 Cuts and Negation
if_then_e!se(P,QR) Either P and Q or not P and R.
if_then_else(P,Q,R) P, , Q. if_then_else(P,Q,R) - R.
Program 11.10 If-then-else statement
Let us investigate the use of cut to express the if-then-else control structure. Program 11.10 defines the relation if _then_else(P,Q,R). Declaratively, the relation is true if P and Q are true, or not P and R are true. Operationally, we prove P and, if successful, prove Q, else prove R.
The utility of a red cut to implement this solution is self-evident. The alternative to using a cut is to make explicit the condition under which R is run. The second clause would read
if_then_else(P,Q,R) - not P, R.
This could be expensive computationally. The goal P will have to be com- puted a second time in the determination of not.
We have seen so far two kinds of red cuts. One küid is built into the program, as m the definitions of not and . A second kind was a green cut that became red when conditions in the programs were removed. However, there is a third kind of red cut. A cut that is introduced into a program as a green cut that just improves efficiency can turn out to be a red cut that changes the program's meaning.
For example, consider trying to write an efficient version of member that does not succeed several times when there are multiple copies of an element in a list. Taking a procedural view, one might use a cut to avoid backtracking once an element is found to be a member of a list. The corresponding code is
member(X,[XIXs]) - I. member(X,[YIYs]) - member(X,Ys).
Adding the cut indeed changes the behavior of the program. However, it is now not an efficient variant of member, since, for example, the query member(X, [1,2,3])? gives only one solution, X=1. It is a variant of member_check, given as Program 7.3, with the explicit condition X Y omitted, and hence the cut is red.
206 Chapter 11
Exercises for Section 11.4
Discuss where cuts could be placed in Program 9.3 for substi- tute. Consider whether a cut-fail combination would be useful, and whether explicit conditions can be omitted.
Analyze the relation between Program 3.19 for select and the pro- gram obtained by adding a single cut:
select(X,[XIXs],Xs) - !.
select(X, [YIYs], [YIZs]) - select(X,Ys,Zs)
(Hint: Consider variants of select.)
115 Default Rules
Logic programs with red cuts essentially consist of a series of special cases and a default rule. For example, Program 11.6 for not had a special case when the goal G succeeded and a default fact not G used otherwise. The second rule for if_then_else in Program 11.10 is
if_then_else(P,Q,R) - R.
It is used by default if P fails. Using cuts to achieve default behavior is in the logic progranmiing
folklore. We argue, using a simple example, that often it is better to compose an alternative logical formulation than to use cuts for default behavior.
Program 11.11 a is a naive program for determining social welfare pay- ments. The relation pension(Person,Pension) determines which pen- sion, Pension, a person, Person, is entitled to. The first pension rule says that a person is entitled to an invalid's pension if he is an invalid. The second rule states that people over the age of 65 are entitled to an old age pension if they have contributed to a suitable pension scheme long enough, that is, they must be paid_up. People who are not paid up are still entitled to supplementary benefit if they are over 65.
Consider extending Program 11.11 a to include the rule that people re- ceive nothing if they do not qualify for one of the pensions. The proce- dural "solution" is to add cuts after each of the three rules, and an extra default fact
207 Cuts and Negation
pension (X, nothing).
This version is given as Program 11.11b. Program liJib behaves correctly on queries to determine the pension
to which people are entitled, for example, pension(mc_tavish,X)?. The program is not correct, though. The query pension(mc_tavish,noth- ing)? succeeds, which mc_tavish wouldn't be too happy about, and pension(X,old_age_perision)? has the erroneous unique answer X=mc_ tavish. The cuts prevent alternatives being found. Program 11.IIb only works correctly to determine the pension to which a given person is entitled.
A better solution is to introduce a new relation erititlement(X,Y), which is true if X is entitled to Y. lt is defined with two rules and uses Program lilla for pension:
entitlement(X,Y) - pension(X,Y).
entitlement(X,nothing) - not pension(X,Y).
This program has all the advantages of Program 11.11b and neither of the disadvantages mentioned before. It shows that making a person
pension(Person,Pension) Pension is the type of pension received by Person.
pension(X,invalid_pension) invalid(X).
pensiou(X,old_age_pension) over_65(X), paid_up(X).
pension(X,supplementary_benef it) over_65(X).
invalid(mc_tavish).
over_65(mc.tavish) . over_65(mc_donald) . over_65(mc_duff)
paid_up(mc_tavish). paid_up(mc.donald).
Program 11.11a Determining welfare payments
pension(Person,Pension) - Pension is the type of pension receìved by Person.
pension(X,invalid_pension) invalid(X),
pension(X,old_age_perision) over_65(X) , paid_up(X)
pension(X,supplementary_benefït) - over_65(X) , !
pension (X , nothing)
Program hub Determining welfare payments
208 Chapter 11
entitled to nothing as the default rule is really a new concept and should be presented as such.
11.6 Cuts for Efficiency
Earlier in this chapter, we claimed that the efficiency of some Prolog programs could be improved through sparing use of the cut. This sec- tion explores the claim. Two issues are addressed. The first is the mean- ing of efficiency in the context of Prolog. The second is appropriate uses of cut.
Efficiency relates to utilization of resources. The resources used by computations are space and time. To understand Prolog's use of space and time, we need to consider Prolog implementation technology.
The two major areas of memory manipulated during a Prolog computa- tion are the stack and the heap. The stack, called the local stack in many Edinburgh Prolog implementations, is used to govern control flow. The heap, called the global stack in many Edinburgh Prolog implementations, is used to construct data structures that are needed throughout the com- putation.
Let us relate stack management to the computation model of Prolog. Each time a goal is chosen for reduction, a stack frame is placed on the stack. Pointers are used to specify subsequent flow of control once the goal succeeds or fails. The pointers depend on whether other clauses can be used to reduce the chosen goal. Handling the stack frame is simplified considerably if it is known that only one clause is applicable. Technically, a choice point needs to be put on the stack if more than one clause is applicable.
Experience has shown that avoiding placing choice points on the stack has a large impact on efficiency. Indeed, Prolog implementation tech- nology has advanced to the stage that deterministic code, i.e., without choice points, can be made to run almost as efficiently as conventional languages.
Cuts are one way that Prolog implementations know that only one clause is applicable. Another way is by the effective use of indexing. Whether a cut is needed to tell a particular Prolog implementation that only one clause is applicable depends on the particular indexing scheme.
209 Cuts and Negation
In this book, we often use the first argument to differentiate between clauses. Indexing on the first argument is the most common among Pro- log implementations. For effective use, consult your Prolog manual.
Efficient use of space is determined primarily by controlling the growth of the stack. Already we have discussed the advantages of iterative code and last call optimization. Too many frames placed on the stack can cause computations to abort. In practice this is a major concern. Running out of stack space is a common symptom of an infinite loop or running a highly recursive program. For example, Program 3.9 implementing Ack- ermann's function, when adapted for Prolog arithmetic, quickly exhausts an implementation's capacity.
Time complexity is approximated by number of reductions. Thus effi- cient use of time can be determined by analyzing the number of reduc- tions a program makes. In Part I, we analyzed different logic programs by the size of proof trees. In Prolog, size of search tree is a better measure, but it becomes difficult to incorporate Prolog's nondeterminism.
Probably the most important approach to improving time performance is better algorithms Although Prolog is a declarative language, the no- tion of an algorithm applies equally well to Prolog as to other languages. Examples of good and bad algorithms for the same problem, together with their Prolog implementations, have been given in previous chap- ters. Linear reverse using accumulators (Program 3.16b) is clearly more efficient than naive reverse (Program 3.16a). Quicksort (Program 3.22) is better than permutation sort (Program 3.20).
Besides coming up with better algorithms, several things can be done to influence the performance of Prolog programs One is to choose a bet- ter implementation. An efficient implementation is characterized by its raw speed, its indexing capabilities, support for tail recursion optimiza- tion, and garbage collection. The speed of logic programniing languages is usually measured in LIPS, or logical inferences per second. A logical inference corresponds to a reduction in a computation. Most Prolog im- plementations claim a LIPS rating. The standard benchmark, by no means ideal, is to time Program 3.16a, naive reverse, reversing a list. There are 496 reductions for a list of 30 elements.
Once the implementation is fixed, the programs themselves can be tuned by
u Good goal ordering, where the rule is "fail as early as possible"
210 Chapter 11
Exploitation of the indexing facility, by ordering arguments appropri- ately
Elimination of nondeterminism using explicit conditions and cuts
Let us elaborate on the third item and discuss guidelines for using cut. As discussed, Prolog implementations will perform more efficiently if they know a predicate is deterministic. The appropriate sparing use of cut is primarily for saying that predicates are deterministic, not for controlling backtracking.
The two basic principles for using a cut are
Make cuts as local as possible.
Place a cut as soon as it is known that the correct clause has been chosen.
Let us illustrate the principles with the quicksort program, Program 3.22. The recursive clause is as follows
quicksort([XIXsI ,Ys) -
partition(Xs,X,Littles,Bigs), quicksort(Littles,Ls),
quicksort(Bigs,Bs), append(Ls, [XIBs] ,Ys).
We know there is only one solution for the partition of the list. Rather than place a cut in the clause for quicksort, the partition predicate should be made deterministic. This is in accordance with the first princi- ple.
One of the partition clauses is
partition([XIXs] ,Y, EXILs] ,Bs) -
X Y, partition(Xs,Y,Ls,Bs).
1f the clause succeeds, then no other will be applicable. But the cut
should be placed before the recursive call to partition rather than after, according to the second principle.
Where and whether to place cuts can depend on the Prolog implemen- tation being used. Cuts are needed only if Prolog does not know the determinism of a predicate. If, for example, indexing can determine that only one predicate is applicable, no cuts are needed. In a system without indexing, cuts would be needed for the same program.
Having discussed appropriate use of cuts, we stress that adding cuts to a program should typically be done after the program runs correctly.
211 Cuts and Negation
A common misconception is that a program can be fixed from giving extraneous answers and behaving incorrectly by adding cuts. This is not so. Prolog code should be debugged as declaratively as possible, a topic we discuss in Chapter 13. Only when the logic is correct should efficiency be addressed.
The final factor that we consider in evaluating the efficiency of Prolog programs is the creation of intermediate data structures, which primarily affects use of the heap. Minimizing the number of data structures being generated is a subject that has not received much attention in the Prolog literature. We analyze two versions of the predicate sublist(Xs,Ys) to illustrate the type of reasoning possible.
The two versions of sublist that we consider involve Program 3.13 for calculating prefixes and suffixes of lists. We must also specify the comparison with respect to a particular use. The one chosen for the analysis is whether a given list is a sublist of a second given list. The first clause that follows denotes a sublìst as a prefix of a suffix, and the second clause defines a sublist as a suffix of a prefix:
sublist(Xs,AsXsBs) - suffix(XsBs,AsXsBs), prefix(Xs,XsBs)
sublist(Xs,AsXsBs) - prefix(AsXs,AsXsBs), suffix(Xs,AsXs)
Although both programs have the same meaning, there is a difference in the performance of the two programs. If the two arguments to sub- list are complete lists, the first clause simply goes down the second list, returning a suffix, then goes down the first list, checking if the suffix is a prefix of the first list. This execution does not generate any new interme- diate data structures. On the other hand, the second clause creates a new list, which is a prefix of the second list, then checks ifthis list is a suffix of the first list. If the check fails, backtracking occurs, and a new prefix of the first list is created.
Even though, on the average, the number of reductions performed by the two clauses is the same, they are different in their efficiency. The first clause does not generate new structures (does not cons, in Lisp jargon). The second clause does. When analyzing Lisp programs, it is common to examine the consing performance in great detail, and whether a program conses or not is an important efficiency consideration. We feel that the issue is important for Prolog programs, but perhaps the state of the art of studying the performance of large Prolog programs has not matured enough to dictate such analyses.
212 Chapter 11
11.7 Background
The cut was introduced in Marseilles Prolog (Colmerauer et al., 1973) and was perhaps one of the most influential design decisions in Pro- log. Colmerauer experimented with several other constructs, which cor- responded to special cases of the cut, before coming up with its full definition.
The terminology green cuts and red cuts was introduced by van Emden (1982), in order to try to distinguish between legitimate and illegitimate uses of cuts. Alternative control structures, which are more structured then the cut, are constantly being proposed, but the cut still remains the workhorse of the Prolog programmer. Some of the extensions are if-then- else constructs (O'Keefe, 1985) and notations for declaring that a relation is functional, or deterministic, as well as "weak-cuts," "snips," remote- cuts (Chikayama, 1984), and not itself, which, as currently implemented, can be viewed as a structured application of the cut.
The controversial nature of cut has not been emphasized in this book. A good starting place to read about some of cut's problems, and the variation in its implementation, is Moss (1986). Many of the difficulties arise from the scope of the cut, and how cuts interact with the system predicates for control such as conjunction, disjunction, and the meta- variable facility. For example, two versions of call have been suggested, one that blocks the cut and one that does not. Further discussion of cut can be found in O'Keefe (1990), including an exposition on when cut should be used.
Some Prologs provide if_then_else(P,Q,R) under the syntax P - Q; R and an abridged if-then form P - Q. Whether to include if-then-else in Standard Prolog has been a controversial issue. The trade-off is con- venience for some programming tasks versus thorny semantic anoma- lies. This issue has been raised several times on the USENET newsgroup comp.lang.prolog. Relevant comments were collected in the May 1991 is- sue of the Newsletter of the Association for Logic Progranmiing, Volume 4, No. 2.
The cut is also the ancestor of the commit operator of concurrent logic languages, which was first introduced by Clark and Gregory (1981) in their Relational Language. The commit cleans up one of the major drawbacks of the cut, which is destroying the modularity of clauses.
213 Cuts and Negation
The cut is asymmetric, because it eliminates alternative clauses below the clause in which it appears, but not above. Hence a cut in one clause affects the meaning of other clauses. The commit, on the other hand, is symmetric and therefore cannot implement negation as failure; it does not destroy the modularity of clauses.
The pioneering work on Prolog implementation technology was in D.H.D. Warren's Ph.D. thesis (1977). Warren later added tail recursion optimization to his original DEC-10 compiler (1986). Tail recursion op- timization was implemented concurrently by Bruynooghe (1982) in his Prolog system. A motley collection of papers on Prolog implementations can be found in Campbell (1984).
Most current compilers and implementation technology are based on the WAM (Warren Abstract Machine), published as a somewhat cryptic technical report (Warren, 1983). Readers seriously interested in program efficiency need to understand the WAM. The best places to start reading about the WAM are Maier and Warren (1988) and AIt-Kaci (1991).
References to negation in logic programming can be found in Sec- tion 5.6. Implementations of a sound negation as failure rule in dialects of Prolog can be found in Prolog-II (van Caneghem, 1982) and MU-Prolog (Naish, 1985a).
The program for same_var and its argument for correctness are due to O'Keefe (1983).
Program hub for pension is a variant of an example due to Sam Steel for a Prolog course at the University of Edinburgh - hence the Scottish flavor. Needless to say, this is not intended as, nor is it an accurate expression, of the Scottish or British social welfare system.
12 Extra-Logical Predicates
There is a class of predicates in Prolog that lie outside the logic program- mirig model, and are called extra-logical predicates. These predicates achieve a side effect in the course of being satisfied as a logical goal. There are basically three types of extra-logical system predicates: pred- icates concerned with I/O, predicates for accessing and manipulating the program, and predicates for interfacing with the underlying operating system. Prolog I/O and program manipulation predicates are discussed in this chapter. The interface to the operating system is too system- dependent to be discussed in this book.
12.1 Input/Output
A very important class of predicates that produces side effects is that concerned with I/O. Any practical programming language must have a mechanism for both input and output. The execution model of Prolog, however, precludes the expression of I/O within the pure component of the language.
The basic predicate for input is read(X). This goal reads a term from the current input stream, usually from the terminal. The term that has been read is unified with X, and read succeeds or fails depending on the result of unification.
The basic predicate for output is write (X). This goal writes the term X on the current output stream, as defined by the underlying operating system, usually to the terminal. Neither read nor write give alternative solutions on backtracking.
216 Chapter 12
writeln([XIXs]) - write(X), writeln(Xs).
writeln(C J) - nl.
Program 12.1 Writing a list of terms
The normal use of read is with a variable argument X, which acquires the value of the first term in the current input stream. The instantiation of X to something outside the program lies outside the logical model, since each time the procedure is called, read(X) succeeds with a (pos- sibly) different value for X. Read attempts to parse the next term on the input stream. If it fails, it
prints an error message on the terminal. There is an asymmetry between the extra-logical nature of read and
write. If all calls to write were replaced with the goal true, which always succeeded once, the semantics of the program would be unaffected. That is not true for read.
Early Prolog implementations did not concentrate on input and output facilities, providing the basic predicates read and write, or their equiva- lents, and little else. More recent Prolog implementations have a wider range of formatted I/O options, some of which have been adopted in Standard Prolog. In this book, the emphasis is not on I/O, and so we re-. strict outselves to basic predicates and some simple utilities described in the rest of this section. For more elaborate I/O, consult your particular Prolog manual.
A useful utility is a predicate writeln(Xs), analogous to the Pascal command, which writes the list of terms Xs as a line of output on the cur- rent output stream. It is defined in Program 12.1. The predicate writein uses the builtin predicate nl, which causes the next output character to be on a new line. As an example of its use, executing the conjunctive goal (X=3, writeln(['The value of X is ',X]) produces the output
The value of X is 3.
Note the use of the quoted atom 'The value of X is '. Both read and write operate at the term level. A lower level for I/O is the character level. Edinburgh Prolog assumed that characters were represented by ASCII codes. Standard Prolog takes a broader perspective to support such character sets as Kanji. The basic output predicate is put_char (Char),
217 Extra-Logical Predicates
read_word_list (Words) - Words is a list of words read from the input stream via side effects.
read_word_list (Words) -
get_char(FirstChar),
road_words (FirstChar ,Words).
read_words (Char, [Word I Words]) - word_char (Char)
read_word (Char Word, NextChar),
read_words (NextChar ,Words).
read_words (Char ,Words) -
fill_char (Char),
get_char(NextCliar),
read_words (NextChar ,Words).
read_words(Char,[ ]) - erid_of_words_char(Char).
read_word(Char Word, NextChar) -
word_chars(Char,Chars ,NextChar),
atom_list(Word,Chars).
word_chars (Char, [Char I Chars] FinalChar)
word_char(Char), ,
get_char(NextChar),
word_chars (NextChar Chars ,FinalChar).
word_chars(Char,[ ],Char) -
not word_char(Char).
Program 12.2 Reading in a list of words
which outputs the character Char on the current output stream. Stan- dard Prolog allows you to specify the output stream, but we do not give examples here. The basic input predicate at the character level is get_ char(Char), which reads a character C from the current input stream and then unifies C with Char.
Program 12.2 defines read_word_list (Words), a utility predicate for reading in a list of words, Words, from the current input, terminated by an end-of-words character, for example a period. Specific definitions of the predicates word_char/i, fill_char/i, and end_of_words_char/i need to be added. It can be used to allow freer form input. In Pro- gram 12.2, words can be separated by arbitrarily many fill characters.
218 Chapter 12
The predicate read_word_list reads a character, FirstChar, and calls read_words(FirstChar,Words). This predicate does one of three ac- tions, depending on what FirstChar is. If FirstChar is a word character, then the next word is found. Word characters in Standard Prolog are up- percase and lowercase letters, underscores, and digits. The second action is to ignore filling characters, and so the next character is read, and the program continues recursively. Finally, if the character denoting the end of the words is reached, the program terminates and returns the list of words.
It is important that the program must always read a character ahead and then test what it should do. If the character is useful, for example, a word character, it must be passed down to be part of the word. Otherwise characters can get lost when backtracking. Consider the following read and process loop:
process([ ])
get_char (C), end_of_words_char(C).
process([WIWordsl) -
get_char(C), word_char(C), get_word(C,W), process(Words).
If the first character in a word is not an end_of_words_char, the first clause will fail, and the second clause will cause the reading of the next character.
Returning to Program 12.2, the predicate read_word (Char ,Word, NextChar) reads a word Word given the current character Char and re- turns the next character after the word, NextChar. The list of characters composing the word is found by word_chars/3 (with the same argu- ments as read_word). The word is created from the list of characters using the system predicate atom_list/2. In word_chars there is the same property of looking ahead one character, so that no character is lost.
Predicates such as fill_char/i and word_char/i exemplify data ab- straction in Prolog.
Exercise for Section 12.1
(i) Extend Program 12.2 to handle a wider range of inputs, for example, numbers.
219 Extra-Logical Predicates
12.2 Program Access and Manipulation
So far programs have been assumed to be resident in computer memory, without discussion of how they are represented or how they got there. Many applications depend on accessing the clauses in the program. Fur- thermore, if programs are to be modified at runtirne, there must be a way of adding (and deleting) clauses.
The first Prologs, implemented as simple interpreted systems, classi- fied predicates as builtm and static or user-defined and dynamic. The subsequent development of compilers and libraries require a more so- phisticated classification.
Each user-defined predicate is either dynamic or static. The procedure of a dynamic predicate can be altered, whereas the procedure of a static predicate cannot. Builtin predicates are assumed to be static. The system predicates introduced in this section apply only to dynamic predicates and will probably cause error messages if applied to static predicates. In this book, we assume all predicates are dynamic unless otherwise specified. In many Prologs, declarations are needed to make a predicate dynamic.
The system predicate for accessing a program is clause (Head,Body). The goal clause (Head,Body) must be called with Head instantiated. The program is searched for the first clause whose head unifies with Head. The head and body of this clause are then unified with Head and Body. On backtracking, the goal succeeds once for each unifiable clause in the procedure. Note that clauses in the program cannot be accessed via their body.
Facts have the atom true as their body. Conjunctive goals are repre- sented using the binary furictor , . The actual representations can be easily abstracted away, however.
Consider Program 3.12 for member:
member(X, [XJXs]).
member(X,[YIYs]) - member(X,Ys)
The goal clause (member (X,Ys) ,Body) has two solutions: tYs=[XXs] Body=true} and (Ys=[YYsl] ,Bodymember(X,Ysl)}. Note that a fresh copy of the variables appearing in the clause is made each time a unifi- cation is performed. In terms of the meta-logical primitives freeze and
220 Chapter 12
melt, the clause is stored in frozen form in the program. Each call to clause causes a new melt of the frozen clause. This is the logical coun- terpart of the classic notion of reentrant code.
System predicates are provided both to add clauses to the program and to remove clauses. The basic predicate for adding clauses is as- sertz (Clause), which adds Clause as the last clause of the correspond- ing procedure. For example, assertz(father(haran,lot))? adds the father fact to the program. When describing rules an extra level of brackets is needed for technical reasons concerning the precedence of terms. For example, assertz((parent(X,Y) - f ather(X,Y))) is the correct syntax.
There is a variant of assertz, asserta, that adds the clause at the beginning of a procedure.
If Clause is uninstantiated (or if Clause has the form HB with H uninstantiated), an error condition occurs.
The predicate retract (C) removes from the program the first clause in the program unifying with C. Note that to retract a clause such as a - b,c,d, you need to specify retract((a - C)). A call to retract may only mark a clause for removal, rather than physically removing it, and the actual removal would occur only when Prolog's top-level query is solved. This is for implementation reasons, but may lead to anomalous behavior in some Prologs.
Asserting a clause freezes the terms appearing in the clause. Retracting the same clause melts a new copy of the terms. In many Prologs this is exploited to be the easiest way of copying a term. Standard Prolog, however, provides a builtin predicate copy_term/2 for this purpose.
The predicates assert and retract introduce to Prolog the possibil- ity of programming with side effects. Code depending on side effects for its successful execution is hard to read, hard to debug, and hard to rea- son about formally. Hence these predicates are somewhat controversial, and using them is sometimes a result of intellectual laziness or incompe- tence. They should be used as little as possible when programming Many of the programs to be given in this book can be written using assert and retract, but the results are less clean and less efficient. Further, as Pro- log compiler technology advances, the inefficiency in using assert and retract will become more apparent.
It is possible, however, to give logical justification for some limited uses of assert and retract. Asserting a clause is justified, for exam-
221 Extra-Logical Predicates
pie, if the clause already logically follows from the program. In such a case, adding it will not affect the meaning of the program, since no new consequences can be derived. Perhaps program efficiency will improve, as some consequences could be derived faster. This use is exemplified in the lemma construct, introduced m Section 12.3.
Similarly, retracting a clause is justified if the clause is logically re- dundant. In this case, retracting constitutes a kind of logical garbage collection, whose purpose is to reduce the size of the program.
12.3 Memo-Functions
Memo-functions save the results of subcomputations to be used later in a computation. Remembering partial results is impossible within pure Prolog, so memo-functions are implemented using side effects to the program Programming in this way can be considered bottom-up pro- gramming.
The prototypical memo-function is lemma (Goal). Operationally, it at- tempts to prove the goal Goal and, if successful, stores the result of the proof as a lemma. It is implemented as
lemina(P) P, asserta((P - !)).
The next time the goal P is attempted, the new solution will be used, and there will be no unnecessary recomputation. The cut is present to prevent the more general program being used. Its use is justified only if P does not have multiple solutions.
Using lemmas is demonstrated with Program 12.3 for solving the Tow- ers of Hanoi problem. The performance of Program 3.31 in solving the problem is dramatically improved. It is well known that the solution of the Towers of Hanoi with N disks requires 2N moves. For example, ten disks require 1,023 moves, or in terms of Program 3.31, 1,023 calls of hanoi(1 ,A,B,C,Xs). The overall number of general calls of hanoi/5 is significantly more.
The solution to the Towers of Hanoi repeatedly solves subproblems moving the identical number of disks. A memo-function can be used to recall the moves made in solving each subproblem of moving a smaller
222 Chapter 12
hanoi (N,A,B,C,Moves) - Moves is the sequence of moves required to move N disks from peg A to peg B using peg C as an intermediary according to the rules of the Towers of Hanoi puzzle.
hanoi(1,A,B,C,[A to B]).
hanoi(N,A,B,C,Moves) -
N > 1,
Ni is N-1,
lemma(hanoi(N1,A,C,B,Msl)),
hanoi(N1,C,B,A,Ms2),
append(Msl,[A to BIMs2],Moves).
lemrna(P) - P, asserta((P - D). Testing
test_hanoi(N,Pegs,Moves)
hanoi(N,A,B,C,Moves), Pegs = [A,B,C].
Program 12.3 Towers of Hanoi using a memo-function
number of disks. Later attempts to solve the subproblem can use the computed sequence of moves rather than recomputing them.
The idea is seen with the recursive clause of hanoi in Program 12.3. The first call to solve hanoi with N - i disks is remembered, and can be used by the second call to hanoi with N - i disks.
The program is tested with the predicate test_hanoi (N,Pegs,Moves). N is the number of disks, Pegs is a list of the three peg names, and Moves is the list of moves that must be made. Note that in order to take advantage of the memo-functions, a general problem is solved first. Only when the solution is complete, and all memo-functions have recorded their results, are the peg names instantiated.
Exercise for Section 12.3
(i) Two players take turns to say a number between i and 3 inclusive. A sum is kept of the numbers, and the player who brings the sum to 20 wins. Write a program to play the game to win, using memo- functions.
223 Extra-Logical Predicates
12.4 Interactive Programs
A common form of a program requiring side effects is an interactive ioop. A command is read from the terminal, responded to, and the next com- mand read. Interactive loops are implemented typically by while loops in conventional languages. Program 12.4 gives the basic skeleton of such programs, where a command is read, then echoed by being written on the screen.
The read/echo ioop is invoked by the goal echo. The heart of the pro- gram is the relation echo (X), where X is the term to be echoed. The pro- gram assumes a user-defined predicate last_input/i, which succeeds if the argument satisfies the termination condition for input. If the termi- nation condition is satisfied by the input, the loop terminates; otherwise the term is written and a new term is read.
Note that the testing of the term is separate from its reading. This is necessary to avoid losing a term: terms cannot be reread. The same phenomenon occurred in Program 12.2 for processing characters. The character was read and then separately processed.
Program 12.4 is iterative and deterministic. lt can be run efficiently on a system with tail recursion optimization, always using the same small amount of space.
We give two examples of programs using the basic cycle of reading a term, and then processing it. The first is a line editor. The second interactive program is a shell for Prolog commands, which is essentially a top-level interpreter for Prolog in Prolog.
The first decision in writing a simple line editor in Prolog is how to represent the file. Each line in the file must be accessible, together with the cursor position, that is the current position within the file. We use a structure file (Bef ore,After), where Before is a list of lines before the cursor, and After is a list of lines after the cursor. The cursor position is
echo - read(X), echo(X).
echo(X) - last_input(X), !.
echo(X) - write(X), nl, read(Y), , echo(Y).
Program 12.4 Basic interactive ioop
224 Chapter 12
edit - edit(file([ ],[ D). edit(File) -
write_prompt, read(Command), edit (File,Command).
edit(File,exit) -
edit (File ,Comnand) -
apply(Command,File,Filel), , edit(Filel).
edit (File, Command) -
writeln([Command,' is not applicable']), !, edit(File).
apply(up,file([XIXs] ,Ys) ,file(Xs, [XIYs]))
apply(up(N) ,file(Xs,Ys) ,file(Xsl,Ysl)) -
N > O, up(N,Xs,Ys,Xsl,Ysl). apply(down,file(Xs,[YIYs]),file([YIXs],Ys)). apply(insert(Line),file(Xs,Ys),file(Xs, [LinelYs])).
apply(delete,file(Xs, [YIYs]) ,file(Xs,Ys)).
apply(print,file([XIXs],Ys),file([XIXs],Ys)) -
write(X), nl.
apply(print(*),file(Xs,Ys),file(Xs,Ys)) -
reverse(Xs,Xsl), write_file(Xsl) , write_file(Ys)
up(N,[ ],Ys,[ ],Ys). up(O,Xs,Ys,Xs,Ys). up(N, [XIXs] ,Ys,Xsl,Ysi) -
N > O, Nl is N-1, up(Nl,Xs,[XIYs],Xsl,Ysl). write_file([XIXs]) -
write(X), nl, write_file(Xs).
write_file([ 1). write_prompt - write('»'), nl.
Program 12.5 A line editor
restricted to be at the end of some line. The lines before the cursor will be in reverse order to give easier access to the lines nearer the cursor. The basic loop accepts a command from the keyboard and applies it to produce a new version of the file. Program 12.5 is the editor.
An editing session is invoked by edit, which initializes the file be- ing processed to the empty file, file ( [ ] , [ ])). The interactive loop is controlled by edit (File). It writes a prompt on the screen, using write_prompt, then reads and processes a command. The process- ing uses the basic predicate edit(File,Command), which applies the command to the file. The application is performed by the goal ap- ply(Command,File,Filel), where Fuel is the new version of the file
225 Extra-Logical Predicates
after the command has been applied. The editing continues by calling edit/i on Fuel. The third edit/2 clause handles the case when no command is applicable, indicated by the failure of apply. In this case, an appropriate message is printed on the screen and the editing contin- ues. The editing session is terminated by the command exit, which is separately tested for by edit/2.
Let us look at a couple of apply clauses, to give the flavor of how commands are specified. Particularly simple are commands for moving the cursor. The riause
apply(up,file([XIXs] ,Ys) ,file(Xs, [XIYs]))
says that we move the cursor up by moving the line immediately above the cursor to be immediately below the cursor. The command fails if the cursor is at the top of the file. The command for moving the cursor down, also shown in Program 12.5, is analogous to moving the cursor up.
Moving the cursor up N lines rather than a single line involves using an auxiliary predicate up/5 to change the cursor position in the file. Issues of robustness surface in its definition. Note that apply tests that the argument to up is sensible, i.e., a positive number of lines, before up is invoked. The predicate up itself handles the case when the number of lines to be moved up is greater than the number of lines in the file. The command succeeds with the cursor placed at the top of the file. Extending the editor program to move a cursor down N lines is posed as an exercise at the end of this section.
Other commands given in Program 12.5 insert and delete lines. The command for insert, insert (Line), contains an argument, namely the line to be inserted. The command for delete is straightforward. It fails if the cursor is at the bottom of the screen. Also in the editor are com- mands for printing the line above the cursor, print, and for printing the whole file, print(*).
The editor commands are mutually exclusive. Only one apply clause is applicable for any command. As soon as an apply goal succeeds, there are no other possible alternatives. Prolog implementations that support indexing would find the correct clause immediately and leave no choice points. Imposing determinism via exploitation of indexing is a little dif- ferent than adding explicit cuts, as described in Section 11.1, where the cuts would have been applied directly to the apply facts themselves. The difference between the two approaches is merely cosmetic. Note that a
226 Chapter 12
shell - shell_prompt, read(Goal), shell(Goal).
shell(exit) -
shell(Goal) -
ground(Goal), !, shell_solve_ground(Goal), shell.
shell(Goal) - shell_solve (Goal), shell.
shell_solve(Goal) -
Goal, write(Goal), nl, fail.
shell_solve (Goal) -
write('No (more) solutions'), nl.
shell_solve_ground(Goal)
Goal, ! , write('Yes'), nl.
shell_solve_ground(Goal)
writeYNo'), nl.
shell_prompt - write('Next command? ').
Program 12.6 An interactive shell
cut is still needed in the second edit clause to indicate that successful execution of a command and reporting of an error message are mutually exclusive.
A possible extension to the editor is to allow each command to handle its own error message. For example, suppose you wanted a more helpful message than "Command not applicable" when trying to move up when at the top of the file. This would be handled by extending the apply clause for moving up in the file.
We shift from editors to shells. A shell accepts commands from a terminal and executes them. We illustrate with an example of a shell for answering Prolog goals. This is presented as Program 12.6.
The shell is invoked by shell. The code is similar to the editor. The shell gives a prompt, using shell_prompt, then reads a goal and tries to solve it using shell (Goal). A distmction is made between solving ground goals, where a yes/no answer is given, and solving nonground goals, where the answer is the appropriately instantiated goal. These two cases are handled by shell_solve_ground and shell_solve, respec- tively. The shell is terminated by the goal exit.
227 Extra-Logical Predicates
Both shell_solve_ground and shell_solve use the meta-variable fa- cility to call the goal to be solved. The success or failure of the goal determines the output message. These predicates are the simplest exam- ples of meta-interpreters, a subject discussed in Chapter 17.
The shell_solve procedure shows an interesting solve-write-fail com- bination, which is useful to elicit all solutions to a goal by forced back- tracking. Since we do not wish the shell to fail, an alternative clause is provided, which succeeds when all solutions to the goal are exhausted. lt is interesting to note that it is not possible to collect all solutions to goals in a straightforward way without using some sort of side effect. This is explained further in Chapter 16 on second-order programming.
The shell can be used as a basis for a logging facility to keep a record of a session with Prolog. Such a facility is given as Program 12.7. This new shell is invoked by log, which calls the basic interactive predicate shell (Flag) with Flag initialized to log. The flag takes one of two val- ues, log or nolog, and indicates whether the output is currently being logged.
The logging facility is an extension of Program 12.6. The principal predicates take an extra argument, which indicates the current state of logging. Two extra commands are added, log and nolog, to turn logging on and off.
The flag is used by the predicates concerned with I/O. Each message written on the screen must also be written in the logging file. Also, each goal read is inserted in the log to increase the log's readability. Thus calls to read in Program 12.6 are replaced by a call to shell_read, and calls to write replaced by calls to shell_write.
The definition of shell_write specifies what must be done:
shell_write(X,nolog) - write(X).
shell_write(X,log) - write(X), file_write([X] , 'prolog.log').
If the flag is currently nolog, the output is written normally to the screen. If the flag is log, an extra copy is written to the file prolog.log. The predicate file_write (X,File) writes the line X to file File.
The remaining two predicates in Program 12.7, f ile_write/2 and close_logging_file, involve interacting with the underlying file sys- tem. Appropriate commands from Standard Prolog are given, and the reader is referred to a Prolog manual for more information.
228 Chapter 12
log - shell(log).
shell(Flag) -
shell_prompt, shell_read(Goal,Flag), shell(Goal,Flag).
shell (exit ,Flag) -
close_logging_file.
shell(nolog,Flag) -
shell(nolog).
shell(log,Flag) -
shell (log).
shell (Goal Flag)
ground(Goal), , shell_solve_ground(Goal,Flag), shell(Flag).
shell (Goal ,Flag) -
shell_solve(Goal,Flag), shell(Flag).
shell_solve (Goal ,Flag)
Goal, shell_write(Goal,Flag), nl, fail.
shell_solve (Goal ,Flag) -
shell_writeYNo (more) solutions' Flag), nl.
shell_solve_ground(Goal Flag) -
Goal, ! , shell_write('Yes' Flag), nl.
shell_solve_ground(Goal ,Flag) -
shell_write('No' ,Flag), nl.
shell_prompt - write('Next command? ').
shell_read(X,log) - read(X),
file_write(['Next command? ',X) , 'prolog.log').
shell_read(X,nolog) - read(X).
shell_write(X,nolog) '- write(X).
shell_write(X,log) - write(X), file_write(X, 'prolog.log').
file_write(X,File) - write_term(File,Term,[ 1). close_logging_file - close('prolog.log').
Program 12.7 Logging a session
229 Extra-Logical Predicates
Exercises for Section 12.4
(i) Extend Program 125, the editor, to handle the following com- mands:
Move the cursor down N lines,
Delete N lines,
Move to a line containing a given term,
Replace one term by another,
Arty command of your choice.
(II) Modify the logging facility, Program 12.7, so that the user can spec- ify the destmation file of the logged output.
123 Failure-Driven Loops
The interactive programs in the previous section were all based on tail re- cursive loops. There is an alternative way of writing loops in Prolog that are analogous to repeat loops in conventional languages. These loops are driven by failure and are called failure-driven loops. These ioops are useful only when used in conjunction with extra-logical predicates that cause side effects. Their behavior can be understood only from an opera- tional point of view.
A simple example of a failure-driven ioop is a query Goal, write (Goal), nl, fail?, which causes all solutions to a goal to be written on the screen. Such a loop is used in the shells of Programs 12.6 and 12.7.
A failure-driven loop can be used to define the system predicate tab(N) for printing N blanks on the screen. It uses Program 8.5 for be- tween:
tab(N) - between(1,N,I), put.char(' '), fail.
Each of the interactive programs in the previous section can be rewrit- ten using a failure-driven loop. The new version of the basic interactive loop is given as Program 12.8. lt is based on a nonterminating system
230 Chapter 12
echo - repeat, read(X), echo(X),
echo(X) last_input(X), !.
echo(X) write(X), nl, fail.
repeat.
repeat - repeat.
Program 12.8 Basic interactive repeat ioop
consult(File) - The clauses of the program in the ifie File are read and asserted.
consult(File) - openOEile,read,DD), consult_loop(DD), close(DD).
consult_loop(DD) - repeat, read(Clause), process(Clause,DD), !.
process(Clause,DD) - at_end_of_streani(DD).
process(Clause,DD) - assertz(Clause), fail.
Program 12.9 Consulting a file
predicate repeat, which can be defined by the minimal recursive proce- dure in Program 12.8. Unlike the Program 12.4 goal, the goal echo (X) fails unless the termination condition is satisfied. The failure causes backtracking to the repeat goal, which succeeds, and the next term is read and echoed. The cut in the definition of echo ensures that the repeat loop is not reentered later.
Failure-driven loops that use repeat are called repeat loops and are the analogue of repeat loops from conventional languages. Repeat loops are useful in Prolog for interacting with the outside system to repeatedly read and/or write. Repeat loops require a predicate that is guaranteed to fail, causing the iteration to continue, unless the loop should be ter- minated. The goal echo(X) in Program 12.8 serves that function, only succeeding when the last input is reached. A useful heuristic for building repeat loops is that there should be a cut in the body of the clause with the repeat goal, which prevents a nonterminating computation were the ioop to be reentered via backtracking.
We use a repeat loop to define the system predicate consult (File) for reading in a file of clauses and asserting them. Program 12.9 contains its definition. The system predicates open/3 and close/i are used for opening and closing an input file, respectively.
231 Extra-Logical Predicates
Tail recursive loops are preferable to repeat loops because the latter have no logical meaning. In practice, repeat loops are often necessary to run large computations, especially on Prolog implementations without tail recursion optimization or garbage collection. Explicit failure typically initiates some implementation-dependent reclamation of space.
Exercise for Section 12.5
(i) Write your own version of the builtin predicate abolish(F,N) that retracts all the clauses for the procedure F of arity N.
12.6 Background
I/O has never really blended well with the rest of the language of Pro- log. Its standard implementation, with side effects, relies solely on the procedural semantics of Prolog and has no connection to the underlying logic programming model. For example, if an output is issued on a fail- ing branch of a computation, it is not undone upon backtracking. If an input term is read, it is lost on backtracking, as the input stream is not backtrackable.
Concurrent logic languages attempt to remedy the problem and to in- tegrate I/O better with the logic programming model by identifying the I/O streams of devices with the logical streams in the language (Shapiro, 1986). Perpetual recursive processes can produce or consume incremen- tally those potentially unbounded streams.
Self-modifying programs are a bygone concept in computer science. Modern programming languages preclude this ability, and good assem- bly language practice also avoids such programming tricks. lt is ironic that a programming language attempting to open a new era in computer programming opens the front door to such arcane techniques, using the predicates assert and retract.
These program manipulation predicates of Prolog were devised ini- tially as a low-level mechanism for loading and reloading programs, im- plemented in DEC-10 Prolog by the consult and reconsult predicates. However, like any other feature of a language, they ended up being used for tasks that, we believe, were not intended by their original designers.
232 Chapter 12
Reluctantly, we must acknowledge that assert and retract are part of Prolog, and clarify the anomalies. Attempts have been made in this direction. Inconsistencies between different Prolog implementations are discussed in Moss (1986). The best way of handling retracts seems to be the logical update view presented in Lindholm and O'Keefe (1987).
The discussion of static and dynamic predicates comes from the Stan- dard Prolog draft (Scowen, 1991).
The program for the Towers of Hanoi was shown to us by Shmuel Safra. Memo-functions in the context of artificial intelligence were proposed by Donald Michie (1968).
The line editor is originally due to Warren (1982b).
13 Program Development
Software engineering considerations are as relevant for programming in logic programming languages as in procedural languages. Prolog is no different from any other language in its need for a methodology to build and maintain large programs A good progranirning style is im- portant, as is a good program development methodology. This chapter discusses programming style and layout and program development, and introduces a method called stepwise enhancement for systematic con- struction of Prolog programs.
13.1 Programming Style and Layout
One basic concern in composing the programs in this book has been to make them as declarative as possible to increase program clarity and readability. A program must be considered as a whole. Its readability is determined by its physical layout and by the choice of names appear- ing in it. This section discusses the guidelines we use when composing programs.
An important influence in making programs easy to read is the naming of the various objects in the program. The choice of all predicate names, variable names, constants, and structures appearing in the program af- fect readability. The aim is to emphasize the declarative reading of the program.
We choose predicate names to be a word (or several words) that names relations between objects in the program rather than describing what the
234 Chapter 13
program is doing. Coining a good declarative name for a procedure does not come easily.
The activity of programming is procedural. It is often easier to name procedurally than declaratively (and programs with procedural names usually run faster :-). Once the program works, however, we often revise the predicate names to be declarative. Composing a program is a cyclic activity in which names are constantly being reworked to reflect our improved understanding of our creation, and to enhance readability by us and others.
Mnemonic variable names also have an effect on program readability. A name can be a meaningful word (or words) or a standard variable form such as Xs for lists.
Variables that appear only once in a clause can be handled separately. They are in effect anonymous, and from an implementation viewpoint need not be named. Standard Prolog supports a special syntactic con- vention, a single underscore, for referring to anonymous variables. Using this convention, Program 3.12 for member would be written
member(X, [XI_]).
member(X, 1_lYs]) member(X,Ys).
The advantage of the convention is to highlight the significant variables for unification. The disadvantage is related; the reading of clauses be- comes procedural rather than declarative.
We use different syntactic conventions for separating multiple words in variable names and predicate functors. For variables, composite words are run together, each new word starting with a capital letter. Multiple words in predicate names are linked with underscores. Syntactic conven- tions are a matter of taste, but it is preferable to have a consistent style.
The layout of individual clauses also has an effect on how easily pro- grams can be understood. We have found the most helpful style to be
f oo((Arguments))
bari((Argumentsi)),
bar2 ((Arguments2)),
bar((Arguments)).
235 Program Development
The heads of all clauses are aligned, the goals in the body of a clause are indented and occupy a separate line each. A blank line is inserted between procedures, but there is no space between individual clauses of a procedure.
Layout in a book and the typography used are not entirely consistent with actual programs. If all the goals in the body of a clause are short, then have them ori one line. Occasionally we have tables of facts with more than one fact per line.
A program can be self-documenting if sufficient care is taken with these two factors and the program is sufficiently simple. Given the nat- ural aversion of programmers to comments and documentation, this is very desirable.
In practice, code is rarely self-documenting and comments are needed. One important part of the documentation is the relation scheme, which can be presented before the clauses defining that relation, augmented with further explanations if necessary. The explanations used in this book define the relation a procedure computes. It is not always easy to come up with a precise, declarative, natural language description of a relation computed by a logic program. However, the inability to do so usually indicates that the programmer does not fu[ly understand the creation, even if the creation actually works. Hence we encourage the use of the declarative documentation conventions adopted in this book. They are a good means of commurncating to others what a program defines as well as a discipline of thought, enabling programmers to think about and reflect on their own creations.
13.2 Reflections on Program Development
Smce programming in pure Prolog is as close to writing specifications as any practical programming language has gotten, one might hope that pure Prolog programs would be bug-free. This, of course, is not the case. Even when axiomatizing one's concepts and algorithms, a wide spectrum of bugs, quite similar to ones found in conventional languages, can be encountered.
Stating it differently, for any formalism there are sufficiently com- plex problems for which there are no self-evidently correct formulations
236 Chapter 13
of solutions. The difference between low-level and high-level languages, then, is only the threshold after which simple examination of the pro- gram is insufficient to determine its correctness.
There are two schools of thought on what to do on such an occasion. The "verification" school suggests that such complex programs be ver- ified by proving that they behave correctly with respect to an abstract specification. It is not clear how to apply this approach to logic programs, since the distance between the abstract specification and the program is much smaller then in other languages. If the Prolog axiomatization is not self-evident, there is very little hope that the specification, no matter in what language it is written, would be.
One might suggest using full first-order logic as a specification formal- ism for Prolog. It is the authors' experience that very rarely is a specifi- cation in full first-order logic shorter, simpler, or more readable then the simplest Prolog program defining the relation.
Given this situation, there are weaker alternatives. One is to prove that one Prolog program, perhaps more efficient though more complex, is equivalent to a simpler Prolog program, which, though less efficient, could serve as a specification for the first. Another is to prove that a pro- gram satisfies some constraint, such as a "loop invariant," which, though not guaranteeing the program's correctness, increases our confidence in it.
In some sense, Prolog programs are executable specifications. The al- ternative to staring at them, trying to convince ourselves that they are correct, is to execute them, and see if they behave in the way we want. This is the standard testing and debugging activity, carried out in pro- gram development fri any other progranmiing language. All the classical methods, approaches, and common wisdom concerning program testing and debugging apply equally well to Prolog.
What is the difference, then, between program development in conven- tional, even symbolic languages and Prolog?
One answer is that although Prolog programming is "just" program- ming, there is some improvement in ease of expression and speed of de- bugging compared to other lower-level formalisms - we hope the reader has already had a glimpse of it.
Another answer is that declarative programming clears your mind. Said less dramatically, progranmiing one's ideas in general, and program- ming in a declarative and high-level language in particular, clarifies one's
237 Program Development
thoughts and concepts. For experienced Prolog programmers, Prolog is not just a formalism for coding a computer, but also a formalism in which ideas can be expressed and evaluated - a tool for thinking.
A third answer is that the properties of the high-level formalism of logic may eventually lead to practical program development tools that are an order of magnitude more powerful then the tools used today. Examples of such tools are automatic program transformers, partial- evaluators, type inference programs, and algorithmic debuggers. The lat- ter are addressed in Section 17.3, where program diagnosis algorithms and their implementation in Prolog are described.
Unfortunately, practical Prolog programming environments incorpo- rating these novel ideas are not yet widely available. In the meantime, a simple tracer, such as explained in Section 17.2, is most of what one can expect. Nevertheless, large and sophisticated Prolog programs can be developed even using the current Prolog environments, perhaps with greater ease than ¡n other available languages.
The current tools and systems do not dictate or support a specific program development methodology. However, as with other symbolic programming languages, rapid prototyping is perhaps the most natural development strategy. In this strategy, one has an evolving, usable pro- totype of the system in most stages of the development. Development proceeds by either rewriting the prototype program or extending it. An- other alternative, or complementary, approach to program development is "think top-down, implement bottom-up." Although the design of a sys- tem should be top-down and goal-driven, its implementation proceeds best if done bottom-up. In bottom-up programming each piece of code written can be debugged immediately. Global decisions, such as repre- sentation, can be tested in practice on small sections of the system, and cleaned up and made more robust before most of the programming has been done. Also, experience with one subsystem may lead to changes in the design of other subsystems.
The size of the chunks of code that should be written and debugged as a whole varies and grows as the experience of the programmer grows. Ex- perienced Prolog programmers can write programs consisting of several pages of code, knowing that what is left after writing is done is mostly simple and mundane debugging. Less experienced programmers might find it hard to grasp the functionality and interaction of more then a few procedures at a time.
238 Chapter 13
We would like to conclude this section with a few moralistic state- ments. For every programming language, no matter how clean, elegant, and high-level, one can find programmers who will use it to write dirty, contorted, and unreadable programs. Prolog is no exception. However, we feel that for most problems that have an elegant solution, there is an elegant expression of that solution in Prolog. It is a goal of this book to convey both this belief and the tools to realize it in concrete cases, by showing that aesthetics and practicality are not necessarily opposed or conflicting goals. Put even more strongly, elegance is not optional.
13.3 Systematizing Program Construction
The pedagogic style of this book is to present well-constructed programs illustrating the important Prolog programming techniques. The examples are explained in sufficient detail so that readers can apply the techniques to construct similar programs to meet their own programming needs. Implicitly, we are saying that Prolog programming is a skill that can be learned by observing good examples and abstracting the principles.
Learning by apprenticeship, observing other programs, is not the only way. As experience with programming in Prolog accumulates, more sys- tematic methods of teaching Prolog programming are emerging. The emergence of systematic methods is analogous to the emergence of structured programniing and stepwise refinement in the early 1970s af- ter sufficient experience had accumulated in writing programs in the computer languages of the 19 SOs and 1960s.
In this section, we sketch a method to develop Prolog programs. The reader is invited to reconstruct for herself how this method could be ap- plied to develop the programs in Parts III and IV of this book. Underlying the method is a desire to provide more structure to Prolog programs so that software components can be reused and large applications can be routinely maintained and extended.
Central to the method is identifying the essential flow of control of a program. A program embodying a control flow is called a skeleton. Extra goals and arguments can be attached to a skeleton. The extra goals and arguments are entwined around the central flow of control and perform additional computations. The program containing the extra arguments
239 Program Development
and goals is called an enhancement of the skeleton. Building an enhance- ment from a skeleton will be called applying a technique.
For example, consider Program 8.6a for summing a list of numbers, reproduced here:
sumlist([XIXs],Sum) - sumlist(Xs,XsSum), Sum is X+XsSum.
sumlist([ 1,0).
The control flow embodied ni the suinlist program is traversing the list of numbers. The skeleton is obtained by dropping the second ar- gument completely, restricting to a predicate with one argument, and removing goals that only pertain to the second argument. This gives the following program, which should be identifiable as Program 3.11 definrng a list.
list([XIXs]) - list(Xs).
list([ 1).
The extra argument of the sunilist program calculates the sum of the numbers in the list. This form of calculation is very common and appeared in several of the examples in Chapter 8.
Another enhancement of the list program is Program 8.11 calculating the length of a list. There is a clear similarity between the programs for length and sumuist. Both use a similar technique for calculating a number, in one case the sum of the numbers in the list, in the second the length of the list.
length([XIXs],N) - length(Xs,N1), N is N1+1.
length([ 1,0).
Multiple techniques can be applied to a skeleton. For example, we can apply both sunirning elements and counting elements in one pass to get the program sum_length:
suni_length([XIXs] ,Sum,N) -
suxnlist(Xs,XsSuni,N1), Sum is X+XsSum, N is N1+1.
sum_length([ 1,0,0).
Intuitively, it is straightforward to create the sum_length program from the programs for sumuist and length. The arguments are taken directly and combined to give a new program. We call this operation com- position. In Chapter 18, a program for composition is presented.
240 Chapter 13
Another example of a technique is adding a pair of arguments as an accumulator and a final result. The technique is informally described in Section 7.5. Applying the appropriate version of the technique to the list skeleton can generate Program 8.6b for sumlist or the iterative version of length, which is the solution to Exercise 8.3(vii).
Identifying control flows of programs may seem contradictory to the ideal of declarative progranmiing espoused in the previous section. How- ever, at some level programming is a procedural activity, and describ- ing well-written chunks of code is fine. It is our belief that recognizing patterns of programs makes it easier for people to develop good style. Declarativeness is preserved by ensuring wherever possible that each en- hancement produced be given a declarative reading.
The progranuning method called stepwise enhancement consists of three steps:
Identify the skeleton program constituting the control flow. Create enhancements using standard programming techniques. Compose the separate enhancements to give the final program.
We illustrate stepwise enhancement for a simple example - calculat- ing the union and intersection of two lists of elements. For simplicity we assume that there are no duplicate elements in the two lists and that we do not care about the order of elements in the answer.
A skeleton for this program follows. The appropriate control flow is to traverse the first list, checking whether each element is a member or not of the second list. There will be two cases:
skel([XIXs],Ys) - mernber(X,Ys), skel(Xs,Ys).
skel([XIXs],Ys) - nonmember(X,Ys), skel(Xs,Ys).
skel([ ],Ys).
To calculate the union, we need a third argument, which can be built top-down in the style discussed in Section 7.5. We consider each clause in turn. When an element in the first list is a member of the second list, it is not included in the union. When an element in the first list is not a member of the second list, it is included in the union. When the first list is empty, the union is the second list. The enhancement for union is given as Program 13.1.
241 Program Development
union(Xs,Ys,Us) Us is the union of the elements ¡n Xs and Ys.
union([XIXs] ,Ys,Us) .- member(X,Ys), union(Xs,Ys,Us). union([XIXs],Ys,[XIUs]) nonmember(X,Ys), unïon(Xs,Ys,Us). union([ ],Ys,Ys).
Program 13.1 Finding the union of two lists
intersect (Xs, Ys,Is) - Is is the intersection of the elements in Xs and Ys.
intersect([XIX$],Ys,[XIIs]) - mernber(X,Ys), intersect(Xs,Ys,Is). intersect([XIXS],Ys,Is) - nonniember(X,Ys), intersect(Xs,Ys,Is). intersect([ 1,Ys,[ 1).
Program 13.2 Finding the intersection of two lists
union_intersect (Xs, Ys, Us,Is) Us and Is are the union and intersection, respectively, of the elements in Xs and Ys.
union_intersect([XIXs] ,Ys,Us, EXils]) - inernber(X,Ys) , union_intersect(Xs,Ys,Us,Is)
union_intersect([XIXs] ,Ys, [XIUs] ,Is) nonniember(X,Ys), union_intersect(Xs,Ys,Us,Is)
union_intersect([ ],Ys,Ys,[ 1)
Program 13.3 Finding the union and intersection of two lists
The intersection, given as Program 13.2, is determined with a similar technique. We again consider each clause in turn. When an element in the first list is a member of the second list, it is included in the intersection. When an element in the first list is not a member of the second list, it is not included in the intersection. When the first list is empty, so is the intersection.
Calculating both the union and the intersection can be determined in a single traversal of the first list by composing the two enhancements. This program is given as Program 13.3.
Developing a program is typically straightforward once the skeleton has been decided. Knowing what skeleton to use is less straightforward
242 Chapter 13
and is learned by experience. Experience is necessary for any design task. By splitting up the program development into three steps, however, the design process is simplified and given structure.
A motivation behind giving programs structure, as is done by stepwise enhancement, is to facilitate program maintenance. It is easy to extend a program by adding new techniques to a skeleton, and it is possible to improve programs by changing skeletons while maintaining techniques. Further, the structure makes it easy to explain a program.
Skeletons and techniques can be considered as constituting reusable software components. This will be illustrated in Chapter 17, where the same skeleton meta-interpreter is useful both for program debugging and for expert system shells.
Having raised software engineering issues such as maintainability and reusabiity, we conclude this chapter by examining two other issues that must be addressed if Prolog is to be routinely used for large software projects. The place of specifications should be clarified, and modules are necessary if code is to be developed in pieces.
It is clear from the previous section that we do not advocate using first-order logic as a specification language. Still, it is necessary to have a specification, that is, a document explaining the behavior of a program sufficiently so that the program can be used without the code having to be read. We believe that a specification should be the primary form of documentation and be given for each procedure in a program.
A suggested form for a specification is given in Figure 13.1. It consists of a procedure declaration, effectively giving the name and arity of the predicate; a series of type declarations about the arguments; a relation scheme; and other important information such as modes of use of the predicate and multiplicities of solutions in each mode of use. We discuss each component in turn.
Types are emerging as important in Prolog programs. An untyped lan- guage facilitates rapid prototyping and interactive development, but for more systematic projects, imposing types is probably worthwhile.
The relation scheme is a precise statement in English that explains the relation computed by the program. All the programs in this book have a relation scheme. It should be stressed that relation schemes must be precise statements. We believe that proving properties of programs will proceed in the way of mathematics, where proofs are given by precise statements in an informal language.
243 Program Development
procedure p(Tj,T2 .....T) Types: T1: type I
T2: type 2
T: type n Relation scheme:
Modes of use:
Multiplicities of' solution:
Figure 13.1 Template for a specification
Prolog programs inherit from logic programs the possibility of be- ing multi-use. In practice, multi-use is rare. A specification should state which uses are guaranteed to be correct. That is the purpose of the modes of use component in Figure 13.1. Modes of use are specified by the instantiation state of arguments before and after calls to the predi- cate.
For example, the most common mode of use of Program 3.15 for ap- pend(Xs,Ys,Zs) for concatenating two lists Xs and Ys to produce a list Zs is as follows. Xs and Ys are instantiated at the time of call, whereas Zs is not, and all three arguments are instantiated after the goal suc- ceeds. Calling append/3 with all three arguments instantiated is a dif- ferent mode of use. A common convention, taken from DEC-10 Prolog is to use + for an instantiated argument, - for an uninstantiated argu- ment, and? for either. The modes for the preceding use of append are append(+, +, ) before the call and append(+, +, +) after the call.
More precise statements can be made by combining modes with types. The mode of use of the current example becomes the following: Before the call the first two arguments are complete lists and the third a vari- able; after the call all three arguments are complete lists.
Multiplicities are the number of solutions of the predicate, and should be specified for each mode of use of the program. lt is useful to give both the minimum and maximum number of solutions of a predicate. The multiplicities can be used to reason about properties of the program.
Modules are primarily needed to allow several people to work on a project. Several programmers should be able to develop separate compo- nents of a large system without worrying about undesirable interactions
244 Chapter 13
such as conflict of predicate names. What is needed is a mechanism for specifying what is local to a module and which predicates are imported and exported.
Current Prolog systems provide primitive facilities for handling mod- ules. The current systems are either atom-based or predicate-based, de- pending on what is made local to the module. Directives are provided for specifying imports and exports. Experience is growing in using existing module facilities, which will be translated into standards for modules that will ultimately be incorporated into Standard Prolog. The current draft on modules in Standard Prolog is in too much flux to describe here. The user needing modules should consult the relevant Prolog manual.
Exercises for Section 13.3
Enhance Program 13.3 to build the list of elements contained in the first list but not in the second list.
Write a program to solve the following problem. Given a binary tree T with positive integers as values, build a tree that has the same structure as T but with every node replaced by the maximum value in the tree. It can be accomplished with one traversal of the tree. (Hint: Use Program 3.23 as a skeleton.)
Write a program to calculate the mean and mode of an ordered list of numbers in one pass of the list.
13.4 Background
Commenting on Prolog programming style has become more prevalent in recent Prolog textbooks. There are useful discussions in both Ross (1989) and O'Keefe (1990). The latter book also introduces program schemas, which have parallels with skeletons and techniques.
Stepwise enhancement has emerged from ongoing work at Case West- ern Reserve University, first in the COMPOSERS group and more recently in the ProSE group. Examples of decomposing Prolog programs into skeletons and techniques are given in Sterling and Kirschenbaum (1993) and presented in tutorial form in Deville, Sterling, and Deransart (1991).
245 Program Development
Underlying theory is given in Power and Sterling (1990) and Kirschen- baum, Sterlmg, and fain (1993). An application of structuring Prolog programs using skeletons and techniques to the inductive inference of Prolog programs can be found in Kirschenbaum and Sterling (1991).
Automatic incorporation of techniques into skeletons via partial evalu- ation has been described in Lakhotia (1989).
The discussion on specifications for Prolog programs is strongly influ- enced by Deville (1990).
Exercise 13.3(11) was suggested by Giles Kahn. The example is orig- inally due to Bird. Exercise 13.3(iii) emerged through interaction with Marc Kirschenbaum. Solutions to both exercises are given in Deville, Ster- ling, and Deransart (1991).
Leonardo Da Vinci. Study of a Woman 's Hands folded over her Breast. Silver- point on pink prepared paper, heightened with white. About 1478. Windsor Castle, Royal Library.
1H Advanced Prolog Programming Techniques
The expressive power and high-level nature of logic programming can be exploited to write programs that are not easily expressed in conven- tional programming languages. Different problem-solving paradigms can be supported, and alternative data construction and access mechanisms can be used.
The simple Prolog programs of the previous part are examples of the use of basic programming techniques, reinterpreted ni the context of logic programming This part collects more advanced techniques that have evolved ni the logic programming community and exploit the spe- cial features of logic programs. We show how they can be used to advan- tage.
14 Nondeterniinistic Prograniming
One feature of the logic programming computation model lacking in con- ventional programming models is nondetermiriism. Nondeterminism is a technical concept used to define, in a concise way, abstract computa- tion models. However, in addition to being a powerful theoretical con- cept, nondeterminism is also useful for defining and implementmg algo- rithms. This chapter shows how, by thinking nondeterministically, one can construct concise and efficient programs.
Intuitively, a nondeterministic machine can choose its next operation correctly when faced with several alternatives. True nondeterministic machines cannot be realized but can be simulated or approximated. In particular, the Prolog interpreter approximates the nondeterministic be- havior of the abstract interpreter of logic programs by sequential search and backtracking, as explained in Chapter 6. However, the fact that non- determinism is only simulated without being "really present" can be ab- stracted away in many cases in favor of nondeterministic thinking in much the same way as pointer manipulation details involved in unifica- tion can be abstracted away in favor of symbolic thinking.
14.1 Generate-and-Test
Generate-and-test is a common technique in algorithm design and pro- gramming Here is how generate-and-test works for problem solving. One process or routine generates candidate solutions to the problem, and an- other process or routine tests the candidates, trying to find one or all candidates that actually solve the problem.
250 Chapter 14
It is easy to write logic programs that, under the execution model of Prolog, implement the generate-and-test technique. Such programs typi- cally have a conjunction of two goals, in which one acts as the generator and the other tests whether the solution is acceptable, as in the following clause:
f ind(X) - generate(X), test(X).
This Prolog program would actually behave like a conventional, procedu- ral, generate-and-test program. When called with f ind(X)?, generate (X) succeeds, returning some X, with which test (X) is called. If the test goal fails, execution backtracks to generate (X), which generates the next element. This continues iteratively until the tester successfully finds a solution with the distinguishing property or until the generator has ex- hausted all alternative solutions.
The programmer, however, need not be concerned with the generate- and-test cycle and can view this technique more abstractly, as an instance of nondetermirilstic programming. In this nondeterministic program the generator guesses correctly an element in the domain of possible solu- tions, and the tester simply verifies that the guess of the generator is correct.
A good example of a program with multiple solutions and com- monly used as a generator is Program 3.12 for member. The query mem- ber(X, [a,b,c])? will yield the solutions X=a, X=b, and X=c successively as required. Thus member can be used to nondeterministically choose the correct element of a list in a generate-and-test program.
Program 14.1 is a simple example of generate-and-test using mem- ber as a generator. The program identifies parts of speech of a sen- tence. We assume that a sentence is represented as a list of words and that there is a database of facts giving the parts of speech of particular words. Each part of speech is a unary predicate whose argument is a word, for example, noun (man) indicates that man is a noun. The relation verb(Sentence,Word) is true if Word is a verb in sentence Sentence. The analogous meanings are intended for noun/2 and article/2. The query verb([a,man,loves,a,woman],V)? finds the verb V'loves in the sentence using generate-and-test. Words in the sentence are generated by member and tested to see if they are verbs.
251 Nondeterministic Programming
verb(Sentence,Verb) - Verb is a verb in the list of words Sentence.
verb(Sentence ,Word) - mernber(Word,Sentence), verb(Word).
noun(Sentence,Word) - member(Word,Sentence), noun(Word).
article(Sentence,Word) - member(Word,Sentence) , article(Word)
Vocabulary
noun (man) . noun (woman)
article (a). verb(loves).
member(X,Xs) '- SeeProgram3.12.
Program 14.1 Finding parts of speech in a sentence
Another simple example is testing whether two lists have an element in common. Consider the predicate intersect (Xs,Ys), which is true if Xs and Ys have an element in common:
intersect(Xs,Ys) - member(X,Xs), member(X,Ys).
The first member goal in the body of the clause generates members of the first list, which are then tested to see whether they are in the second list by the second member goal Thinking nondeterrninistically, the first goal guesses an X in Xs, and the second verifies that the guess is a member of Ys.
Note that when executed as a Prolog program, this clause effectively implements two nested loops, The outer ioop iterates over the elements of the first list, and the irmer loop checks whether the chosen element is a member of the second list. Hence this nondetermiriistic logic program achieves, under the execution model of Prolog, a behavior very similar to the standard solution one would compose for this problem in Fortran, Pascal, or Lisp.
The definition of member in terms of append,
member(X,Xs) - append(As, [X lBs] ,Xs).
is itself essentially a generate-and-test program. The two stages, how- ever, are amalgamated by the use of unification. The append goal gen- erates splits of the list, and immediately a test is made whether the first element of the second list is X.
Typically, generate-and-test programs are easier to construct than pro- grams that compute the solution directly, but they are also less efficient.
252 Chapter 14
A standard technique for optimizing generate-and-test programs is to "push" the tester inside the generator as deeply as possible. Ultimately, the tester is completely intertwined with the generator, and only correct solutions are generated.
Let us consider optimizing generate-and-test programs by pushing the tester into the generator. Program 3.20 for permutation sort is another example of a generate-and-test program. The top level is as follows:
sort(Xs,Ys) - permutation(Xs,Ys), ordered(Ys).
Abstractly, this program guesses nondeterministically the correct permu- tation via permutation(Xs,Ys), and ordered checks that the permuta- tion is actually ordered.
Operationally, the behavior is as follows. A query involving sort is re- duced to a query involving permutation and ordered. A failure-driven loop ensues. A permutation of the list is generated by permutation and tested by ordered. If the permuted list is not ordered, the execution backtracks to the permutation goal, which generates another permuta- tion to be tested. Eventually an ordered permutation is generated and the computation terminates.
Permutation sort is a highly inefficient sorting algorithm, requiring time super-exponential in the size of the list to be sorted. Pushing the tester into the generator, however, leads to a reasonable algorithm. The generator for permutation sort, permutation, selects an arbitrary ele- ment and recursively permutes the rest of the list. The tester, ordered, verifies that the first two elements of the permutation are in order, then recursively checks the rest. If we view the combined recursive permuta- tion and ordered goals as a recursive sorting process, we have the basis for insertion sort, Program 3.21. To sort a list, sort the tail of the list and insert the head of the list into its correct place in the order. The arbitrary selection of an element has been replaced by choosing the first element.
Another example of the advantage of intertwining generating and test- ing can be seen with programs solving the N queens problem.
The N queens problem requires the placement of N pieces on an N- by-N rectangular board so that no two pieces are on the same line: hori- zontal, vertical, or diagonal. The original formulation called for 8 queens to be placed on a chessboard, and the criterion of not being on the same line corresponds to two queens not attacking each other under the rules of chess. Hence the problem's name.
253 Nondeterministic Programming
QII Q Q
Figure 14.1 A solution to the 4 queens problem
queens(N,Queens) Queens is a placement that solves the N queens problem, represented as a permutation of the list of numbers 11,2,... ,N1.
queens(N,Qs) rane(1,N,Ns), permutation(Ns,Qs), safe(Qs).
safe(Qs) - The placement Qs is safe.
safe([QIQs]) - safe(Qs), not attack(Q,Qs). safe([ ]). attack(X,Xs) - attack(X,1,Xs). attack(X,N,[YIYs]) - X is Y+N X is Y-N. attack(X,N,[YIYs]) Nl is N+l, attack(X,Nl,Ys). permutation(Xs,Ys) - See Program 3.20. rarige(M,N,Ns) - See Program 8.12.
Program 14.2 Naive generate-and-test program solving N queens
The program has been well studied in the recreational mathematics lit- erature. There is no solution for N = 2 and N 3, and a unique solution up to reflection for N = 4, shown in Figure 14.1. There are 88 solutions for N = 8, or 92, depending on strictness with symmetries.
Program 14.2 is a simplistic program solving the N queens problem. The relation queen (N,Qs) is true if Qs is a solution to the N queens prob- lem. Solutions are specified as a permutation of the list of the numbers i to N. The first element of the lìst is the row number to place the queen in the first column, the second element indicates the row number to place the queen in the second column, etc. Figure 14.1 indicates the solution 12,4,1,3] to the 4 queens problem. This specification of solutions, and
254 Chapter 14
the program generating them, has implicitly incorporated the observa- tion that any solution to the N queens problem will have a queen on each 'row and a queen on each colunm.
The program behaves as follows. The predicate range creates a list Ns of the numbers from i to N. Then a generate-and-test cycle begins. The permutation predicate generates a permutation Qs of Ns, which is tested to see whether it is a solution to the problem with the predi- cate safe(Qs). This predicate is true if Qs is a correct placement of the queens. Since two queens are not placed on the same row or colunm, the predicate need only check whether two queens attack each other along a diagonal. Safe is defined recursively. A list of queens is safe if the queens represented by the tail of the list are safe and the queen represented by the head of the list does not attack any of the other queens. The def- inition of attack(Q , Qs) uses a neat encapsulation of the interaction of diagonals. A queen is on the same diagonal as a second queen N columns away if the second queen's row number is N units greater than, or N units less than, the first queen's row number. This is expressed by the first clause of attack/3 in Program 14.2. The meaning of attack(Q,Qs) is that queen Q attacks some queen in Qs. The diagonals are tested itera- tively until the end of the board is reached.
Program 14.2 cannot recognize when solutions are symmetric. The program gives two solutions to the query queens(4,Qs)?, namely Qs=[2,4,1,3] and Qs=[3,1,4,2].
Although it is a well-written logic program, Program 14.2 behaves inef- ficiently. Many permutations are generated that have no chance of being solutions. As with permutation sort, we improve the program by pushing the tester, in this case safe, into the generator.
Instead of testing the complete permutation, that is, placing all the queens, each queen can be checked as it is being placed. Program 14.3 computes solutions to the N queens problem by placing the queens one at a time. It also proceeds by generating and testing, in contrast to inser- tion sort, which became a deterministic algorithm by the transformation. The generator in the program is select and the tester is attack, or more precisely its negation.
The positions of the previously placed queens are necessary to test whether a new queen is safe. Therefore the final solution is built upward using an accumulator. This is an application of the basic technique de- scribed in Section 7.5. A consequence of using an accumulator is that the queens are placed on the right-hand edge of the board. The two solu-
255 Nondeterministic Programming
queens(N,Queens) - Queens is a placement that solves the N queens problem, represented as a permutation of the list of numbers [1,2.....NJ.
queens(N,Qs) - range(1,N,Ns), queeris(Ns,[ ],Qs). queens (UriplacedQs , SaÍeQs , Qs)
select (Q ,Unplacedqs ,UnplacedQsl), not attack(Q,SaÍeQs), queens (UnplacedQsl, [QISafeQs] ,Qs).
queens([ ],Qs,Qs). select(X,Xs,Ys) - SeeProgram3.19. attack(X,Xs) - See Program 14.2.
Program 14.3 Placing one queen at a time
Figure 14.2 A map requiring four colors
tions to the query queens(4,Qs)? are given in the opposite order to the solutions given by Program 14.2.
The next problem is to color a planar map so that no two adjoining re- gions have the same color. A famous conjecture, an open question for a hundred years, was proved in 1976, showing that four colors are suffi- cient to color any planar map. Figure 14.2 gives a simple map requiring four colors to be colored correctly. This can be proved by enumeration of the possibilities. Hence four colors are both necessary and sufficient.
Program 14.4, which solves the map-coloring problem, uses the generate-and-test programming technique extensively. The program im- plements the following nondeterministic iterative algorithm:
For each region of the map, choose a color, choose (or verify) colors for the neighboring regions from the
remaining colors.
256 Chapter 14
color_map (Map,Colors) - Map is colored with Colors so that no two neighbors have the same color. The map is represented as an adjacency-list of regions
region(Name,Color,Neighbors), where Name is the name of the region, Color is its color, and Neighbors are the colors of its neighbors.
color_map([RegionlRegions] ,Colors) -
color_region(Region,Colors),
color_map(Regions Colors).
color_map( E I Colors).
color_region (Region, Colors) - Region and its neighbors are colored using Colors so that the region's color is different from the color of any of its neighbors.
color_region(region(Name,Color,Neighbors) Colors) -
select (Color Colors ,Colorsl),
members (Neighbors ,Colorsi).
select(X,Xs,Ys) See Program 3.19.
members(Xs,Ys) See Program 7.6.
Program 14.4 Map coloring
A data structure is needed to support the algorithm. The map is repre- sented as a list of regions. Each region has a name, a color, and a list of colors of the adjoining regions. The map in Figure 14.2, for example, is represented as
[region(a,A, [B,C,D]) ,region(b,B, [A,C,E]),
region(c,C, [A,B,D,E,F]) ,region(d,D, [A,C,F]),
region(e,EJB,C,F]),region(f,F,[C,D,E])].
The sharing of variables is used to ensure that the same region is not colored with two different colors by different iterations of the algorithm.
The top-level relation is color_map (Map,Colors), where Map is repre- sented as before, and Colors is a list of colors used to color the map. Our colors are red, yellow, blue, and white. The heart of the algorithm is the definition of color_region(Region,Colors):
color_region(region(Nanie,Color,Neighbors) ,Colors) -
select (Color,Colors,Colorsl), members(Neighbors,Colorsl).
257 Nondeterministic Programming
Test data test_color(Name,Map)
map (Name , Map)
colors (Naine,Colors) color_map (Map,Colors)
map(test, [region(a,A, [B,C,D]),region(b,B, [A,C,E]), region(c,C,[A,B,D,E,F]),region(d,D,[A,C,F]), region(e,E, [B,C,F]),region(f,F, [C,D,E])]).
map(west_europe, [region(portugal ,P, [E]), region(spain,E, [F,PI), region(fraiice,F,[E,I,S,B,WG,L]), region(belgiuxn,B,[F,H,L,WG]), regïon(holland,H,[B,WG]), region(west_germa.ny,WG,[F,A,S,H,B,L]), region(luxembourg,L,[F,B,WG]), region(italy,I,[F,A,S]), region(switzerland,S,[F,I,A,WG]), region(austria,A,[I,S,WG])]).
colors(X, [red,yellow,blue,white])
Program 14.3 Test data for map coloring
Both the select and members goals can act as generators or testers, depending on whether their arguments are instantiated.
Overall, the effect of the program is to instantiate a data structure, the map. The calls to select and members can be viewed as specifying local constraints. The predicates either generate by instantiating arguments in the structure or test whether instantiated values satisfy local constraints. Program 14.5 tests the map coloring solution.
Instantiating a data structure designed especially for a problem is a particularly effective means of implementing generate-and-test solutions. Unification and failure to unify control the building of the final solution structure, avoiding creation of unnecessary intermediate data structures. Since unification is supported well by Prolog implementations, solutions are found quickly. Exercise 14.1(iv) assigns the task of designing a data structure that can be instantiated to solve the N queens problem. The resulting program solves the N queens problem much more quickly than Program 14.3.
Our final example is solving a logic puzzle. The behavior of the pro- gram is similar to the map-coloring program. The logic puzzle consists of some facts about some small number of objects that have various at- tributes. The minimum number of facts is given about the objects and attributes, to yield a unique way of assigning attributes to objects.
258 Chapter 14
Here is an example that we use to describe the technique of solving logic puzzles.
Three friends came first, second, and third iii a programming competi- tion. Each of the three has a different first name, likes a different sport, and has a different nationality.
Michael likes basketball and did better than the American. Simon, the Israeli, did better than the tennis player. The cricket player came first.
Who is the Australian? What sport does Richard play? Logic puzzles such as this one are elegantly solved by instantiating
the values of a suitable data structure and extracting the solution val- ues. Each clue is translated into a fact about the data structure. This can be done before the exact form of the data structure is determined using data abstraction. Let us analyze the first clue: "Michael likes basketball and did better than the American." Two distinct people are referred to. One is named Michael, whose sport is basketball, and the other is Amer- ican. Further, Michael did better than the American. If we assume the structure to be instantiated is Friends, then the clue is expressed as the conjunction of goals
did_better(Nanl,Man2,Friends), f irst_name(Manl,michael),
sport(Manl,basketball), nationality(Man2,arnerican),
Similarly, the second clue can be translated to the conditions
did_better(Nani,Man2,Friends), f irst_name(Manl,sinion),
nationality(Manl,israeli), sport(Man2,tennis),
and the third clue to the conditions
f irst(Friends,Man), sport(Man,cricket).
A framework for solving puzzles is given as Program 14.6. The rela- tion computed is solve_puzzle(Puzzle,Solution), where Solution is the solution to Puzzle. The puzzle is represented by the structure puz- zle(Clues,Queries,Solution), where the data structure being instan- tiated is incorporated into the clues and queries, and the values to be extracted are given by Solution.
The code for solve_puzzle is trivial. All it does is successively solve each clue and query, which are expressed as Prolog goals and are exe- cuted with the meta-variable facility.
The clues and queries for our example puzzle are given in Program 14.7. We describe the structure assumed by the clues to solve the puzzle.
259 Nondetermiriistic Programming
solve_puzzle(Puzzle,Solution) - Solution is a solution of Puzzle, where Puzzle is puzzle (Clues, Queries,Solution).
solve_puzzle (puzzle(Clues,Queries,Solution) ,Solution) -
solve (Clues)
solve (Queries)
solve([CluelClues]) -
Clue, solve(Clues).
solve([ I).
Program 14.6 A puzzle solver
Each person has three attributes and can be represented by the structure friend(Name,Country,Sport). There are three friends whose order in the progranmiing competition is significant. This suggests an ordered sequence of three elements as the structure for the problem, ì.e., the list
[friend(Nl,C1,S1),friend(N2,C2,S2),friend(N3,C3,S3)].
The programs defining the conditions did_better, first_name, na tionality, sport, and first are straightforward, and are given in Program 14.7.
The combination of Programs 14.6 and 14.7 works as a giant generate- and-test. Each of the did_better and member goals access people, and the remaining goals access attributes of the people. Whether they are generators or testers depends on whether the arguments are instanti- ated or not. The answer to the complete puzzle, for the curious, is that Michael is the Australian, and Richard plays tennis.
The puzzle given in Program 14.7 is simple An interesting question is how well does the framework of Program 14.6 scale. A good example of a larger puzzle is given in Exercise 14.1(vi). Is the framework adequate for such a puzzle?
The short answer is yes. Prolog is an excellent language for solving logic puzzles. However, care must be taken when formulating the clues and queries. For example, the predicate member is often essential to spec- ify inclivìduals, as is done to formulate the query in Program 14.7. lt may be tempting to become systematic and begin the puzzle solution by spec- ifying all individuals by member goals. This can lead to very inefficient programs because too many choice-points are set up. In general, implicit checking of a condition is usually more efficient. Another observation is
260 Chapter 14
Test data
test_puzzle (Name ,Solution) -
structure (Name ,Structure),
clues (Name ,Structure, Clues)
queries (Name ,Structure, Queries ,Solution),
solve_puzzle (puzzle (Clues, Queries ,Solution) ,Solution).
structure(test, [friend(N1,C1,S1),friend(N2,C2,S2),friend(N3,C3,S3)]).
clues(test ,Friends,
[(did_better (ManiCluel ,Man2Cluel ,Friends), 'h Clue i
first_name (ManiCluel michael), sport(ManlCluei basketball),
nationality (Man2Cluel american)),
(did_better(ManlClue2,Man2Clue2,Friends), X Clue 2
first_name (ManlClue2, simon), nationality(ManlClue2, israeli),
sport (Man2Clue2 ,tennis)),
(first(Friends,ManClue3), sport(ManClue3,cricket)) X Clue 3
])
queries (test, Friends,
[ member(Qi,Friends),
first_name(Q1,Name),
nationality(Qi australian), 'h Query i
member(Q2,Friends),
first_naine(Q2,richard)
sport(Q2,Sport) X Query 2
[['The Australian is ', Name] ['Richard plays ', Sport]]
did_better(A,B, [A,B,C]).
did_better(A,C, [A,B,C]).
did_better(B,C,[A,B,C]).
first_name(friend(A,B,C) ,A).
nationality(friend(A,B,C) B).
sport (friend(A,B,C) ,C).
f irst ( [X IXs] , X)
Program 14.7 A description of a puzzle
261 Nondeterministic Programming
that the order of the goals in the queries can significantly affect run- ning time. It is best to worry about this once the problem formulation is correct. Determining appropriate goal order is a skill easily learned by experience.
Another tip concerns negative clues, such as "John is not the tailor." These clues are best regarded as specifying two separate individuals, John and the tailor, rather than as setting up a negative condition about one individual. The predicate select can be used instead of member to guarantee that individuals are different.
Exercises for Section 14.1
(i) Write a program to compute the integer square root of a natu- ral number N defined to be the number I such that j2 < N, but (I + 1)2 > N. Use the predicate between/3, Program 85, to generate successive natural numbers on backtracking.
(II) Write a program to solve the stable marriage problem (Sedgewick, 1983), stated as follows:
Suppose there are N men and N women who want to get married. Each man has a list of all the women in his preferred order, and each woman has a list of all the men in her preferred order. The problem is to find a set of marriages that is stable. A pair of marriages is unstable if there are a man and woman who prefer each other to their spouses. For example, consider the pair of marriages where David is married to Paula, and Jeremy is married to Judy. If David prefers Judy to Paula, and Judy prefers David to Jeremy, the pair of marriages is unstable. This pair would also be unstable if Jeremy preferred Paula to Judy, and Paula preferred Jeremy to David. A set of marriages is stable if there is no pair of unstable marriages.
Your program should have as input lists of preferences, and pro- duce as output a stable set of marriages. It is a theorem from graph theory that this is always possible. Test the program on the follow- ing five men and five women with their associated preferences:
avraham: chana tamar zvia ruth sarah binyamin: zvia chana ruth sarah tamar chaim: chana ruth tamar sarah zvia david: zvia ruth chana sarah tamar elazar: tamar ruth chana zvia sarah
262 Chapter 14
zvia: elazar avraham david binyamin chaim chana: david elazar binyamin avraham chaim ruth: avraham david binyamin chaim elazar sarah: chaim binyamin david avraham elazar tamar: david binyamin chaim elazar avraham
Use Program 14.4 to color the map of Western Europe. The coun- tries are given in Program 14.5.
Design a data structure for solving the N queens problem by instan- tiation. Write a program that solves the problem by instantiating the structure.
(y) Explain why the following program solves the N queens problem:
queens (N, Qs)
gen_list(N,Qs), place_queens(N,Qs,Ups,Downs).
gen_list(O,[ 1). gen_list(N,[QIL]) - N > O, Nl is N-1, gen_list(N1,L).
place_queens (O,Qs ,Ups,Downs).
place_queens(I,Qs,Ups, [DlDowns]) -
I > O, Il is I-1,
place_queens(I1,Qs, [UlUps] ,Downs),
place_queen(I ,Qs,Ups,Downs).
place_queen(Q, [QIQs] , [QiUpsi , [QlDowns]).
place_queen(Q, [Q1IQs], [UlUps] , [DlDowns] -
place_queen(Q,Qs ,Ups,Downs).
(vi) Write a program to solve the following logic puzzle. There are five houses, each of a different color and inhabited by a man of a differ- ent nationality, with a different pet, drink, and brand of cigarettes.
The Englishman lives in the red house.
The Spaniard owns the dog.
Coffee is drunk in the green house.
The Ukrainian drinks tea.
263 Nondeterministic Programming
The green house is immediately to the right (your right) of the ivory house.
The Winston smoker owns snails.
Kools are smoked in the yellow house.
Milk is drunk in the middle house.
The Norwegian lives in the first house on the left.
The man who smokes Chesterfields lives in the house next to the man with the fox.
Kools are smoked in the house next to the house where the horse is kept.
(1) The Lucky Strike smoker drinks orange juice.
The Japanese smokes Parliaments.
The Norwegian lives next to the blue house.
Who owns the Zebra? Who drinks water?
(vii) Write a program to test whether a graph is planar using the algo- rithm of Hoperoft and Tarjan (Deo, 1974; Even, 1979).
14.2 Don't-Care and Don't-Know Nondeterminism
Two forms of nondeterminism are distinguished in the logic program- ming literature. They differ in the nature of the choice that must be made among alternatives. For don't-care nondeterminism, the choice can be made arbitrarily. In terms of the logic programming computation model, any goal reduction will lead to a solution, and it does not matter which particular solution is found. For don't-know nondeterminism, the choice matters but the correct one is not known at the time the choice is made.
Most examples of don't-care nondeterminism are not relevant for the Prolog programmer. A prototypical example is the code for minimum. Program 3.7 is the standard, incorporating a limited amount of don't-care nondetermïriism, namely, when X and Y are the same:
264 Chapter 14
minimum(X,Y,X) - X Y.
minimum(X,Y,Y) - Y X.
In Section 7.4, we termed this redundancy and advised against its use. On the other hand, programs exhibiting don't-know nondeterminism
are common. Consider the program for testing whether two binary trees are isomorphic (Program 3.25, reproduced here). Each clause is indepen- dently correct, but given two isomorphic binary trees, we don't know which of the two recursive clauses should be used to prove the isomor- phism. Operationally, only when the computation terminates success- fully do we know the correct choice:
isotree(void,void).
isotree(tree(X,L1,R1),tree(X,L2,R2))
isotree(L1,L2), isotree(R1,R2).
isotree(tree(X,L1,R1),tree(X,L2,R2)) -
isotree(L1,R2), isotree(L2,R1).
Composing Prolog programs exhibiting either form of nondeterminism can be indistinguishable from composing deterministic programs. Each clause is written independently. Whether inputs match only one clause or several is irrelevant to the programmer. Indeed this is seen from the multiple uses that can be made of Prolog programs. With arguments in- stantiated in one way, the program is deterministic; with another pattern of instantiation, the program is nondeterministic. For example, append/3 is deterministic if called with its first two arguments instantiated, while it is generally nondeterministic if called with the third argument instan- tiated and the first two arguments uninstantiated.
The behavior of Prolog programs seemingly having don't-know nonde- terminism such as isotree is known. A given logic program and a query determine a search tree, as discussed in Chapter 5, which is searched depth-first by Prolog. Writing a program possessing don't-know nonde- terminism is really specifying a depth-first search algorithm for solving the problem.
We consider this viewpoint in a little more detail with a particular example: finding whether two nodes in a graph are connected. Figure 14.3 contains two graphs that will be used to test our ideas. The left- hand one is a tree, while the right-hand one is not, containing a cycle. Trees, or more generally, directed acyclic graphs (DAGs), behave better than graphs with cycles, as we will see in our example programs.
265 Nondeterministic Programming
Figure 14.3 Directed graphs
connected(X,Y) Node X is connected to node Y, given an edge/2 relation describing a DAG.
connected(X,X) conriected(X,Y) - edge(X,N), connected(N,Y). Data
edge (a, b) edge (a , c) edge(a,d) edge (a, e) edge(d,j) edge(c,f) edge (c , g) edge(f,h). edge(e,k). edge (f , i) edge (z, y) edge (y, z) edge (z , x) edge (y, u) edge(z,v).
Program 14.8 Connectivity in a finite DAG
Our first program is a small modification of a logic program of Section 2.3. Program 14.8 defines the relation connected(X,Y), which is true if two nodes in a graph, X and Y, are connected. Edges are directed; the fact edge(X,Y) states that a directed edge exists from X to Y. Declaratively the program is a concise, recursive specification of what it means for nodes in a graph to be connected. Interpreted operationally as a Prolog program, it is the implementation of an algorithm to find whether two nodes are connected using depth-first search.
The solutions to the query connected(a,X)? using the data from the left-hand graph in Figure 14.3 gives as values for X, a, b, c, f, h, i, g, d, j, e, k. Their order constitutes a depth-first traversal of the tree.
Program 14.9 is an extension of this simple program that finds a path between two nodes. The predicate path(X,Y,Path) is true if Path is
266 Chapter 14
path(X,Y,Path) - Path is a path between two nodes X and Y in the DAG defined by the relation edge/2.
path(X,X, [X]).
path(X,Y,[XIP]) - edge(X,N), path(N,Y,P).
Program 14.9 Finding a path by depth-first search
connected(X,Y) - Node X is connected to node Y in the graph defined by edge/2.
connected(X,Y) - connected(X,Y, [X]).
connected(X,X,Visited).
connected(X,Y,Visited) - edge(X,N), not member(N,Visited), connected(N,Y, [NiVisited]).
Program 14.10 Connectivity in a graph
a path from the node X to the node Y in a graph. Both endpoints are included in the path. The path is built downward, which fits well with the recursive specification of the connected relation. The ease of computing the path is a direct consequence of the depth-first traversal. Extending a breadth-first traversal to find the path is much more difficult. Sections 16.2 and 20.1 show how it can be done.
Depth-first search, dfs, correctly traverses any finite tree or DAG (di- rected acyclic graph). There is a problem, however, with traversing a graph with cycles. The computation can become lost in an infinite loop around one of the cycles. For example, the query connected(x,Node)?, referring to the right-hand graph of Figure 14.3 gives solutions Node=y, Node=z, and Node=x repeatedly without reaching u or y.
The problem is overcome by modifying connected. An extra argument is added that accumulates the nodes visited so far. A test is made to avoid visiting the same node twice. This is shown in Program 14.10.
Program 14.10 successfully traverses a finite directed graph depth- first. The pure Prolog program needed for searching finite DAGs must be extended by negation in order to work correctly. Adding an accumulator of paths visited to avoid entering loops effectively breaks the cycles in the graph by preventing traversal of an edge that would complete a cycle.
267 Nondeterrninistic Programming
q
Figure 14.4 Initial and final states of a blocks world problem
The program is not guaranteed to reach every node of an infinite graph. To do so, breadth-first search is necessary. This is discussed further in Section 16.2.
This section is completed with a program for building simple plans in the blocks world. The program is written nondeterministically, essen- tially performing a depth-first search. It combines the two extensions mentioned before - keeping an accumulator of what has been traversed, and computing a path.
The problem is to form a plan in the blocks world, that is, to specify a sequence of actions for restacking blocks to achieve a particular con- figuration. Figure 14.4 gives the initial state and the desired final state of a blocks world problem. There are three blocks, a, b, and c, and three places, p, q, and r. The actions allowed are moving a block from the top of a block to a place and moving a block from one block to another. For an action to succeed, the top of the moved block must be clear, and also the place or block to which it is being moved must be clear.
The top-level procedure of Program 14.11 solving the problem is transform(Statel,State2,Plaiì). A plan of actions, Plan, is produced that transforms Statel into State2 when executed.
States are represented by a list of relations of the form on(X,Y), where X is a block and Y is a block or place. They represent the facts that are true in the state. For example, the initial and final states in Figure 14.4 are, respectively, [on(a,b) ,on(b,p) ,on(c,r)] and [on(a,b) ,on(b,c) ,on(c,r)]. The state descriptions are ordered in the sense that the on relation for a precedes that of b, which precedes the on relation for c. The state descriptions allow easy testing of whether a block or place X is clear in a given state by checking that there is no relation of the form on(A,X). The predicates clear/2 and on/3 in Pro- gram 14.11 take advantage of this representation.
268 Chapter 14
transform (State 1,State2,Plan) Plan is a plan of actions to transform Statel into State2.
transform(Statel,State2,Plan) -
transform(Statel,State2, [Statel] Plan).
transform(State,State,Visited,[ 1). transform(Statel,State2,Visited, [ActionlActions]) -
legal_action(Action,Statel),
update(Action,Statel ,State),
not member(State,Visited),
transform(State,State2, [State IVisited] ,Actions).
legal_action(to_place(Block,Y,Place) ,State)
on(Block,Y,State), clear(Block,State),
place (Place), clear(Place,State).
legal_action(to_block(Blockl,Y,Block2) State) -
on(Blockl,Y,State), clear(Blockl,State), block(Block2),
Blocki Block2, clear(Block2,State)
clear(X,State) - not member(on(A,X),State).
on(X,Y,State) - member(on(X,Y) ,State).
update(to_block(X,Y,Z) ,State,Statei) -
substitute(on(X,Y),on(X,Z),State,Statel).
update(to_place(X,Y,Z),State,Statel) -
substitute(on(X,Y),on(X,Z),State,Statel).
substitute(X,Y,Xs,Ys) - See Exercise 3.3(i).
Program 14.11 A depth-first planner
The nondeterministic algorithm used by the planner is given by the recursive clause of transforin/4 in the program:
While the desired state is not reached, find a legal action, update the current state, check that it has not been visited before.
There are two possible actions, moving to a block and moving to a place. For each, the conditions for which it is legal must be specified, and a method given for updating the state as a result of performing the action.
Program 14.11 successfully solves the simple problem given as Pro- gram 14.12. The first plan it produces is horrendous, however:
269 Nondeterministic Programming
[to_place(a,b,q) ,to_block(a,q, c) ,to_place(b,p,q) ,to_place(a, c
to_block(a,p,b) ,to_place(c ,r,p) ,to_place(a,b,r) ,to_block(a,r, c),
to_place(b,q,r),to_place(a,c,q),to_block(a,q,b),to_place(c,p,q),
to_place(a,b,p),to_block(a,p,c),to_place(b,r,p),to_place(a,c,r),
to_block(b,p,a) ,to_place(c,q,p) ,to_block(b,a,c) ,to_place(a,r,q),
to_block(b,c,a),to_place(c,p,r),to_block(b,a,c),to_place(a,q,p),
to._block (a , p , b) ]
Block a is first moved to q, then to c. After that, block b is moved to q, block a is moved to p and b, and after 20 more random moves, the final configuration is reached.
lt is easy to incorporate a little more intelligence by first trying to achieve one of the goal states. The predicate legal_action can be re- placed by a predicate choose_action(Actiorj,Statel,State2). A sim- pie definition suffices to produce intelligent behavior in our example problem:
choose_action(Action,Statel,State2) -
suggest(Action,State2), legal_action(Action,Statel).
choose_action(ActionState1,State2) -
legal_action(Action,Statel).
suggest (to_place (X,Y,Z) ,State)
member(on(X,Z) State), place(Z).
suggest (to_block (X,Y,Z) ,State)
mernber(on(X,Z) ,State), block(Z).
The first plan now produced is [to_place(a,b,q),to_block(b,p,c), to_block (a, q, b)]
Testing and data
test_plan(Narne,Plan) - initial_state(Name,I), fïnal_state(Narne,F), transform(I,F,Plan).
ïnitial_state(test,[on(a,b),on(b,p),on(c,r)]). final_state(test, [on(a,b) ,on(b,c) ,on(c,r)]) block(a). block(b). block(c). place(p). place(q). place(r).
Program 14.12 Testing the depth-first planner
270 Chapter 14
Exercises for Section 14.2
Apply Program 14.11 to solve another simple blocks world prob- lem.
Modify Program 14.11 to solve the following planning problem. Consider a simplified computer consisting of a single accumula-
tor and a large number of general purpose registers. There are four instructions: load, store, add and subtract. From the initial state where the accumulator is empty, registerl contains the value cl, register2 contains c2, register3 contains c3 and register4 contains c4, achieve a final state where the accumulator contains
(cl - c2) + (c3 - c4)
(cl - c2) + (cl - c2)
cl, and registerl contains cl + (c2 - c3), and register2 contains c2 - c3.
14.3 Artificial Intelligence Classics: ANALOGY, EUZA, and McSAM
"The best way to learn a subject is to teach it" is a cliche commonly repeated to new teachers. An appropriate analogue for new programmers is that the best way to understand a program is to rewrite or extend it. In this spirit, we present logical reconstructions of three AI programs. Each is clear, understandable, and easily extended. The exercises at the end of the section encourage the reader to add new facts and rules to the programs.
The three programs chosen are the ANALOGY program of Evans for solving geometric analogy questions from intelligence tests; the ELIZA program of Weizenbaum, which simulates or rather parodies conversa- tion; and McSAM, a microversion of SAM, a program for "understanding" stories from the Yale language group. Each logical reconstruction is ex- pressed very simply. The nondeterminism of Prolog allows the program- mer to ignore the issues of search.
Consider the task of solving the geometric analogy problems typically used in intelligence tests. Several diagrams are presented in a prototypi-
271 Nondeterministic Programming
¡sto A B
as
Figure 14.5 A geometric analogy problem
cal problem. Diagrams A, B, and C are singled out from a list of possible answers and the following question is posed: "A is to B as C is to which one of the 'answer' diagrams?" Figure 14.5 gives a simple problem of this type.
Here is an intuitive algorithm for solving the problem, where terms such as find, apply, and operation are left unspecified:
Find an operation that relates A to B. Apply the operation to C to give a diagram X. Find X, or its nearest equivalent, among the answers.
In the problem in Figure 14.5, the positions of the square and triangle are swapped (with appropriate scaling) between diagrams A and B. The "obvious" answer is to swap the square and the circle in diagram C. The resultant diagram appears as no. 2 in the possible answers.
Program 14.13 is a simple program for solving analogy problems. The basic relation is analogy (Pain , Pair2 ,Answers), where each Pair is of the form X is_to Y. To parse the program, is_to must be declared as an infix operator. The two elements in Pair i bear the same relation as the two elements in Pair2, and the second element in Pair2 appears in Answers. The definition of analogy implements the intuitive algorithm:
A 3
analogy(A is_to B,C is_to X,Answers) match(A,B,Dperation),
match(C,X,Operation), member(X,Answers).
o ¡sto C
272 Chapter 14
analogy (Pairl,Pair2,Answers) - An analOgy holds between the pairs of figures Pain and Pair2. The second element of Pair2 is one of the possible Answers.
analogy(A is_to B,C is_to X,Answers) -
match(A,B4Operation),
match(C,X,Operation),
member (X,Answers).
match(inside(Figurei ,Figure2) , inside(Figure2,Figurel) ,invert).
match(above(Figurel ,Figure2) ,above(Figure2,Figurel) ,invert).
Program 14.13 A program solving geometric analogies
A fundamental decision is how the figures in the problem should be represented. In Program 14.13, figures are represented as Prolog terms. Diagram A in Figure 14.5 is a square inside a triangle and is represented as the term inside(square,triangle).
The relation between two diagrams is found by the predicate match (A,B4Operation). This relation is true if Operation matches A and B. The operation invert is used to solve our example problem. The behav- ior of invert is to swap its arguments.
The predicate match is used in two distinct ways in this program. The first time it produces an operation matching two given diagrams The second time it produces a second diagram given an operation and a di- agram. These details are, however, irrelevant from the nondeterministic viewpoint. Finding the answer is implemented simply, checking whether a given diagram appears in the list of answers. Note that the second match goal and the member goal can be swapped. The behavior then be- comes guessing an answer using member as a generator, and verifying that the guess has the same relation to diagram C as A does to B. Pro- gram 14.14 tests the analogy program.
Exercise 14.3(i) poses three additional problems to be solved by anal- ogy, which can be expressed within the framework of Program 14.13. The representation of pictures by terms, and the description of operations becomes increasingly ad hoc. Indeed, much of the "intelligence" of the program is embedded in the representation. The original ANALOGY pro- gram did not assume as much knowledge. It took a line drawing and tried to recognize the objects. Triangles, squares, and so on, were not assumed to be the primitives.
273 Nondeterministic Programming
Testing and data
test_analogy(Nanie,X) figures(Name,A,B,C) answers (Naine ,Answers), analogy(A is_to B,C is_to X,Answers).
figures(testl,inside(square,triangle),inside(triangle,square) inside (circle , square)
answers(testl, [inside(circle,triangle) ,inside(square,circle) inside (triangle ,square)])
Program 14.14 Testing ANALOGY
1 am unhappy. » How long have you been unhappy?
Six months. Can you help me. » What makes you think I help you?
You remind me of my father and brother. >> Please tell me more about father.
I like teasing father. >> Does anyone else in your family like teasing father?
No, only me. >> Please go on.
Figure 14.6 Sample conversation with ELIZA
The next Al classic considered is a simplified version of ELIZA. The pro- gram aims to simulate a conversation. A user types in a sentence, and ELIZA responds with an appropriate question or comment. ELIZA does not understand in any real sense, responding to its input by recognizing word patterns and replying by using a corresponding response pattern. To make the patterns of responses more credible, a psychiatrist setting is adopted. A sample interaction with ELIZA is given in Figure 14.6. Com- puter responses are preceded by ».
The heart of ELIZA is a procedure for matching the input sentence against a pattern. The resulting match is applied to another pattern to de- termine the program reply. The pair of patterns can be considered a stim- ulus/response pair, where the input is matched against the stimulus and
274 Chapter 14
the output generated from the response. A typical stimulus/response pair is
I am (statement> How long have you been (statement>?
Using this pair, the response of the program to the input statement "I am unhappy" will be the question "How long have you been unhappy?" The (statement> can be viewed as a slot to be filled.
Program 14.15 is a simple version of ELIZA. It implements the follow- ing algorithm:
Read the input. While the input is not bye,
choose a stimulus/response pair, match the input to the stimulus, generate the reply from the response and the above match, output the response, read the next input.
The stimulus/response pairs are represented as facts of the form pat- tern(Stimulus,Response), where both Stimulus and Response are lists of words and slots. Slots in the patterns are represented by integers. The predicate match(Pattern,Table ,Words) is used for both the second and third steps of the algorithm. It expresses a relation between a pattern Pattern, a list of words Words, and a table Table, where the table records how the slots in the pattern are filled. A central part of the match proce- dure is played by a nondeterministic use of append to break up a list of words. The table is represented by an incomplete data structure, dis- cussed in more detail in Chapter 15. The missing procedure lookup/3 is given in Section 15.3. The reply is generated by reply(Words). which is a modified version of Program 12.1 for writeln that leaves spaces between words.
The final program presented in this section is Micro SAM or McSAM. It is a simplified version of the SAM (Script Applier Mechanism) program developed in the natural language group at Yale University. The aim of McSAM is to "understand" stories. Given a story, it finds a relevant script and matches the individual events of the story against the patterns in the script. In the process, events in the script not explicitly mentioned in the story are filled in.
275 Nondeterministic Programming
eliza Simulates a conversation via side effects.
eliza read_word_list(Input), eliza(Input), !. eliza([bye])
reply(['Goodbye. I hope I have helped you']). eliza(Input) -
pattern(Stirnulus ,Response) match(Stimulus,Dictionary,Input), match(Response,Dictionary,Output), reply(Output), read_word_list ( Input 1)
eliza(Inputl). match(Pattern,Dictionary,Words) -
Pattern matches the list of words Words, and matchings are recorded in the Dictionary.
match([NlPattern] ,Dictionary,Target) integer(N), lookup(N,Dictionary,LeftTarget), append(Leftlarget ,RightTarget ,Target), match(Pattern,Dictionary,RightTarget).
match([WordlPattern] ,Dictionary, [Word Target]) - atom(Word) , match(Pattern,Dictionary,Target).
match([ ] ,Dictionary, E ]) lookup(Key,Dictionary,Value) - See Program 15.8. pattern (Stimulus,Response)
Response is an applicable response pattern to the pattern Stimulus. pattern([ï,am,1],['How',long,have,you,been,l,?]). pattern([1,you,2,me],['What',makes,you,think,'I',2,you,?]). pattern([i,like,l],['Does',anyone,else,in,your,farnïly,like,l,?]). pattern(Ei,feel,l],E'Do',you,often,feel,that,way,?]). pattern([1,X,2],['Please',tell,me,rnore,about,X,.])
important (X) pattern ( [1] , E' Please ' , go , on ,
important (father). important (mother). important(son). important(sïster). important (brother). important (daughter). reply([HeadlTail]) - write(Head), write(' '), reply(Tail). reply([ ]) - nl. read_word_list(Xs) - See Program 12.2.
Program 14.15 ELIZA
276 Chapter 14
Input: John went to Leones, ate a hamburger, and left.
Output: John went to Leones. He was shown from the door to a seat. A waiter brought Joim a hamburger, which John ate by mouth. The waiter brought Joim a check, and John left Leones for another place.
Figure 14.7 A story filled in by McSAM
Both the story and the script are represented in terms of Schank's theory of conceptual dependency. For example, consider the input story in Figure 14.7, which is used as an example in our version of McSAM. The English version
"John went to Leones, ate a hamburger, and left"
is represented in the program as a list of lists:
E [ptrans, john, john, Xl, leones],
[ingest, X2, hamburger, X3],
[ptrans, Actor, Actor, X4, X5] ]
The first element in each list, ptrans and ingest, for example, is a term from conceptual dependency theory. The representation of the story as a list of lists is chosen as a tribute to the original Lisp version.
Programming McSAM in Prolog is a triviality, as demonstrated by Program 14.16. The top-level relation is rncsain(Story,Script), which expands a Story into its "understood" equivalent according to a rele- vant Script. The script is found by the predicate f ind(Story,Script, Defaults). The story is searched for a nonvariable argument that trig- gers the name of a script. In our example of John visiting Leones, the atom leones triggers the restaurant script, indicated by the fact trig- ger(leones,restaurant) in Program 14.17.
The matching of the story to the script is done by match(Script, Story), which associates lines in the story with lines in the script. Re- maining slots in the script are filled in by name_defaults(Defaults). The "output" is
[ptrans ,jo,john,place1,leones]
[ptrans , j ohn,john, door ,seat]
[mtrans , john ,waiter, hamburger]
277 Nondeterministic Programming
mcsam(Story,Script) - Script describes Story.
mcsaxnCStory,Script) find(Story,Script ,Defaults), match(Script ,Story) name_deau1ts(Defau1ts).
find(Story,Script ,Defaults) - f iliLer (Slot , Story) trigger(Slot.Name), script (Name,Script ,Defaults).
match(Script,Story) - Story is a subsequence of Script.
match(Script, E J) matcii([LinelScript] ,ELine Story]) - match(Script,Story) match([LinelScript],Story) match(Script,Story). filler (Slot,Story)
Slot is a word in Story.
filler(Slot ,Story) rnember([ActionlArgs] ,Story), member(Slot ,Args), nonvar(Slot)
name_defaults(De faults) Unifies default pairs in Defaults.
naine_defaults([ ]). name_defaults([[N,N]IL]) - name_defaults(L). name_defaults([[N1,N2]IL]) - Nl N2, na.me_defaults(L).
Program 14.16 McSAM
[ingest ,john,hamburger, [mouth,john]1
[atrans ,john, check, john,waiter]
[ptrans ,john, j ohn,leones, place2].
Its translation to English is given in Figure 14.7. The work done on the original McSAM was all in the searching and
pattern matching This is accomplished in Prolog by nondeterministic programming and unification.
278 Chapter 14
Testing and data test_mcsam(Name ,TinderstoodStory) -
story(Name,Story), mcsam(Story,UnderstoodStory).
story(test, [[ptrans, john, john, Xl, leones]
[ingest, X2, hamburger, X3],
[ptrans, Actor, Actor, X4, X5] ]).
script (restaurant,
[ptrans, Actor, Actor, EarlierPlace, Restaurant]
[ptrans, Actor, Actor, Door, Seat],
[mtrans, Actor, Waiter, Food]
[ingest, Actor, Food, [mouth, Actor] ],
[atrans, Actor, Money, Actor, Waiter]
[ptrans, Actor, Actor, Restaurant, Gone] ],
[Actor, customer] , [EarlierPlace, placet]
[Restaurant, restaurant] , [Door, door]
[Seat, seat] , [Food, meal] , [Waiter, waiter]
[Money, check], [Gone, place2] ] ).
trigger(leones restaurant)
trigger(waiter,restaurant).
Program 14.17 Testing McSAM
Exercises for Section 14.3
Extend ANALOGY, Program 14.13, to solve the three problems in Figure 14.8.
Extend ELIZA, Program 14.15, by adding new stimulus/response patterns.
If the seventh statement in Figure 14.6 is changed to be "I like teasing my father," ELIZA responds with "Does any one else in your family like teasing my father." Modify Program 14.15 to "fix" this behavior, changing references such as I, my, to you, your, etc.
Rewrite McSAM to use structures.
(y) Reconstruct another AI classic. A good candidate is the general problem solver GPS.
279 Nondeterministic Programming
(iii) Given:
D
7 OA
9 10
D D D
11 12 13 14 15
Figure 14.8 Three analogy problems
Given: L14 Lì
B
EÇDo1 Given:
280 Chapter 14
14.4 Background
Applying Prolog to generate-and-test problems has been very common. Many researchers have discussed the behavior of Prolog in solving the N queens problem and map coloring. A good discussion of how Prolog handles the N queens problem can be found in Elcock (1983). The N queens program given in Exercise 14.1(v), the fastest of which we are aware, is due to Thomas Fruewirth. A classification of generate-and-test programs in Prolog is given in Bansal and Sterling (1989).
Several researchers have used Prolog's behavior on generate-and- test problems as a reason to investigate alternative control of logic programs. Suggestions for improvement include co-routining incorpo- rated in IC-Prolog (Clark and McCabe, 1979) and intelligent backtracking (Bruynooghe and Pereira, 1984). Neither have been widely adopted into Prolog.
Other examples of solving puzzles by instantiating structures are given in a book by Evan Tick (1991) comparing Prolog program performance with concurrent logic progranmiing languages.
The zebra puzzle, Exercise 14.1(iv) did the rounds on the Prolog Digest in the early 1980s. It was used as an unofficial benchmark to test both the speed of Prolog implementations and the ability of Prolog program- mers to write clear code. The description of clues given in Program 14.7 was influenced by one of the solutions. The framework of Program 14.6 was tested extensively by Steven Kaminski in a course project at Case Western Reserve University. He took the first 20 puzzles of an avail- able puzzle book and solved them using the framework. Although very much a Prolog novice, he was able to use Prolog fairly easily to find so- lutions. His experience highlighted some interesting points, namely, how to handle negative information and the undesirability of too many choice points with redundant calls to select and member.
The definitive discussion of don't-care and don't-know nondetermin- ism in logic programming appears in Kowalski (1979a).
Program 14.11 for planning is a variant of an example from Kowalski (1979a). The original planning program in Prolog was WARPLAN (Warren, 1976), reproduced in Coelho et al. (1980). Exercise 14.2(li) was adapted from descriptions of WARPLAN's abilities in Coelho and Cotta (1988).
281 Nondeterministic Programming
ANALOGY constituted the Ph.D. thesis of Thomas Evans at MIT in the rnid-1960s. A good description of the program appears in Semantic Infor- mation Processing (Minsky, 1968). Evans's program tackled many aspects of the problem that are made trivial by our choice of representation, for example, identifying that there are triangles, squares, and circles in the diagrams. Our version, Program 14.13, emerged from a discussion group of Leon Sterling with a group of epistemics students at the University of Edinburgh.
ELIZA was originally presented in Weizenbaum (1966). Its performance led people to believe that a limited form of the Turing test had been passed. Weizenbaum, its author, was horrified by people's reactions to the program and to Al more generally, and he wrote an impassioned plea against taking the program too seriously (Weizenbaum, 1976). Our ver- sion, Program 14.15, is a slight variant of a teaching program attributed to Alan Bundy, Richard O'Keefe, and Henry Thompson, which was used for AI courses at the University of Edinburgh.
McSAM is a version of the SAM program, which was tailored for teaching AI programming (Schank and Riesbeck, 1981). Our version, Program 14.16, is due to Ernie Davis and Ehud Shapiro. More informa- tion about conceptual dependency can be found in Schank and Abelson (1977).
A rational reconstruction of GPS, suggested in Exercise 14.3(v), was shown to us by George Ernst.
15.1 Difference-Lists
Consider the sequence of elements 1,2,3. It can be represented as the difference between pairs of lists. It is the difference between the lists [1,2,3,4,5] and [4,51, the difference between the lists [1,2,3,8] and [8], and the difference between [1,2,3] and [J. Each of these cases is an instance of the difference between two incomplete lists [1,2,3 Xs] and Xs.
We denote the difference between two lists as a structure As\Bs, which is called a difference-list. As is the head of the difference-list and Bs the
15 Incomplete Data Structures
The programs presented so far have been discussed in terms of relations between complete data structures. Powerful programming techniques emerge from extending the discussion to incomplete data structures, as demonstrated in this chapter.
The first section discusses difference-lists, an alternative data struc- ture to lists for representing a sequence of elements. They can be used to simplify and increase the efficiency of list-processing programs. In some respects, difference-lists generalize the concept of accumulators. Data structures built from the difference of incomplete structures other than lists are discussed in the second section. The third section shows how tables and dictionaries, represented as incomplete structures, can be built incrementally during a computation. The final section discusses queues, an application of difference-lists.
284 Chapter 15
tail. In this example [1,2,3Xs]\Xs is the most general difference-list repre- senting the sequence 1,2,3, where [1,2,3Xs] is the head of the difference- list and Xs the tail.
Logical expressions are unified, not evaluated. Consequently the bi- nary functor used to denote difference-lists can be arbitrary. Of course, the user must be consistent in using the same functor in any one pro- gram. Another common choice of functor besides \ is -. The functor for difference-lists can also be omitted entirely, the head and the tail of the difference-list becoming separate arguments in a predicate. While this last choice has advantages from a perspective of efficiency, we use the functor \ throughout for clarity.
Lists and difference-lists are closely related. Both are used to repre- sent sequences of elements. Any list L can be trivially represented as a difference-list L\[ J. The empty list is represented by any difference-list whose head and tail are identical, the most general form being As\As.
Difference-lists are an established logic programming technique. The use of difference-lists rather than lists can lead to more concise and efficient programs. The improvement occurs because of the combining property of difference-lists. Two incomplete difference-lists can be con- catenated to give a third difference-list in constant time. In contrast, lists are concatenated using the standard append program in time linear in the length of the first list.
Consider Figure 15.1. The difference-list Xs\Zs is the result of append- ing the difference-list Ys\Zs to the difference-list Xs\Ys. This can be expressed as a single fact. Program 15.1 defines a predicate append_ dl(As,Bs,Cs), which is true if the difference-list Cs is the result of appending the difference-list Bs to the difference-list As. We use the suf- fix _dl to denote a variant of a predicate that uses difference-lists.
A necessary and sufficient condition characterizing when two differ- ence-lists As\Bs and Xs\Ys can be concatenated using Program 15.1 is that Bs be unifiable with Xs. In that case, the two difference-lists are com- patible. If the tail of a difference-list is uninstantiated, it is compatible with any difference-list. Furthermore, in such a case Program 15.1 would concatenate it in constant time. For example, the result of the query append_dl([a,b,cIXs]\Xs, [1,2]\[ J ,Ys)? is (Xs=[1,2] ,Ys=[a,b,c,
1,2]\[ ]).
Difference-lists are the logic prograniming counterpart of Lisp's rplacd, which is also used to concatenate lists in constant time and save consing
285 Incomplete Data Structures
Xs\Zs
Figure 15.1 Concatenating difference-lists
append_dI(As,Bs,Cs) The difference-list Cs is the result of appending Bs to As, where As and Bs are compatible difference-lists.
append_dl(Xs\Ys, Ys\Zs, Xs\Zs).
Program 15.1 Concatenating difference-lists
(allocating new list-cells). There is a difference between the two: the for- mer are free of side effects and can be discussed in terms of the abstract computation model, whereas rplaccl is a destructive operation, which can be described only by reference to the machine representation of S- expressions.
A good example of a program that can be improved by using differ- ence-lists is Program 9.la for flattening a list. It uses double recursion to flatten separately the head and tail of a list of lists, then concatenates the results. We adapt that program to compute the relation f latten_ dl(Xs,Ys), where Ys is a difference-list representing the elements that appear in a list of lists Xs in correct order. The direct translation of Program 9.la to use difference-lists follows:
flatten_dl([XIXs] ,Ys\Zs) -
f latten_dl(X,As\Bs), f latten_dl(Xs,Cs\Ds),
append_dl (As\Bs , Cs\Ds , Ys\Zs).
flatten_dl(X, [XIXs]\Xs) -
conistant(X), X[ J.
flatten_dl([ ] ,Xs\Xs).
Xs Xs\Ys
Ys Ys\Zs
Zs
286 Chapter 15
flatten (Xs,Ys) Ys is a flattened list containing the elements in Xs.
flatten(Xs,Ys) - flatten_dl(Xs,Ys\[ 1). flatten_dl([XIXs] ,Ys\Zs) -
flatten_dl(X,Ys\Ysl), flatten_dl(Xs,Ysl\Zs)
flatten_dl(X, [XIXs]\Xs) -
constant(X), XL[ ].
flatten_diCE J ,Xs\Xs)
Program 15.2 Flattening a list of lists using difference-lists
The doubly recursive clause can be simplified by unfolding the append_ dl goal with respect to its definition in Program 15.1. Unfolding is dis- cussed in more detail in Chapter 18 on program transformation. The result is
flatten_dl([XIXs] ,As\Ds) -
flatten_dl(X,As\Bs), flatten_dl(Xs,Bs\Ds).
The program for flatten_cl! can be used to implement flatten by ex- pressing the connection between the desired flattened list and the difference-list computed by f latt en_dl as follows:
flatten(Xs,Ys) - flatten_dl(Xs,Ys\[ 1).
Collecting the program and renaming variables yields Program 15.2. Declaratively Program 15.2 is straightforward. The explicit call to ap-
pend is made unnecessary by flattening the original list of lists into a difference-list rather than a list. The resultant program is more efficient, because the size of its proof tree is linear in the number of elements in the list of lists rather than quadratic.
The operational behavior of programs using difference-lists, such as Program 15.2, is harder to understand. The flattened list seems to be built by magic.
Let us investigate the program in action. Figure 15.2 is a trace of the query flatten([[a] , [b, [c]]] ,Xs)? with respect to Program 15.2.
The trace shows that the output, Xs, is built top-down (in the terminol- ogy of Section 7.5). The tail of the difference-list acts like a pointer to the end of the incomplete structure. The pomter gets set by unification. By using these "pointers" no intermediate structures are built, in contrast to Program 9.la.
287 Incomplete Data Structures
flatten([[a] , [b,[c]]] ,Xs)
flatten_dl([[a] , [b, [cl]] ,Xs\[ 1) flatten_dl([a] ,Xs\Xsl)
flatten_dl(a,Xs\Xs2) Xs [aIXs2]
constant (a)
a [] flatten_dl([],Xs2\Xsl) Xs2 = Xsl
flatten_dl([[b,[c]]],Xsl\[ 1) flatten_dl([b, [cl] ,Xsl\Xs3)
flatten_dl(b,Xsl\Xs4) Xsl = [bIXs4]
constant (b)
b [] flatten_dl([[c]] ,Xs4\Xs3)
flatten_dl([c] ,Xs4\Xs5)
flatten_dl(c,Xs4\Xs6) Xs4 [cIXs6]
constant (c)
c [] flatten_dl([ ],Xs6\Xs5) Xs6 Xs5
flatten_dl([ 1 ,Xs5\Xs3) Xs5 Xs3
flatten_dl([ 1 ,Xs3\[ 1) Xs3 [ I Output: Xs = [a,b,c]
Figure 15.2 Tracing a computation using difference-lists
The discrepancy between clear declarative understanding and difficult procedural understanding stems from the power of the logical variable. We can specify logical relations implicitly and leave their enforcement to Prolog. Here the concatenation of the difference-lists has been expressed implicitly, and it is mysterious when it happens in the program.
Building structures with difference-lists is closely related to building structures with accumulators. Loosely, difference-lists build structures top-down, while accumulators build structures bottom-up. Exercise 9.1(i) asked for a doubly recursive version of flatten that avoided the call to append by using accumulators. A solution is the following program:
flatten(Xs,Ys) - flatten(Xs,[ ],Ys).
flatten([XIXsJ,Zs,Ys) -
flatten(Xs,Zs,Ysl), flatten(X,Ysl,Ys).
flatten(X,Xs, [XIXs]) -
constant(X), X[ ]. flatten([ I ,Xs,Xs).
288 Chapter 15
reverse(Xs,Ys) - Ys is the reversal of the list Xs.
reverse(Xs,Ys) - reverse_dl(Xs,Ys\[ 1). reverse_dl([XIXs] ,Ys\Zs) -
reverse_dl (Xs,Ys\ [XIZs]).
reverse_dl( E ] ,Xs\Xs).
Program 15.3 Reverse with difference-lists
The similarity of this program to Program 15.2 is striking. There are only two differences between the programs. The first difference is syntactic. The difference-list is represented as two arguments, but in reverse order, the tail preceding the head. The second difference is the goal order in the recursive clause of flatten. The net effect is that the flattened list is built bottom-up from its tail rather than top-down from its head.
We give another example of the similarity between difference-lists and accumulators. Program 15.3 is a translation of naive reverse (Pro- gram 3.16a) where lists have been replaced by difference-lists, and the append operation has been unfolded away.
When are difference-lists the appropriate data structure for Prolog pro- grams? Programs with explicit calls to append can usually gain in effi- ciency by using difference-lists rather than lists. A typical example is a doubly recursive program where the final result is obtained by append- ing the outputs of the two recursive calls. More generally, a program that independently builds different sections of a list to be later combined is a good candidate for using difference-lists.
The logic program for quicksort, Program 3.22, is an example of a doubly recursive program where the final result, a sorted list, is obtained from concatenating two sorted sublists. It can be made more efficient by using difference-lists. All the append operations involved in combining partial results can be performed implicitly, as shown in Program 15.4.
The call of quicksort_di by quicksort is an initializing call, as for flatten in Program 15.2. The recursive clause is the quicksort algorithm interpreted for difference-lists where the final result is pieced together implicitly rather than explicitly. The base clause of quicksort_di states that the result of sorting an empty list is the empty difference-list. Note the use of unification to place the partitioning element X after the smaller
289 Incomplete Data Structures
quicksort (List,SortedList) - SortedList is an ordered permutation of List.
quicksort(Xs,Ys) quicksortd1(Xs,Ys\[ 1). quicksort_dl([XIXs] ,Ys\Zs) -
partition(Xs,X,Littles,Bigs)
quicksort_dl(Littles , Ys\ [X lYsi])
quicksort_dl(fligs,Ysl\Zs)
quicksort_dl([ ] ,Xs\Xs)
partition(Xs,X,Ls,Bs) See Program 3.22.
Program 15.4 Quicksort using difference-lists
elements Ys and before the bigger elements Ysi in the call quicksort_ dl(Littles,Ys\[XYs1]).
Program 15.4 is derived from Program 3.22 in exactly the same way as Program 15.2 is derived from Program 9.la. Lists are replaced by difference-lists and the append_dl goal unfolded away. The initial call of quicksort_dl by quicksort expresses the relation between the desired sorted list and the computed sorted difference-list.
An outstanding example of using difference-lists to advantage is a solu- tion to a simplified version of Dijkstra's Dutch flag problem. The problem reads: "Given a list of elements colored red, white, or blue, reorder the list so that all the red elements appear first, then all the white elements, followed by the blue elements. This reordering should preserve the orig- inal relative order of elements of the same color." For example, the list [red(1),white(2),blue(3),red(4),white(5)] should be reordered to [red(1) ,red(4) ,white(2) ,white(5) ,blue(3)].
Program 15.5 is a simple-minded solution to the problem that collects the elements in three separate lists, then concatenates the lists. The basic relation is dut ch (Xs ,Ys), where Xs is the original list of colored elements and Ys is the reordered list separated into colors.
The heart of the program is the procedure distribute, which con- structs three lists, one for each color. The lists are built top-down. The two calls to append can be removed by having distribute build three distinct difference-lists instead of three lists. Program 15.6 is an appro- priately modified version of the program.
The implicit concatenation of the difference-lists is done in the ini- tializing call to distribute_dls by dutch. The complete list is finally
290 Chapter 15
dutch (Xs,RedsWhitesBlues) - Reds WhitesBlues is a list of elements of Xs ordered by color: red, then white, then blue.
dutch(Xs ,RedsWhitesBlues) -
distribute(Xs,Reds,Whites,Blues),
append(Whites,Blues ,WhitesBlues),
append (Reds, WhitesBlues , RedsWhitesBlues).
distribute(Xs,Reds, Whites,Blues) - Reds, Whites, and Blues are the lists of the red, white, and blue elements in Xs, respectively.
distribute([red(X)IXs] ,[red(X) Reds] ,Whites,Blues) -
distribute(Xs,Reds,Whites,Blues).
distribute([white(X)IXs] ,Reds,[white(X)IWhites] Blues) -
distribute(Xs,Reds,Whites,Blues).
distribute([blue(X)IXs] ,Reds,Whites, [blue(X)jBlues]) -
distribute(Xs,Reds,Whites,Blues).
distribute([ l,E], E ] [1). append(Xs,Ys,Zs) - See Program 3.15.
Program 15.5 A solution to the Dutch flag problem
"assembled" from its parts with the satisfaction of the base clause of distribute_dis.
The Dutch flag example demonstrates a program that builds parts of the solution independently and pieces them together at the end. It is a more complex use of difference-lists than the earlier examples.
Although it makes the program easier to read, the use of an explicit constructor such as \ for difference-lists incurs noticeable overhead in time and space. Using two separate arguments to represent the difference-list is more efficient. When important, this efficiency can be gained by straightforward manual or automatic transformation.
Exercises for Section lSd
Rewrite Program 15.2 so that the final list of elements is in the reverse order to how they appear in the list of lists.
Rewrite Programs 3.27 for preorder(Tree,List), inorder(Tree, List) and postorder(Tree,List), which collect the elements oc-
291 Incomplete Data Structures
dutch (Xs,RedsWhitesBlues) Reds WhitesBlues is a list of elements of Xs ordered by color: red, then white, then blue.
dutch(Xs ,RedsWhitesBlues) -
distribute_dis (Xs , RedswhitesBlues\Whitesßlues,
WhitesBlues\Blues,Blues\[ 1). distribute_dis (Xs,Reds, Whites,Blues)
Reds, Whites, and Blues are the difference-lists of the red, white, and blue elements in Xs, respectively.
distribute_dls( [red(X) I Xs]
[red(X) IReds]\Redsl,Whìtes,Blues)
distribute_dls(Xs,Reds\Redsl,Whites,Blues).
distrïbute_dls([white(X) IXs]
Reds, [whïte(X)Iwhites]\Whitesl,Blues)
distribute_dls(Xs,Reds,Whites\Whitesl Blues)
distribute_dls( [blue (X) I Xs]
Reds,Whites, [blue(X) IBlues]\Bluesl) -
distribute_d1s(Xs,Reds,.ihites ,Blues\Bluesl)
distrïbute_dls([ ],Reds\Reds,Whites\Whites,Blues\Blues).
Program 15.6 Dutch flag with difference-lists
currmg in a binary tree, to use difference-lists and avoid an explicit call to append.
(iii) Rewrite Program 12.3 for solving the Towers of Hanoi so that the list of moves is created as a difference-list rather than a list.
i 5.2 Difference-Structures
The concept underlying difference-lists is the use of the difference be- tween incomplete data structures to represent partial results of a compu- tation. This can be applied to recursive data types other than lists. This section looks at a specific example, sum expressions.
Consider the task of normalizing sum expressions. Figure 15.3 con- tains two sums (u + b) + (C + ci) and (ci + (b + (c + cl))). Standard Prolog syntax brackets the term a + b + c as ((a + b) + c). We describe a pro- cedure converting a sum into a normalized one that is bracketed to the right. For example, the expression on the left in Figure 15.3 would be
292 Chapter 15
/\ + +/\ /\
a b cd
Figure 15.3 Unriormalized and normalized sums
normalize(Sum,NormalizedSum) - NormalizedSum is the result of normalizing the sum expression Sum.
normalize(Exp,Norm) normalize_ds(Exp,Norm++O).
normalize_ds (A+B , Norm++Space) -
normalize_ds (A, Norm++Normß), normalize_ds (B, NormB++Space).
normalize_ds(A, (A+Space)++Space)
constant (A)
Program 15.7 Normalizing plus expressions
converted to the one on the right. Such a procedure is useful for doing algebraic simplification, facilitating writing programs to test whether two expressions are equivalent.
We introduce a difference-sum as a variant of a difference-list. A difference-sum is represented as a structure El ++ F2, where El and F2 are incomplete normalized sums. lt is assumed that ++ is defined as a binary infix operator. It is convenient to use O to indicate an empty sum.
Program 15.7 is a program for normalizing sums. The relation scheme is normalize (Exp Norm), where Norm is an expression equivalent to Exp that is bracketed to the right and preserves the order of the constants appearing in Exp.
This program is similar in structure to Program 15.2 for flattening lists using difference-lists. There is an initialization stage, where the difference-structure is set up, typically calling a predicate with the same name but different arity or different argument pattern. The base case passes out the tail of the incomplete structure, and the goals in the body
293 Incomplete Data Structures
of the recursive clause pass the tail of the first incomplete structure to be the head of the second.
The program builds the normalized sum top-down. By analogy with the programs usmg difference-lists, the program can be easily modified to build the structure bottom-up, which is Exercise (ii) at the end of this section.
The declarative reading of these programs is straightforward. Opera- tionally the programs can be understood in terms of building a structure incrementally, where the "hole" for further results is referred to explic- itly. This is entirely analogous to difference-lists.
Exercises for Section 15.2
(i) Define the predicate normalized_sum(Expressiori), which is true if Expression is a normalized sum.
(ii) Rewrite Program 15.7 so that
The normalized sum is built bottom-up;
The order of the elements is reversed.
(iii) Enhance Program 15.7 so that numbers appearing in the addends are added together and returned as the first component of the nor- malized sum, For example, (3 + X) + 2 + (y + 4) should be normal- ized to 9 + (X + y).
(iv) Write a program to normalize products using difference-products, defined analogously to difference-sums.
15.3 Dictionaries
A different use of incomplete data structures enables the implementa- tion of dictionaries. Consider the task of creating, using, and maintaining a set of values indexed under keys. There are two main operations we would like to perform: looking up a value stored under a certain key, and entering a new key and its associated value. These operations must en- sure consistency - for example, the same key should not appear twice
294 Chapter 15
lookup (Key,Dictionary, Value) - Dictionary contams Value mdexed under Key. Dictionary is represented as an incomplete list of pairs of the form (Key,Value).
lookup(Key, [(Key,Value) IDict] ,Value).
lookup(Key, [(Keyl,Valuel)IDict] ,Value) -
Key Keyl, lookup(Key,Dict,Value).
Program 15.8 Dictionary lookup from a list of tuples
with two different values. It is possible to perform both operations, look- ing up values of keys, and entering new keys, with a single simple proce- dure by exploiting incomplete data structures.
Consider a linear sequence of key-value pairs. Let us see the advan- tages of using an incomplete data structure for its representation. Pro- gram 15.8 defines the relation lookup(Key,Dictionary,Value) which is true if the entry under Key in the dictionary Dictionary has value Value. The dictionary is represented as an incomplete list of pairs of the form (Key,Value).
Let us consider an example where the dictionary is used to remember phone extensions keyed under the names of people. Suppose that Dict is initially instantiated to [(arnold, 8881) , (barry,4513) , (cathy, 5950) Xs]. The query lookup(arnold,Dict,N)? has as answer N=8881 and is used for finding Arnold's phone number. The query lookup(barry, Dict,4513)? succeeds, checking that Barry's phone number is 4513.
The entry of new keys and values is demonstrated by the query lookup(david,Dict,1199)?. Syntactically this appears to check David's phone number. Its effect is different. The query succeeds, instantiating Dict to [(arnold,8881),(barry,4513),(cathy,5950),(david,1199) Xsl]. Thus lookup has entered a new value. What happens if we check Cathy's number with the query lookup
(cathy,Dict,5951)?, where the number is incorrect? Rather than en- tering a second entry for Cathy, the query fails because of the test Key Keyl.
The lookup procedure given in Program 15.8 completes Program 14.15, the simplified ELIZA. Note that when the program begins, the dictionary is empty, indicated by its being a variable. The dictionary is built up
295 Incomplete Data Structures
lookup (Key,Dictionary, Value) - Dictionary contains Value indexed under Key. Dictionary is represented as an ordered binary tree.
lookup(Key,dict(Key,X,Left,Right),Value)
!, X Value.
lookup(Key,dict(Keyl,X,Left,Right),Value) -
Key < Keyl, lookup(Key,Left,Value).
lookup(Key,dict(Keyl,X,Left,Right),Value)
Key > Keyl, lookup(Key,Right,Value).
Program 1.9 Dictionary lookup in a binary tree
during the matching against the stimulus half of a stimulus-response pair. The constructed dictionary is used to produce the correct response. Note that entries are placed in the dictionary without their values being krown: a striking example of the power of logical variables. Once an integer is detected, it is put in the dictionary, and its value is determined later.
Searching linear lists is not very efficient for a large number of key- value pairs. Ordered binary trees allow more efficient retrieval of infor- mation than linear lists. The insight that an incomplete structure can be used to allow entry of new keys as well as to look up values carries over to binary trees.
The binary trees of Section 3.4 are modified to be a four-place structure dict(Key,Value,Left,Right), where Left and Right are, respectively, the left and right subdictionaries, and Key and Value are as before. The functor dict is used to suggest a dictionary.
Looking up in the dictionary tree has a very elegant definition, simi- lar in spirit to Program 15.8. It performs recursion on binary trees rather than on lists, and relies on unification to instantiate variables to dictio- nary structures. Program 15.9 gives the procedure lookup(Key,Dictio- nary,Value), which as before both looks up the value corresponding to a given key and enters new values.
At each stage, the key is compared with the key of the current node. If it is less, the left branch is recursively checked; if it is greater, the right branch is taken. If the key is non-numeric, the predicates < and > must be generalized. The cut is necessary in Program 15.9, in contrast to
296 Chapter 15
freeze(A,B) - Freeze term A into B.
freeze(A,B) - copy_term(A,B), numbervars(B4O,N).
me!t_new(A,B) - Melt the frozen term A into B.
melt_new(A,B) - melt(A,B,Dictionary),
melt('$VAR' (N) ,X,Dictionary)
lookup(N,Dictionary,X).
melt(X,X,Dictionary)
constant (X)
melt (X, Y ,Dictionary)
compound(X),
f unctor (X , F , N)
f unctor (Y ,F , N)
melt(N,X,Y,Dictionary).
melt(N,X,Y,Dictionary) -
N > O,
arg(N,X,ArgX),
melt(ArgX,ArgY,Dictionary),
arg(N,Y,ArgY),
Nl is N-1,
melt (Nl ,X,Y,Dictionary).
melt (O,X,Y,Dictionary).
numbervars(Term,Nl ,N2) '- See Program 10.8.
lookup(Key,Dictionary,Value) See Program 15.9.
Program 13.10 Meltmg a term
Program 15.8, because of the nonlogical nature of comparison operators, which will give errors if keys are not instantiated.
Given a number of pairs of keys and values, the dictionary they deter- mine is not unique. The shape of the dictionary depends on the order in which queries are posed to the dictionary.
The dictionary can be used to melt a term that has been frozen using Program 10.8 for numbervars. The code is given as Program 15.10. Each melted variable is entered into the dictionary, so that the correct shared variables will be assigned.
297 Incomplete Data Structures
15.4 Queues
An interesting application of difference-lists is to implement queues. A queue is a first-in, first-out store of information. The head of the difference-list represents the beginning of the queue, the tail represents the end of the queue, and the members of the difference-list are the ele- ments in the queue. A queue is empty if the difference-list is empty, that is, if its head and tail are identical.
Maintaining a queue is different from maintaining a dictionary. We consider the relation queue(s), where a queue processes a stream of commands, represented as a list S. There are two basic operations on a queueenqueuing an element and dequeuing an elementrepresented, respectively, by the structures enqueue (X) and dequeue (X), where X is the element concerned.
Program 15.11 implements the operations abstractly. The predicate queue(s) calls queue(S,Q), where Q is initialized to an empty queue. queue/2 is an interpreter for the stream of enqueue and dequeue com- mands, responding to each command and updating the state of the queue accordingly. Enqueuing an element exploits the incompleteness of the tail of the queue, instantiating it to a new element and a new tail, which is passed as the updated tail of the queue. Clearly, the calls to enqueue and dequeue can be unfolded, resulting in a more concise and efficient, but perhaps less readable, program.
queue(S) S is a sequence of enqueue and dequeue operations, represented as a list of terms enqueue(X) and dequeue(X).
queue(s) queue(S,Q\Q).
queue([enqueue(X) XsJ ,Q)
eriqueue(X,Q,Q1), queue(Xs,Q1).
queue([dequeue(X)IXs] ,q)
dequeue(X,Q,Q1), queue(Xs,Q1).
queue([ ],Q).
enqueue(X,Qh\[XIQt] ,Qh\Qt).
dequeue(X, [XIQh]\Qt,Qh\Qt).
Program 1S.11 A queue process
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flatten(Xs,Ys) - Ys is a flattened list containing the elements in Xs.
flatten(Xs,Ys) - flatten_q(Xs,Qs\Qs,Ys).
flatten_q([XIXs) ,Ps\[XslQs] ,Ys) -
flatten_q(X,Ps\Qs,Ys).
flatten_q(X, [QIPs]\Qs, [XIYs]) -
constant(X), X[ I, flatten_q(Q,Ps\Qs,Ys). flatten_q([ ],Q,Ys) -
non_empty(Q), dequeue(X,Q,Q1), flatten_q(X,Q1,Ys).
flatten_q([ ],[ I\[ ],[ 1). non_empty([ ]\[ 1) - !, fail. non_empty (Q)
dequeue(X, [XIQh]\Qt,Qh\Qt).
Program 15.12 Flattening a list using a queue
The program terminates when the stream of commands is exhausted. It can be extended to insist that the queue be empty at the end of the commands by changing the base fact to
queue([ I,Q) empty(Q).
A queue is empty if both its head and tail can be instantiated to the empty list, expressed by the fact empty ( [ ] \ E ]). Logically, the clause empty (Xs\Xs) would also be sufficient; however, because of the lack of the occurs check in Prolog, discussed in Chapter 4, it may succeed erroneously on a nonempty queue, creating a cyclic data structure.
We demonstrate the use of queues in Program 15.12 for flattening a list. Although the example is somewhat contrived, it shows how queues can be used. The program does not preserve the order of the elements in the original list.
The basic relation is flatten_q(Ls,Q,Xs), where Ls is the list of lists to be flattened, Q is the queue of lists waiting to be flattened, and Xs is the list of elements in Ls. The initial call of f latten_q/3 by f latten/2 initializes an empty queue. The basic operation is enqueuing the tail of the list and recursively flattening the head of the list:
flatten_q(EXIXs],Q,Ys)enqueue(Xs,Q,Q1), flatten._q(X,Q1,Ys).
299 Incomplete Data Structures
The explicit call to enqueue can be omitted and incorporated via unifica- tion as follows:
flatten_q([XIXs] ,Qh\[XsIQt] ,Ys) - flatten_q(X,Qh\Qt,Ys).
If the element being flattened is a constant, it is added to the output structure being built top-down, and an element is dequeued (by unifying with the head of the difference-list) to be flattened in the recursive call:
flatten_q(X, [QIQh]\Qt, [XIYs]) -
constant(X), X[ ] , flatten_q(Q,Qh\Qt,Ys).
When the empty list is being flattened, either the top element is de- queued
flatten_q([ ],Q,Ys)
non_empty(Q), dequeue(X,Q,Q1), flattenq(X,Q1,Ys).
or the queue is empty, and the computation terminates:
flatten_q([ ],[ ]\[ ],[ ]).
A previous version of Program 15.12 incorrectly expressed the case when the list was empty, and the top element was dequeued as
flatten_q([ ],[QIQh]\Qt,Ys) - flatten_q(Q,Qh\Qt,Ys).
This led to a nonterminating computation, since an empty queue Qs\Qs unified with [QIQh] \Qt and so the base case was never reached.
Let us reconsider Program 15.11 operationally. Under the expected use of a queue, enqueue(X) messages are sent with X determined and de- queue (X) with X undetermined. As long as more elements are enqueued than dequeued, the queue behaves as expected, with the difference be- tween the head of the queue and the tail of the queue being the elements in the queue. However, if the number of dequeue messages received ex- ceeds that of enqueue messages, an interesting thing happens - the content of the queue becomes negative. The head runs ahead of the tail, resulting in a queue containing a negative sequence of undetermined el- ements, one for each excessive dequeue message.
It is interesting to observe that this behavior is consistent with the as- sociativity of appending of difference-lists. If a queue qs\[X1,X2,X3Qs] that contains minus three undetermined elements has the queue [a, b, c,d,ejXs]\Xs that contains five elements appended to it, then the result
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will be the queue [d, e Xs] \Xs with two elements, where the "negative" elements X1,X2,X3 are unified with a,b,c.
15.5 Background
Difference-lists have been in the logic programming folklore since its inception. The fifst description of them in the literature is given by Clark and Tarniund (1977).
The automatic transformation of simple programs without difference- lists to programs with difference-lists, for example, reverse and f lat- ten, can be found in Bloch (1984).
Section 15.1 implicitly contains an algorithm for converting from a program with explicit calls to append to an equivalent, more efficient program that uses difference-lists to concatenate the elements and which is much more efficient. Care is needed in application of the algorithm. There are excellent discussions of a correct algorithm and the dangers of using difference-lists without the occurs check in Sondergaard (1990) and Marriott and Søndergaard (1993).
There is an interesting discussion of the Dutch flag problem in O'Keefe (1990).
Automatic removal of a functor denoting difference-lists is described in Gallagher and Bruynooghe (1990).
Maintaining dictionaries and queues can be given a theoretical basis as a perpetual process, as described by Warren (1982) and Lloyd (1987).
Queues are particularly important in concurrent logic programming languages, since their input need not be a list of requests but a stream, which is generated incrementally by the processes requesting the ser- vices of the queue.
16 Second-Order Programming
16.1 All-Solutions Predicates
Solving a Prolog query with a program entails finding an instance of the query that is implied by the program. What is involved in finding all instances of a query that are implied by a program? Declaratively, such a query lies outside the logic programming model presented in Chapter 1. lt is a second-order question, since it asks for the set of elements with a certain property. Operationally, it is also outside the pure Prolog computation model. In pure Prolog, all information about a certain branch of the computation is lost on backtracking. This prevents
Chapters 14 and 15 demonstrate Prolog programming techniques based directly on logic programming. This chapter, in contrast, shows pro- grarnming techniques that are missing from the basic logic programming model but can nonetheless be incorporated into Prolog by relying on lan- guage features outside of first-order logic. These techniques are called second-order, since they talk about sets and their properties rather than about individuals.
The first section introduces predicates that produce sets as solutions. Computing with predicates that produce sets is particularly powerful when combined with programming techniques presented in earlier chap- ters. The second section gives some applications. The third section looks at lambda expressions and predicate variables, which allow functions and relations to be treated as "first-class" data objects.
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father(terach,abraham). f ather(haran,lot).
f ather(terach,nachor). f ather(haran,milcah).
f ather(terach,haran). father(haran,yiscah).
f ather(abraham,isaac).
male (abraham). male (haran). female (yiscah).
male (Isaac) male (nachor). female (milcah).
male (lot)
Program 16.1 Sample data
a simple way of using pure Prolog to find the set of all solutions to a query, or even to find how many solutions there are to a given query.
This section discusses predicates that return all instances of a query. We call such predicates all-solutions predicates. Experience has shown that all-solutions predicates are very useful for programming.
A basic all-solutions predicate is f indall(Terrn,Goal,Bag). The pred- icate is true if and only if Bag unifies with the list of values to which a variable X not occurring in Term or Goal would be bound by successive resatisfaction of call (Goal), X=Term? after systematic replacement of all variables in X by new variables.
Procedurally, f indall(Term,Goal,Bag) creates an empty list L, re- names Goal to a goal G, and executes G. If G succeeds, a copy of Term is appended to L, and G is reexecuted. For each successful reexecution, a copy of Term is appended to the list. Eventually, when G fails, Bag is um- fled with L. The success or failure of f Indall depends on the success or failure of the unification.
We demonstrate the use of all-solutions predicates using part of the biblical database of Program 1.1, repeated here as Program 16.1.
Consider the task of finding all the children of a particular father. It is natural to envisage a predicate children(X,Kids), where Kids is a list of children of X. It is immediate to define using f indall, namely,
children(X,Kids) - f indall(Kid,father(X,Kid) ,Kids).
The query children(terach,Xs)? with respect to Program 16.1 pro- duces the answer Xs = [abraham,nachor,haran].
The query f indall(F,father(F,K),Fs)? with respect to Program 16.1 produces the answer F = [terach,haran,terach,haran,terach, haran, abraham]. It would be useful to conceive of this query as asking
303 Second-Order Programming
for_all ( Goal,Condition) For all solutions of Goal, Condition is true.
for_all(Goal,Coudition) -
findall(Condition,Goal,Cases), check(Cases).
check([CaselCases]) Case, check(Cases)
check([ 1).
Program 16.2 Applying set predicates
who is a father and to receive as solution [terach,haran,abraham]. This answer can be obtained by removing duplicate solutions.
Another interpretation can be made of the query f indali(F,father (F,K) ,Fs)?. Instead of having a single solution, all fathers, there could be a solution for each child K. Thus one solution would be K=abrahain, Fs = [terach]; another would be K=lot, Fs = [haran]; and so on.
Standard Prolog provides two predicates that distinguish between these two interpretations. The predicate bagof (Terni ,Goal ,Bag) is like f indall except that alternative solutions are found for the variables in Goal. The predicate setof(Terni,Goal,Bag) is a refinement of bagof where the solutions in Bag are sorted corresponding to a standard order of terms and duplicates removed. If we want to emphasize that the solu- tion should be conceived of as a set, we refer to all-solutions predicates as set predicates.
Another all-solutions predicate checks whether all solutions to a query satisfy a certain condition. Program 16.2 defines a predicate f or ali (Goal,Condition), which succeeds when Condition is true for all values of Goal. It uses the meta-variable facility.
The query for_all(father(X,C),male(C))? checks which fathers have only male children. It produces two answers: X=terach and X=abra- hain.
A simpler, more efficient, but less general version of for_all can be written directly using a combination of nondeterminism and negation by failure. The definition is
forall(Goal,Condition) - not (Goal, not Condition).
It successfully answers a query such as f orall(father(terach,X), male (X))? but fails to give a solution to the query f orall(father(X, C) ,male(C))?.
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find_alLdl (X, Goal,Instances) - Instances is the multiset of instances of X for which Goal is true. The multiplicity of an element is the number of different ways Goal can be proved with it as an instance of X.
f ind_all_di (X ,Goal, Xs) -
assertaY$instance'('$mark')), Goal,
asserta('$instance' (X)), fail.
f ind_all_dl (X ,Goal, Xs\Ys) -
retract('$instance'(X)), reap(X,Xs\Ys),
reap(X,Xs\Ys) -
X '$mark', retract('$instance'(Xl)), !,
reap (Xl, Xs\ [X lYs] )
reapY$mark' ,Xs\Xs).
Program 16.3 Implementing an all-solutions predicate using difference- lists, assert, and retract
We conclude this section by showing how to implement a simple vari- ant of f indall. The discussion serves a dual purpose. It illustrates the style of implementation for all-solutions predicates and gives a utility that will be used in the next section. The predicate f ind_all_ dl(X,Goal,Instances) is true if Instances is the bag (multiset) of instances of X, represented as a difference-list, where Goal is true.
The definition of f ind_all_dl is given as Program 16.3. The program can only be understood operationally. There are two stages to the pro- cedure, as specified by the two clauses for f ind_all_dl. The explicit failure in the first clause guarantees that the second will be executed. The first stage finds all solutions to Goal using a failure-driven loop, as- serting the associated X as it proceeds. The second stage retrieves the solutions.
Asserting $niark is essential for nested all-solutions predicates to work correctly, lest one set should "steal" solutions produced by the other all- solutions predicate.
Exercise for Section 16.1
(i) Define the predicate intersect(Xs,Ys,Zs) using an all-solutions predicate to compute the intersection Zs of two lists Xs and Ys.
305 Second-Order Programming
What should happen if the two lists do not intersect? Compare the code with the recursive definition of intersect.
16.2 Applications of Set Predicates
Set predicates are a significant addition to Prolog. Clean solutions are ob- tained to many problems by using set predicates, especially when other programming techniques, discussed in previous chapters, are incorpo- rated. This section presents three example programs: traversing a graph breadth-first, using the Lee algorithm for finding routes in VLSI circuits, and producing a keyword in context (KWIC) index.
Section 14.2 presents three programs, 14.8, 14.9, and 14.10, for traversing a graph depth-first. We discuss here the equivalent programs for traversing a graph breadth-first.
The basic relation is comiected(X,Y), which is true if X and Y are connected. Program 16.4 defines the relation. Breadth-first search is im- plemented by keeping a queue of nodes waiting to be expanded. The connected clause accordingly calls connected_bfs(Queue,Y), which is true if Y is in the connected component of the graph represented by the nodes in the Queue.
Each call to connected_bf s removes the current node from the head of the queue, finds the edges connected to it, and adds them to the tail of the queue. The queue is represented as a difference-list, and the all- solutions predicate f ind_all_di is used. The program fails when the queue is empty. Because difference-lists are an incomplete data struc- ture, the test that the queue is empty must be made explicitly. Otherwise the program would not terminate.
Consider the edge clauses in Program 16.4, representing the left-hand graph in Figure 14.3. Using them, the query connected(a,X)? gives the values a, b, c, d, e, f, g, j, k, h, i for X on backtracking, which is a breadth- first traversal of the graph.
Like Program 14.8, Program 16.4 correctly traverses a finite tree or a directed acyclic graph (DAG). If there are cycles in the graph, the program will not terminate. Program 16.5 is an improvement over Program 16.4 in which a list of the nodes visited in the graph is kept. Instead of adding all the successor nodes at the end of the queue, each is checked to see if
306 Chapter 16
connected(X,Y) - Node X is connected to node Y in the DAG defined by edge/2 facts.
connected(X,Y) - enqueue(X,Q\Q,Q1), connected_bfs(Q1,Y).
connected_bfs(Q,Y) - empty(Q), !, fail.
connected_bfs(Q,Y) - dequeue(X,Q,Q1), X=Y.
connected_bfs(Q,Y) -
dequeue(X,Q,Qi), enqueue_edges(X,Q1,Q2), connected_bfs(Q2,Y).
enqueue_edges(X,Xs\Ys,Xs\Zs) - find_all_dl(N,edge(X,N),Ys\Zs),
empty([ ]\[ 1). enqueue/3, dequeue/3 - See Program 15.11.
f ind_all_dl(Terrn,Goal ,DList) - See Program 16.3.
Data edge (a, b) edge(a,c). edge(a,d). edge (a » e) edge (f , i)
edge(c,f). edge(c,g). edge (f , h) edge (e , k) edge(d,j).
edge(x,y). edge (y , z) edge (z , x) edge (y , u) edge (z , y)
Program 16.4 Testing connectivity breadth-first in a DAG
it has been visited before. This is performed by the predicate filter in Program 16.5.
Program 16.5 in fact is more powerful than its depth-first equivalent, Program 14.10. Not only will it correctly traverse any finite graph but it will also correctly traverse infinite graphs in which every vertex has finite degree as well. It is useful to summarize what extensions to pure Prolog have been necessary to increase the performance in searching graphs. Pure Prolog correctly searches finite trees and DAGs. Adding negation allows correct searching of finite graphs with cycles, while set predicates are necessary for infinite graphs. This is shown in Figure 16.1.
Calculating the path between two nodes is a little more awkward than for depth-first search. It is necessary to keep with each node in the queue a list of the nodes linking it to the original node. The technique is demon- strated in Program 20.6.
The next example combines the power of nondeterministic program- ming with the use of second-order programming. lt is a program for calculating a minimal cost route between two points in a circuit using the Lee algorithm.
307 Second-Order Programming
connected(X,Y) Node X is connected to node Y in the graph defined by edge/2 facts.
connected (X Y)
enqueue(X,q\Q,Q1), connected_bfs(Q1,Y,[X]).
connected_bfs(Q,Y,Visited) '- ernpty(Q) , , fail.
connected_bfs(Q,Y,Visited) - dequeue(X,Q,Q1), XY.
connected_bfs(Q,Y,Visïted)
dequeue(X,Q,Q1)
findall(N,edge(X,N) ,Edges),
filter(Edges,Vïsited,Visitedl,Q1,Q2)
connected_bfs(Q2,Y,Visitedl)
filter([NINs] ,Visited,Visitedl,Q,Ql) -
member(N,Visited), ! , filter(Ns,Visited,Visitedl,Q,Q1).
filter([NINs] ,Visited,Visitedi,Q,Q2)
not member(N,Visited), ! , enqueue(N,Q,Q1),
filter(Ns, [NiVisited] ,Visitedl,Q1,Q2).
filter([ I ,Visited,Visited,q,Q)
empty([ ]\[ 1). enqueue/3, dequeue/3 See Program 15.11.
Program 16.5 Testing connectivity breadth-first in a graph
Finite trees and DAGs Pure Prolog Finite graphs Pure Prolog + negation Infinite graphs Pure Prolog -- second order + negation
Figure 16.1 Power of Prolog for various searching tasks
308 Chapter 16
Figure 16.2 The problem of Lee routing for VLSI circuits
The problem is formulated as follows. Given a grid that may have obstacles, find a shortest path between two specified points. Figure 16.2 shows a grid with obstacles. The heavy solid line represents a shortest path between the two points A and B. The shaded rectangles represent the obstacles.
We first formulate the problem in a suitable form for programming. The VLSI circuit is modeled by a grid of points, conveniently assumed to be the upper quadrant of the Cartesian plane. A route is a path between two points in the grid, along horizontal and vertical lines only, subject to the constraints of remaining in the grid and not passing through any obstacles.
Points in the plane are represented by their Cartesian coordinates and denoted X-Y. In Figure 16.2, A is 1-1 and B is 5-5. This representation is chosen for readability and utilizes the definition of - as an infix binary
309 Second-Order Programming
operator. Paths are calculated by the program as a list of points from B to A, including both endpoints. In Figure 16.2 the route calculated is [5-5,5-4,5-3,5-2,4-2,3-2,2-2,1-2,1-11, and is marked by the heavy solid line.
The top-level relation computed by the program is lee_route (A, B, Obstacles,Path), where Path is a route (of minimal distance) from point A to point B in the circuit. Obstacles are the obstacles in the grid. The program has two stages. First, successive waves of neighboring grid points are generated, starting from the initial point, until the final point is reached. Second, the path is extracted from the accumulated waves. Let us examine the various components of Program 16.6, the overall program for Lee routing.
Waves are defined inductively. The initial wave is the list [A]. Succes- sive waves are sets of points that neighbor a point in the previous wave and that do not already appear in previous waves. They are illustrated by the lighter solid lines in Figure 16.2.
Wave generation is performed by waves (B, WavesSoFar ,Obstacles, Waves). The predicate waves/4 is true if Waves is a list of waves to the destination B avoiding the obstacles represented by Obstacles and WavesSoFar is an accumulator containing the waves generated so far in traveling from the source. The predicate terminates when the destina- tion is in the current wave. The recursive clause calls next _wave/4, which finds all the appropriate grid points constituting the next wave using the all-solutions predicate f indal 1.
Obstacles are assumed to be rectangular blocks. They are represented by the term obstacle(L,R), where L is the coordinates of the lower left-hand corner and R the coordinates of the upper right-hand corner. Exercise (i) at the end of this section requires modifying the program to handle other obstacles.
The predicate path (A,B,Waves,Path) finds the path Path back from B to A through the Waves generated in the process. Path is built downward, which means the order of the points is from B to A. This order can be changed by using an accumulator in path.
Program 16.6 produces no output while computing the Lee route. In practice, the user may like to see the computation in progress. This can be easily done by adding appropriate write statements to the procedures next_wave and path.
310 Chapter 16
lee_route ( Source,Destination, Obstacles,Path) Path is a minimal length path from Source to Destination that does not cross Obstacles.
lee_route (A, B ,Obstacles, Path)
waves(B,[[A] 1 ]] ,Obstacles,Waves), path (A ,B , Waves , Path)
waves (Destination, WavesSoFar, Obstacles, Waves) - Waves is a list of waves including WavesSoFar (except, perhaps, its last wave) that leads to Destination without crossing Obstacles.
waves(B, [Wave Waves] ,Obstacles,Waves) - member(B,Wave), !. waves(B, [Wave ,LastWavelLastWaves] ,Obstacles,Waves) -
next_wave (Wave, LastWave ,Obstacles, NextWave), waves(B, [NextWave,Wave ,LastWave ILastWaves] ,Obstacles,Waves).
next_ wave ( Wave,Last Wave, Obstacles,NextWave) - Next Wave is the set of admissible points from Wave, that is, excluding points from Last Wave, Wave and points under Obstacles.
next_wave(Wave ,LastWave ,Obstacles ,NextWave) -
findall(X,admissible(X,Wave ,LastWave,Obstacles) ,NextWave).
admissible (X ,Wave, LastWave ,Obstacles) -
adjacent(X,Wave,Obstacles),
not member(X,LastWave),
not member(X,Wave).
adjacent (X,Wave ,Obstacles) -
member (Xl, Wave) neighbor(Xl ,X),
not obstructed(X,Obstacles).
neighbor(Xl-Y,X2-Y) - next_to(X1,X2).
neighbor(X-Yi,X-Y2) - next_to(Yl,Y2).
next_to(X,X1) - Xl is X+l.
next_to(X,X1) - X > O, Xl is Xl.
obstructed (Point ,Obstacles) -
menber(Obstacle,Obstacles), obstructs(Point,Obstacle).
obstructs(X-Y,obstacle(X-Yl,X2-Y2)) - Yl Y, Y Y2.
obstructs(X-Y,obstacle(XlYl,X-Y2)) - Yl Y, Y Y2.
obstructs(X-Y,obstacle(X1-Y,X2-Y2)) '- Xl X, X X2.
obstructs(X-Y,obstacle(Xl-Yl,X2-Y)) Xl X, X X2.
Program 16.6 Lee routing
311 Second-Order Programming
path ( Source,Destination, Waves,Path) Path is a path from Source to Destination going through Waves.
path(A,A,Waves,[A]) - L
path(A,B, [Wave IWaves] , [BiPath]) -
member(Bi ,Wave),
neighbor (B , Bi)
!, path(A,B1,Waves,Path).
Testing and data
test_lee (Naine , Path)
data(Name,A,B,Dbstacles), lee_route(A,B4Obstacles,Path).
data(test,i-1,5-5, [obstacle(2-3,4-5),obstacle(6-6,8-8)]).
Program 16.6 (Continued)
Our final example in this section concerns the keyword in context (KWIC) problem. Again, a simple Prolog program, combining nondeter- ministic and second-order programming, suffices to solve a complex task.
Finding keywords in context involves searching text for all occurrences of a set of keywords, extracting the contexts in which they appear. We consider here the following variant of the general problem: "Given a list of titles, produce a sorted list of all occurrences of a set of keywords in the titles, together with their context."
Sample input to a program is given in Figure 16.3 together with the expected output. The context is described as a rotation of the title with the end of the title indicated by -. In the example, the keywords are algorithmic, debugging, logic, problem, program, programming, prolog, and solving, all the nontrivial words.
The relation we want to compute is kwic(Titles,Kwiclitles) where Titles is the list of titles whose keywords are to be extracted, and Kwic- Titles is the sorted list of keywords in their contexts. Both the input and output titles are assumed to be given as lists of words. A more gen- eral program, as a preliminary step, would convert freer-form input into lists of words and produce prettier output.
The program is presented in stages. The basis is a nondeterministic specification of a rotation of a list of words. It has an elegant definition in terms of append:
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Input: progranmiïng m prolog logic for problem solving logic programming algorithmic program debugging
Output: algorithmic program debugging -, debuggrrg - algorithmic program, logic for problem solving -, logic programming -, problem solving - logic for, program debugging - algorithmic, programming in prolog -, programming - logic, prolog - prograinnhing in, solving - logic for problem
Figure 16.3 Input and output for keyword in context (KWIC) problem
rotate(Xs,Ys) - append(As,Bs,Xs), append(Bs,As,Ys).
Declaratively, Ys is a rotation of Xs if Xs is composed of As followed by Bs, and Ys is Bs followed by As.
The next stage of development involves identifying single words as potential keywords. This is done by isolating the word in the first call to append. Note that the new rule is an instance of the previous one:
rotate(Xs,Ys) -
append(As,[KeyIBsJ,Xs), append([KeyIBs],As,Ys).
This definition also improves the previous attempt by removing the du- plicate solution when one of the split lists is empty and the other is the entire list.
The next improvement involves examining a potential keyword more closely. Suppose each keyword Word is identified by a fact of the form keyword (Word). The solutions to the rotate procedure can be filtered so that only words identified as keywords are accepted. The appropriate version is
rotate_and_f ilter(Xs,Ys) - append(As, [KeyIBs] ,Xs),
keyword(Key), append(rKeylBs] ,As,Ys).
313 Second-Order Programming
kwìc (Titles,KWTitles) - KWTitles is a KWIC index of the list of titles Titles.
kwic(Titles,KWTitles) - setof(Ys,XsI(rnember(Xs,Tit].es), rotate_and_f ilter(Xs,Ys)) ,KWTitles).
rot ate_and_fiiter(Xs, Ys) - Ys is a rotation of the list Xs such that the first word of Ys is significant and - is inserted after the last word of Xs.
rotate_and_filter(Xs,Ys) - append(As, [Key lBs] ,Xs), not insignificant(Key), append([KeylBs] [''lAs] ,Ys).
Vocabulary of insignificant words
insignificant(a). insignif ïcant (the). insignificant(iri) . insigriif icant (for). Testing arid data
test_kwic(Books,Kwic) - titles(Books,Titles) , kwic(Titles,Kwic)
titles(lp, [[logic,for,problem,solving] [logic, programming] [algorithmic, program ,debugging], [programming, in prolog]]).
Program 16.7 Producing a keyword in context (KWIC) index
Operationally rotate_and_filter considers all keys, filtering out the unwanted alternatives. The goal order is important here to maximize program efficiency.
In Program 16.7, the final version, a complementary view to recogniz- ing keywords is taken. Any word Word is a keyword unless otherwise specified by a fact of the form insignificant(Word). Further the proce- dure is augmented to insert the end-of-title mark -, providing the con- text information. This is done by adding the extra symbol in the second append call. Incorporating this discussion yields the clause for rotate_ and_f ilter ni Program 16.7.
Finally, a set predicate is used to get all the solutions. Quantification is necessary over all the possible titles. Advantage is derived from the
314 Chapter 16
behavior of setof in sorting the answers. The complete program is given as Program 16.7, and is an elegant example of the expressive power of Prolog. The test predicate is test_kwic/2.
Exercises for Section 162
Modify Program 16.6 to handle other obstacles than rectangles.
Adapt Program 16.7 for KWIC so that it extracts keywords from lines of text.
Modify rotation of a list so that it uses difference-lists.
Write a program to find a minimal spanning tree for a graph.
(y) Write a program to find the maximum flow in a network design using the Ford-Fulkerson algorithm.
16.3 Other Second-Order Predicates
First-order logic allows quantification over individuals. Second-order logic further allows quantification over predicates. Incorporating this extension into logic programming entails using rules with goals whose predicate names are variables. Predicate names become "first-class" data objects to be manipulated and modified.
A simple example of a second-order relation is the determination of whether all members of a list have a certain property. For simplicity the property is assumed to be described as a unary predicate. Let us define has_property(Xs,P), which is true if each element of Xs has some property P. Extending Prolog syntax to allow variable predicate names enables us to define has_property as in Figure 16.4. Because has_ property allows variable properties, it is a second-order predicate. An example of its use is testing whether a list of people Xs is all male with a query has_property(Xs,male)?.
Another second-order predicate is rnap_list(Xs,P,Ys). Ys is the map of the list Xs under the predicate P. That is, for each element X of Xs there is a corresponding element Y of Ys such that P (X , Y) is true. The
315 Second-Order Programming
has_property([XIXsI,P) P(X), has_property(Xs,P).
has_property([ ] ,P).
map_1it([XIXs] ,P, [YIYsI) - P(X,Y) , map_list(Xs,P,Ys). rnap_list( E I ,P, E I).
Figure 16.4 Second-order predicates
order of the elements in Xs is preserved in Ys. We can use map_list to rewrite some of the programs of earlier chapters. For example, Pro- gram 7.8 mapping English to French words can be expressed as map_ list(Words,dict,Mots). Uke has_property, map_list is easily defined using a variable predicate name. The definition is given in Figure 16.4.
Operationally, allowing variable predicate names implies dynamic con- struction of goals while answering a query. The relation to be computed is not fixed statically when the query is posed but is determined dynam- ically during the computation.
Some Prologs allow the programmer to use variables for predicate names, and allow the syntax of Figure 16.4. lt is unnecessary to com- plicate the syntax however. The tools already exist for implementing second-order predicates. One basic relation is necessary, which we call apply; it constructs the goal with a variable functor. The predicate apply is defined by a set of clauses, one for each functor name and arity. For example, for functor foo of ariry n, the clause is
apply(foo,X1, . . . ,Xn) - foo(X1, . . . ,Xn).
The two predicates in Figure 16.4 are transformed into Standard Prolog in Program 16.8. Sample definitions of apply clauses are given for the examples mentioned in the text.
The predicate apply performs structure inspection. The whole collec- tion of apply clauses can be generalized by using the structure inspec- tion primitive, univ. The general predicate apply(P,Xs) applies predi- cate P to a list of arguments Xs:
apply(F,Xs) - Goal =.. [FIXs], Goal.
We can generalize the function to be applied from a predicate name, i.e., an atom, to a term parameterized by variables. An example is substitut- ing for a value in a list. The relation substitute/4 from Program 9.3
316 Chapter 16
has_property (Xs,P) Each element in the list Xs has property P.
has_property([XIXs] ,P) -
apply(P,X), has_property(Xs,P).
has_property( E I ,P). apply(male,X) male(X).
maplist (Xs,P, Ys) Each element in the list Xs stands in relation P to its corresponding element in the list Ys.
map_list([XIXs],P,[YIYs]) -
apply(P,X,Y), map_list(Xs,P,Ys).
map_list([ ],P,[ J).
apply(dict,X,Y) - dict(X,Y).
Program 16.8 Second-order predicates in Prolog
can be viewed as an instance of map_ list if parameterization is allowed. Namely, map_list(Xs,substitute(Old,New) ,Ys) has the same effect in substituting the element New for the element Old in Xs to get Ys - exactly the relation computed by Program 9.3. In order to handle this correctly, the definition of apply must be extended a little:
apply(P,Xs) -
P =.. Li, append(Li,Xs,L2), Goal .. L2, Goal.
Using apply as part of map_list leads to inefficient programs. For ex- ample, using substitute directly rather than through map_list results in far fewer intermediate structures being created, and eases the task of compilation. Hence these second-order predicates are better used in conjunction with a program transformation system that can translate second-order calls to first-order calls at compile-time.
The predicate apply can also be used to implement lambda expres- sions. A lambda expression is one of the form lambda(X1,.. .,X).Expres- sion. If the set of lambda expressions to be used is known in advance, they can be named. For example, the above expression would be replaced by some unique identifier, f 00 say, and defined by an apply clause:
apply(foo,X1, . . - ,Xn) - Expression.
317 Second-Order Programming
Although possible both theoretically and pragmatically, the use of lambda expressions and second-order constructs such as has_property and map_list is not as widespread in Prolog as in functional program- ming languages like Lisp. We conjecture that this is a combination of cultural bias and the availability of a host of alternative programming techniques. It is possible that the ongoing work on extending the logic prograniniing model with higher-order constructs and integrating it with functional programming will change the picture.
In the meantime, all-solutions predicates seem to be the main and most useful higher-order construct in Prolog.
Exercise for Section 16.3
(i) Write a program performing beta reduction for lambda expressions.
16.4 Background
The discussion of f indall uses the description contained in the Stan- dard Prolog document (Scowen, 1991). An excellent discussion of the all-solutions predicates bagof and setof in Edinburgh Prolog are given in Warren (1982a). Discussions of "rolling your own" set predicates can be found in both O'Keefe (1990) and Ross (1989).
Set predicates are a powerful extension to Prolog. They can be used (in- efficiently) to implement negation as failure and meta-logical type pred- icates (Kahn, 1984). If a goal G has no solutions, which is determined by a predicate such as f indall, then not G is true. The predicate var(X) is implemented by testing whether the goal X=1 ; X=2 has two solutions. Further discussion of such behavior of set predicates and a survey of dif- ferent implementations of set predicates can be found in Naish (1985a).
Further description of the Lee algorithm and the general routing prob- lem for VLSI circuits can be found in textbooks on VLSI, for example, Breuer and Carter (1983). A neat graphic version of Program 16.6 has been written by Dave Broderick.
318 Chapter 16
Recent logic programming research has focused somewhat more on higher-order logic programming. Approaches of note are Lambda-Prolog (Miller and Nadathur, 1986) and HiLog (Chen et al., 1989).
KWIC was posed as a benchmark for high-level programming languages by Penis, and was used to compare several languages. We find the Prolog implementation of it perhaps the most elegant of all.
Our description of lambda expressions is modeled after Warren (1982a). Predicates such as apply and map_list were part of the utili- ties package at the University of Edinburgh. They were fashionable for a while but fell out of favor because they were not compiled efficiently, and no source-to-source transformation tools were available.
17 Interpreters
Meta-programs treat other programs as data. They analyze, transform, and interpret other programs. The writing of meta-programs, or meta- programming, is particularly easy in Prolog because of the equivalence of programs and data: both are Prolog terms. We have already presented some examples of meta-programs, namely, the editor of Program 12.5 and the shell process of Program 12.6. This chapter covers interpreters, an important and useful class of meta-programs, and Chapter 18 dis- cusses program transformation.
17.1 Interpreters for Finite State Machines
The sharp distinction between programs and data present in most com- puter languages is lacking in Prolog. The equivalence of programs and data greatly facilitates the writing of interpreters. We demonstrate the facility in this section by considering the basic computation models of computer science. interpreters for the various classes of automata are very easily written in Prolog.
lt is interesting to observe that the interpreters presented in this sec- tion are a good application of nondetermmistic progranmiing The pro- grams that are presented illustrate typical examples of don't-know non- determinism. The same interpreter can execute both deterministic and nondetermimstic automata because of the nondeterminism of Prolog.
Definition A (nondeterministic) finite automaton, abbreviated NDFA, is a 5-tuple (QX,Ö,I,F), where Q is a set of states, is a set of symbols, 6 is a
320 Chapter 17
accept(Xs) - The string represented by the list Xs is accepted by the NDFA defined by initial/i, delta/3, and final/i.
accept(Xs) - initial(Q), accept(Xs,Q).
accept([XIXs],Q) - delta(Q,X,Q1), accept(Xs,Q1)
accept([ ],Q) - final(Q).
Program 17.1 An interpreter for a nondeterministic finite automaton (NDFA)
mapping from Q x to Q, I is an initial state, and F is a set of final states. If the mapping is a function, then an NDFA is deterministic.
A finite automaton can be specified as a Prolog program by three col- lections of facts. The predicate initial (Q) is true if Q is the initial state. The predicate final (Q) is true if Q is a final state. The most interesting is delta(Q,X,Ql), which is true if the NDFA changes from state Q to state Ql on receipt of symbol X. Note that both the set of states and the set of symbols can be defined implicitly as the constants that appear in the initial, final, and delta predicates.
An NDFA accepts a string of symbols from the alphabet if when started in its initial state, and following the transitions specified by ô, the NDFA ends up in one of the final states. An interpreter for an NDFA must determine whether it accepts given strings of symbols. Program 17.1 is an interpreter. The predicate accept (Xs) is true if the NDFA defined by the collection of initial, final, and delta facts accepts the string represented as the list of symbols Xs.
Figure 17.1 shows a deterministic automaton that accepts the language (ab)*. There are two states, qO and ql. If in state qO an a is received, the automaton moves to state ql. The automaton moves back from ql to qO if a b is received. The initial state is qO, and qO is also the single final state.
To use the interpreter, a specific automaton must be given. Program 17.2 is the realization in Prolog of the automaton in Figure 17.1. The combination of Programs 17.1 and 17.2 correctly accepts strings of al- ternating a's and b's.
If an arc from qO to itself labeled a is added to the automaton in Fig- ure 17.1, we get a new automaton that recognizes the language (a(a*)b)*.
321 Interpreters
Figure 17.1 A simple automaton
initial(qO)
uinal(qO)
delta (qQ , a, qi)
cielta(ql,b,qO).
Program 17.2 An NDFA that accepts the language (ab)*
This automaton is nondeterministic because on receipt of an a in state qO it is not determined which path will be followed. Nondeterminism does not affect the interpreter in Program 17.1. All that is needed to produce the new automaton is to add the fact delta(qO,a,qO) and the combined program will behave correctly.
Another simple computation model is a pushdown automaton that ac- cepts the class of context-free languages. Pushdown automata extend NDFAs by providing a single stack for memory in addition to the in- ternal state of the automaton. Formally, a (nondeterministic) pushdown automaton, abbreviated NPDA, is a 7-tuple (Q,G,&LZ,F) where Q, , I, F are as before, G is the set of symbols that can be pushed onto the stack, Z is the start symbol on the stack, and 6 is changed to take the stack into account.
Specifically, ô is a mapping from Q x x G* to Q x G*. The mapping controls the change of state of the NPDA and the pushing and popping of elements onto and off the stack by the NPDA. In one operation, the NPDA can pop (push) one symbol off (onto) the stack.
Analogously to an NDFA, an NPDA accepts a string of symbols from the alphabet E*, if when started ¡n its initial state and with the starting sym- bol on the stack, and following the transitions specified by ô, the NPDA ends up in one of the final states with the stack empty An interpreter for an NPDA is given as Program 17.3. The predicate accept(Xs) is true if the NDFA defined by the collection of initial, final, and delta facts
322 Chapter 17
accept(Xs) - The string represented by the list Xs is accepted by the NPDA defined by initial/I, delta/5, and final/i.
accept(Xs) - initial(Q), accept(Xs,Q,[ 1). accept([XIXs],Q,S) - delta(Q,X,S,Q1,S1), accept(Xs,Q1,S1).
accept([ ],Q,[ 1) - final(Q). Program 17.3 An interpreter for a nondeterministic pushdown automaton (NPDA)
initial(qO). final(ql).
delta(qO,X,S,qO, [XIS]).
delta(qO,X,S,ql, [XIS]).
delta (qO ,X, S, qi, S)
delta(ql,X, [XIS] ,ql,S).
Program 17.4 An NPDA for palindromes over a finite alphabet
accepts the string represented as the list of symbols Xs. The interpreter is very similar to the interpreter of an NDFA given as Program 17.1. The only change is the explicit manipulation of the stack by the delta predi- cate.
A particular example of an NPDA is given as Program 17.4. This au- tomaton accepts palindromes over a finite alphabet. A palindrome is a nonempty string that reads the same backwards as forwards. Example palindromes are noon, madam, and gleneig. Again, the automaton is specified by initial, final, and delta facts, and the sets of symbols being defined implicitly. The automaton has two states: qO, the initial state when symbols are pushed onto the stack, and qi, a final state when symbols are popped off the stack and compared with the symbols in the input stream. When to stop pushing and start popping is decided nonde- terministically. There are two delta facts that change the state from qO to ql to allow for palindromes of both odd and even lengths.
Programs 17.1 and 17.2 can be combined into a single program for recognizing the language (ab)*. Similarly, Programs 17.3 and 17.4 can be combined into a single program for recognizing palindromes. A program that can achieve this combination is given in Chapter 18.
323 Interpreters
It is straightforward to build an interpreter for a Turing machine writ- ten in a similar style to the interpreters in Programs 17.1 and 17.3. This is posed as Exercise (iii) at the end of this section. Building ari interpreter for Turing machines shows that Prolog has the power of all other known computation models.
Exercises for Section 17.1
Define an NDFA that accepts the language ab*c.
Define an NPDA that accepts the language ab1.
Write an interpreter for a Turing machine
17.2 Meta-Interpreters
We turn now to a class of especially useful interpreters. A meta-inter- preter for a language is an interpreter for the language written in the language itself. Being able to write a meta-interpreter easily is a very pow- erful feature of a programming language. It gives access to the computa- tion process of the language and enables the building of an integrated programming environment. The examples in the rest of this chapter demonstrate the potential of meta-interpreters arid the ease with which they can be written. In this section, we also examine issues in writing meta-interpreters.
Throughout the remainder of this chapter, the predicate solve is used for a meta-interpreter. A suitable relation scheme is as follows. The re- lation solve(Goal) is true if Goal is true with respect to the program being interpreted.
The simplest meta-interpreter that can be written in Prolog exploits the meta-variable facility. Et is defined by a single clause:
solve(A) - A.
This trivial interpreter is only useful as part of a larger program. For example, a version of the trivial interpreter forms the basis for the in- teractive shell given as Program 12.6 arid the logging facility given as Program 12.7. In general, as we suggest here and see in more detail in
324 Chapter 17
solve(Goal) - Goal is true given the pure Prolog program defined by clause/2.
solve (true)
solve((A,B)) - solve(A), solve(B).
salve(A) - clause(A,B), solve(B).
Program 17.5 A meta-interpreter for pure Prolog
Sections 17.3 and 17.4, meta-interpreters are useful and important be- cause of the easily constructed enhancements.
The best known and most widely used meta-interpreter models the computation model of logic programs as goal reduction. The three clauses of Program 17.5 interpret pure Prolog programs. This meta- interpreter, called vanilla, together with its enhancements, is the basis of the rest of this section and Section 17.3.
The interpreter in Program 17.5 can be given a declarative reading. The solve fact states that the empty goal, represented by the constant true, is true. The first solve rule states that a conjunction (A,B) is true if A is true and B is true. The second solve rule states that a goal A is true if there is a clause A - B in the interpreted program such that B is true.
We also give a procedural reading of the three clauses in Program 17.5. The solve fact states that the empty goal, represented in Prolog by the atom true, is solved. The next clause concerns conjunctive goals. It reads: "To solve a conjunction (A,B), solve A and solve B." The general case of goal reduction is covered by the final clause. To solve a goal, choose a clause from the program whose head unifies with the goal, and recursively solve the body of the clause.
The procedural reading of Prolog clauses is necessary to demonstrate that the meta-interpreter of Program 17.5 indeed reflects Prolog's choices of implementing the abstract computation model of logic programming. The two choices are the selection of the leftmost goal as the goal to reduce, and sequential search and backtracking for the nondeterministic choice of the clause to use to reduce the goal. The goal order of the body of the solve clause handling conjunctions guarantees that the leftmost goal in the conjunction is solved first. Sequential search and backtracking comes from Prolog's behavior in satisfying the clause goal.
The hard work of the interpreter is borne by the third clause of Pro- gram 17.5. The call to clause performs the unification with the heads
325 Interpreters
solve(rnember(X, [a,b,c])) clause(inember(X, [a,b,c]) ,B) solve (true)
true Output: X=a
solve (true) clause(true,T) f
clause(member(X, [a,b,c] 6) solve(member(X, [b,c]))
clause(member(X, [b,c]) ,Bl) solve (true) true Output: Xb
solve (true) clause(true,T) f
clause(member(X, [b,c]),B1) solve (member(X, [cl))
clause (member(X, [cl ,B2) solve (true) true Output: X=c
solve (true) clause(true,T) f
clause(rnember(X, [cl ,B2)) solve(member(X, C i))
clause(member(X, E ]),B3) no (more) solutions
Figure 17.2 Tracing the meta-interpreter
{Xa, 13=true
tBmember(X, [b,c])}
{x=b ,B1=true
(B1member(X, [c])}
{X=c ,B2=true}
{B2=member(X, C
of the clauses appearing in the program. It is also responsible for giv- ing different solutions on backtracking. Backtracking also occurs in the conjunctive rule reverting from B to A.
Tracing the meta-interpreter of Program 17.5 solving a goal is instruc- tive. The trace of answering the query solve (member (X, [a, b, c])) with respect to Program 3.12 for member is given in Figure 17.2.
The vanilla meta-interpreter inherits Prolog's representation of clauses using the system predicate clause. Alternative representations of clauses are certainly possible, and indeed have been used by alter- native Prologs. lists are one possible representation. The clause A - B1,B2,...,B can be represented by the clause rule(A, [B1,...,Bn]). In
326 Chapter 17
solve(Goal) - Goal is true given the pure Prolog program defined by clause/2.
solve(Goal) solve(Goal,[ J).
solve([ ],[ 1). solve([ ],[GIGoals]) solve(G,Goals).
solve([AIB] ,Goals) append(B,Goals,Goalsl), solve(A,Goalsl).
solve(A,Goals) - rule(A,B), solve(B,Goals).
Program 17.6 A meta-interpreter for pure Prolog in continuation style
this representation, the empty list represents the empty goal and list construction represents conjunction. This representation is used in Pro- gram 17.6.
A different representation imposes a different form on the meta- interpreter, as illustrated in Program 17.6. Unlike Program 17.5, this version of the vanilla meta-interpreter makes explicit the remaining goals in the resolvent. Enhancements can be written to exploit the fact that the resolvent is accessible during the computation, for example, allowing a more sophisticated computation rule. The behavior of Program 17.6 can be considered as being in continuation style promoted by languages such as Scheme.
Differences in meta-interpreters can be characterized in terms of their granularity, that is the chunks of the computation that are made acces- sible to the programmer. The granularity of the trivial one-clause meta- interpreter is too coarse. Consequently there is little scope for applying the meta-interpreter. It is possible, though not as easy, to write a meta- interpreter that models unification and backtracking. The granularity of such a meta-interpreter is very fine. Working at this fine level is usually not worthwhile. The efficiency loss is too great to warrant the extra ap- plications. The meta-interpreter in Program 17.5, at the clause reduction level, has the granularity most suited for the widest range of applica- tions.
The vanilla meta-interpreter must be extended to handle language fea- tures outside pure Prolog. Builtin predicates are not defined by clauses in the program and need different treatment. The easiest way to incor- porate builtin predicates is to use the meta-variable facility to call them directly. A table of builtin predicates is necessary. In this chapter, we assume a table of facts of the form builtin(Predicate) for each builtin
327 Interpreters
builtin(A is ) builtin(A > B)
builtin(read(X)) buìltin(write(X)).
builtin(integer(X)). builtin(functor(T,F,N)).
builtin(clause(A,B)) builtin(builtin(X)).
Figure 17.3 Fragment of a table of builtin predicates
predicate. Figure 17.3 gives part of that table. A table of builtin predi- cates is provided in some Prologs by another name but is not present in Standard Prolog.
The clause solve(A) builtin(A), A. can be added to the meta- interpreter in Program 17.5 to correctly handle builtin predicates. The resulting program handles four disjoint cases, one per clause, for solving goals: the empty goal, conjunctive goals, builtin goals, and user-defined goals. For compatibility with a number of Prolog systems, the meta- interpreters in the rest of this section contain cuts to indicate that the clauses are mutually exclusive.
The extra solve clause makes the behavior of the builtin predicates in- visible to the meta-interpreter. User-defined predicates that one wants to make invisible can be handled similarly with a single clause. Conversely, there are occasions when builtin predicates for negation and second- order programming should be made visible.
The vanilla meta-interpreter needs to be extended to handle cuts cor- rectly. A naive incorporation of cuts treats them as a builtin predicate, effectively adding a clause solve (!) !. This clause does not achieve the correct behavior of cut. The cut in the clause commits to the current solve clause rather than pruning the search tree.
To achieve correct behavior of cut in a meta-interpreter, one needs to understand scope, that is to which clause the cut commits. The scope of cut, as described in Chapter 11, is the clause in which the cut is a goal in the body. The scope of cut when it is contained within a meta-logical builtin predicate such as conjunction and disjunction is less distinct and varies in different Prologs. If a cut is part of a disjunction, should ex- ecution of the cut commit to the current disjunct or to the clause in which the disjunction is embedded? Handling cut correctly in a mcta- interpreter is tricky and usually relies on technical details of the scope of cut in a particular implementation of Prolog. Incorporating cuts within
328 Chapter 17
solve_trace( Goal) - Goal is true given the Prolog program defined by clause/2. The program traces the proof by side effects.
solve_trace(Goal) - solve_trace(Goal,O).
solve_trace(true,Depth) -
solve_trace((A,B) ,Depth) -
solve_trace(A,Depth), solve_trace(B,Depth).
solve_trace (A ,Depth) -
builtin(A), !, A, display(A,Depth), nl.
solve_trace (A ,Depth)
clause(A,B), display(A,Depth), nl, Depthl is Depth + 1,
solve_trace (B ,Depthl).
display(A,Depth) -
Spacing is 3*Depth, put_spaces(Spacing), write(A).
put_spaces(N) -
between(1,N,I), put_charY '), fail.
put_spaces (N)
between(i,N,I) - SeeProgram8.5.
Program 17.7 A tracer for Prolog
meta-interpreters has been widely studied, and references to solutions are given in Section 17.5.
We apply meta-mterpreters to develop a simple tracer. Program 17.7 handles success branches of computations and does not display failure nodes in the search tree. It is capable of generating the traces presented in Chapter 6.
The basic predicate is solve_trace(Goal,Depth), where Goal is solved at some depth. The starting depth is assumed to be O. The first solve_trace/2 clause in Program 17.7 states that the empty goal is solved at any depth. The second clause indicates that each goal in a con- junct is solved at the same depth. The third clause handles builtins. The final solve_trace/2 clause matches the goal with the head of a program clause, displays the goal, increments the depth, and solves the body of the program clause at the new depth.
The predicate display(Goal,Depth) is an interface for printing the traced goal. The second argument, Depth, controls the amount of inden- tation of the first argument, Goal. Level of indentation correlates with depth in the proof tree.
329 Interpreters
solve(Goal,Tree) - Tree is a proof tree for Goal given the program defined by c!ause/2.
solve(true,true)
solve((A,B) , (ProofA,ProofB)) -
solve(A,ProoÍA), solve(B,ProofB)
solve(A,(Abuiltin)) - builtin(A), 1, A.
solve(A,(AProof)) clause(A,fl), solve(B,Proof).
Program 17.8 A meta-interpreter for building a proof tree
There is subtlety in the goal order of the clause
solve_trace (A ,Depth) -
clause(A,B), display(A,Depth), nl, Depthl is Depth + 1,
solve_trace(B,Depthl).
The display goal is between calls to clause and solve_trace, ensuring that the goal is displayed each time Prolog backtracks to choose another clause. If the clause and display goals are swapped, only the initial call of the goal is displayed.
Using Program 17.7 for the query solve_trace (append(Xs,Ys, [a,b, c]))? with Program 3.15 for append generates a trace like the one pre- sented in Section 6.1. The output messages and semicolons for alterna- tive solutions are provided by the underlying Prolog. There is only one difference from the trace in Figure 6.2. The unifications are already per- formed. Separating out unifications requires explicit representation of unification and is considerably harder.
A simple application of meta-interpreters constructs a proof tree while solving a goal. The proof tree is built top-down. A proof tree is essen- tial for the applications of debugging and explanation in the next two sections.
The basic relation is solve(Goal,Tree), where Tree is a proof tree for the goal Goal, Proof trees are represented by the structure Goal - Proof. Program 17.8 implements solve/2 and is a straightforward en- hancement of the vanilla meta-interpreter. We leave as an exercise for the reader giving a declarative reading of the program.
Here is an example of using Program 17.8 with Program 1.2. The query solve(son(lot,haran) ,Proof)? has the solution
330 Chapter 17
solve( Goal,Certainty) Certainty is our confidence that Goal is true.
solve(true,1) -
solve((A,B),C) -
!, solve(A,C1), solve(B,C2), minimum(C1,C2,C).
solve(A,1) - builtin(A), !, A.
solve(A,C) - clause_cf(A,B,C1), solve(B,C2), C is Cl * C2.
minimum(X,Y,Z) - See Program 11.3.
Program 17.9 A meta-interpreter for reasoning with uncertainty
Proof = (son(lot,haran) -
((f ather(haran,lot)true),
(male(lot)true))).
The query solve (son (X,haran) ,Proof)? has the solution X=lot and the same value for Proof.
Our next enhancement of the vanilla meta-interpreter incorporates a mechanism for uncertainty reasoning. Associated with each clause is a certainty factor, which is a positive real number less than or equal to 1. A logic program with certainties is a set of ordered pairs (Clause,Factor), where Clause is a clause and Factor is a certainty factor.
The simple meta-interpreter in Program 17.9 implements the un- certainty reasoning mechanism. The program is a straightforward en- hancement of the vanilla meta-interpreter. The top-level relation is solve (Goal,Certainty), which is true when Goal is satisfied with cer- tainty Certainty.
The meta-interpreter computes the combination of certainty factors in a conjunction as the minimum of the certainty factors of the conjuncts. Other combining strategies could be accommodated just as easily. Pro- gram 17.9 assumes that clauses with certainty factors are represented using a predicate clause_cf (A, B ,CF).
Program 17.9 can be enhanced to prune computations that do not meet a desired certainty threshold. An extra argument constituting the value of the cutoff threshold needs to be added. The enhanced program is given as Program 17.10. The new relation is solve (Goal,Certainty, Threshold).
The threshold is used in the fourth clause in Program 17.10. The cer- tainty of any goal must exceed the current threshold. If the threshold is
331 Interpreters
solve (Goal, Certainty, Threshold) - Certainty is our confidence, greater than Threshold, that Goal is true.
solve(true,1,T) -
solve( (A , B) ,C , T)
I, solve(A,Cl,T), solve(B,C2,T), minimuin(Ci,C2,C).
solve(A,1,T) .- builtin(A), !, A.
solve(A,C,T) -
clause_cf(A,B,C1), Cl > T, Ti is T/Cl,
solve(B,C2,Tl), C is Cl * C2.
minimuin(X,Y,Z) See Program 11.3.
Program 17.10 Reasoning with uncertainty with threshold cutoff
exceeded, the computation continues. The new threshold is the quotient of the previous threshold by the certainty of the clause.
Exercises for Section 17.2
Write a meta-interpreter to count the number of times a procedure is called in a successful computation.
Write a meta-interpreter to find the maximum depth reached in a computation.
Extend Program 17.6 to give a tracer and build a proof tree.
Extend Program 17.7 for solve_trace/2 to print out failed goals.
(y) Modify Program 17.8 to use a different representation for a proof tree.
17.3 Enhanced Meta-Interpreters for Debugging
Debugging is an essential aspect of programming, even in Prolog. The promise of high-level programming languages is not so much iii the prospect for writing bug-free programs but in the power of the com- puterized tools for supportmg the process of program development. For reasons of bootstrapping and elegance, these tools are best implemented
332 Chapter 17
in the language itself. Such tools are programs for manipulating, analyz- ing, and simulating other programs, or in other words, meta-programs.
This section shows meta-programs for supporting the debugging process of pure Prolog programs. The reason for restricting ourselves to the pure part is clear: the difficulties in handling the impure parts of the language.
To debug a program, we must assume that the programmer has some intended behavior of the program in mind, and an intended domain of application on which the program should exhibit this behavior. Given those, debugging consists of finding discrepancies between the pro- gram's actual behavior and the behavior the programmer intended. Recall the definitions of an intended meaning and a domain from Sec- tion 5.2. An intended meaning M of a pure Prolog program is the set of ground goals on which the program should succeed. The intended domain D of a program is a domain on which the program should ter- minate. We require the intended meaning of a program to be a subset of the intended domain.
We say that A1 is a solution to a goal A if the program returns on a goal A its instance A1. We say that a solution A is true in an intended meaning M if every instance of A is in M. Otherwise it is false in M.
A pure Prolog program can exhibit only three types of bugs, given an intended meaning and an intended domain. When invoked on a goal A in the intended domain, the program may do one of three things:
Fail to terminate Return some false solution AO Fail to return some true solution AO
We describe algorithms for supporting the detection and identification of each of these three types of bugs.
In general, it is not possible to detect if a Prolog program is nonter- minating; the question is undecidable. Second best is to assign some a priori bound on the running time or depth of recursion of the program, and abort the computation if the bound is exceeded. It is desirable to save part of the computation to support the analysis of the reasons for nontermination. The enhanced meta-interpreter shown in Program 17.11 achieves this. It is invoked with a call solve(A,D,Overf low), where A is an initial goal, and D an upper bound on the depth of recursion. The call
333 Interpreters
solve(A,D,Overflow) - A has a proof tree of depth less than D and Overflow equals no_overflow, or A has a branch in the computation tree longer than D, and Overflow contains a list of its first D elements.
solve(true,D,no_overflow) L solve(A,0,overflow([ J)) - L
solve((A,B) ,D,Overf low)
D >0, !, solve (A,D,Overf iowA)
solve_conjunction(OverflowA,B,D,Overflow).
solve(A,D,no_overflow) -
D > 0,
builtiri(A), !, A.
solve(A,D,Overflow)
D > 0,
clause (A ,B)
Dl is D-1,
soive(B,D1,OverflowB)
return_overflow(QverflowB ,A ,Overf low).
solve_conjunction(overfiow(S) ,B,D,overflow(S))
solveconjunctïon(nooverflow,B,D,Overfiow) -
solve (B,D,Overflow)
returnoverflow(no_overflow,A ,no_overf low).
return_overflow(overflow(S),A,overfiow([AIS])).
Program 17.11 A meta-interpreter detecting a stack overflow
succeeds if a solution is found without exceeding the predefined depth of recursion, with Overflow instantiated to no_overflow. The call also succeeds if the depth of recursion is exceeded, but in this case Over- flow contains the stack of goals, i.e., the branch of the computation tree, which exceeded the depth-bound D.
Note that as soon as a stack overflow is detected, the computation returns, without completing the proof. This is achieved by solve_ conjunction and return_overflow.
For example, consider Program 17.12 for insertion sort. When called with the goal solve(isort([2,2],Xs),6,Overf low), the solution re- turned is
334 Chapter 17
isort(Xs,Ys) - Ys is an ordered permutation of Xs. Nontermination program.
isort([XIXs],Ys) - isort(Xs,Zs), insert(X,Zs,Ys).
isort([ ],[ 1).
insert(X,[YIYs],[X,YIYs]) -
X < Y.
insert(X,[YIYs],[YIZs]) - X Y, insert(Y,[XIYs],Zs).
insert(X,[ ],[X]).
Program 17.12 A nonterminating insertion sort
The overflowed stack can be further analyzed, upon return, to diagnose the reason for nontermination. This can be caused, for example, by a loop, i.e., by a sequence of goals G1,G2,. . .,G, on the stack, where G1 and G are called with the same input, or by a sequence of goals that calls each goal with increasingly larger inputs. The first situation occurs in the preceding example. It is clearly a bug that should be fixed in the program. The second situation is not necessarily a bug, and knowing whether the program should be fixed or whether a larger machine should be bought in order to execute it requires further program-dependent information.
The second type of bug is returning a false solution. A program can return a false solution only if it has a false clause. A clause C is false with respect to an intended meaning M if it has an instance whose body is true in M and whose head is false in M. Such an instance is called a counterexam pie to C.
Consider, for example, Program 17.13 for insertion sort. On the goal isort([3,2,1] ,Xs) it returns the solution isort([3,2,1] [3,2,1])
which is clearly false.
Xs = [2,2, 2,2,2,2], Overflow = overflow( [
isort( [2,2] [2,2,2,2,2,2]), insert (2, [2] ,[2,2,2,2,2,2]), insert (2,[2] [2,2,2,2,2]), insert (2,[2] ,[2,2,2,2]), insert (2, [2] ,[2,2,2]), insert (2, [2] ,[2,2])])
335 Interpreters
isort(Xs,Ys) - Buggy insertion sort.
isort(EXIXs],Ys) isort(Xs,Zs), irisert(X,Zs,Ys).
isort([ I,[ I). inert(X, [YIY] , [X,YIYs]) -
X Y.
insert (X, [YIYsJ , [Y IZs]
X > Y, insert(X,Ys,Zs). insert(X,[ I,EX]).
Program 17.13 An incorrect and incomplete insertion sort
The false clause in the program is
insert(X, [YIYs], [X,YIYs]) - X Y.
and a counterexample to it is
insert(2, [1] [2,1]) .- 2 1.
Given a ground proof tree corresponding to a false solution, one can find a false instance of a clause as follows: Traverse the proof tree in postorder. Check whether each node in the proof tree is true. If a false node is found, the clause whose head is the false node and whose body is the conjunction of its sons is a counterexample to a clause in the program. That clause is false and should be removed or modified.
The correctness of this algorithm follows from a simple inductive proof. The algorithm is embedded in an enhanced meta-interpreter, shown as Program 17.14.
The algorithm and its implementation assume an oracle that can an- swer queries concerning the intended meaning of the program. The or- acle is some entity external to the diagnosis algorithm. It can be the programmer, who can respond to queries concerning the intended mean- ing of the program, or another program that has been shown to have the same meaning as the intended meaning of the program under de- bugging. The second situation may occur in developing a new version of a program while using the older version as an oracle. lt can also occur when developing an efficient program (e.g., quicksort), given an ineffi- cient executable specification of it (i.e., permutation sort), and using the specification as an oracle.
336 Chapter 17
false_solution (A,Clause) - If A is a provable false instance, then Clause is a false clause in the program. Bottom-up algorithm.
false_solution(A,Clause)
solve(A,Proof)
false_clause (Proof ,Clause)
solve(Goal,Proof) - See Program 17.8.
false_clause(true,ok)
faise_clause((A,B) ,Clause) -
f aise_clause (A , ClauseA)
check_conjunction(CiauseA ,B ,Clause).
false_clause((AB) ,Clause) - f aise_clause (B , ClauseB)
check_clause (ClauseB , A, B ,Clause).
check_conjunction(ok ,B ,Clause)
f aise_clause (B , Clause)
check_conjunction((AB1) ,B, (ABl)).
check_clause(ok,A,B,Clause) -
query_goal (A ,Answer),
check_answer (Answer, A, B ,Clause).
check_clause((AlBl) ,A,B, (AlBl)).
check_answer(true,A,B,ok). check_answer(false,A,B, (ABl)) -
extract_body (B,Bl).
extract_body (true , true)
extract_body((AB) ,A). extract_body(((AB) ,Bs) , (A,As)) -
extract_body(Bs,As).
query_goal(A,true) -
built in (A)
query_goal (Goal ,Answer) -
not builtin(Goal),
writeln(['Is the goal ',Goal,' true?']),
read (Answer).
Program 17.14 Bottom-up diagnosis of a false solution
337 Interpreters
When invoked with the goal false_solution(isort([3,2,1] ,X) ,C) the algorithm exhibits the following interactive behavior:
false_solution(isort([3,2,1],X),C)?
Is the goal isort([ ],[ ]) true? true. Is the goal insert(1, [ ] , [1]) true? true. Is the goal isort ([1] , [1] ) true? true. Is the goal insert (2, [1] , [2,1]) true? false.
X = [3,2,1], C = insert(2,[1],[2,1]) - 2 1.
This returns a counterexample to the false clause. The proof tree returned by solve/2 is not guaranteed to be ground,
in contrast to the assumption of the algorithm. However, a ground proof tree can be generated by either instantiating variables left in the proof tree to arbitrary constants before activating the algorithm, or by request- ing the oracle to instantiate the queried goal when it contains variables. Different instances might imply different answers. Since the goal of this algorithm is to find a counterexample as soon as possible, the oracle should instantiate the goal to a false instance if it can.
One of the main concerns with diagnosis algorithms is improving their query complexity, i.e., reducing the number of queries they require to diagnose the bug. Given that the human programmer may have to answer the queries, this desire is understandable. The query complexity of the preceding diagnosis algorithm is linear in the size of the proof tree. There is a better strategy, whose query complexity is linear in the depth of the proof tree, not its size. In contrast to the previous algorithm, which is bottom-up, the second algorithm traverses the proof tree top- down. At each node it tries to find a false son. The algorithm recurses with any false son found. If there is no false son, then the current node constitutes a counterexample, as the goal at the node is false, and all its Sons are true.
The implementation of the algorithm is shown in Program 17.15. Note the use of cut to implement ¡mplicit negation in the first clause of f alse_ goal/2 and the use of query_goal/2 as a test predicate.
338 Chapter 17
false_solution (A, Clause) If A is a provable false instance, then Clause is a false clause in the program. Top-down algorithm.
false_solution(A ,Clause) -
solve (A,Proof),
false_goal (Proof ,Clause).
solve(Goal,Proof) - See Program 17.8.
false_goal((A'B) ,Clause) -
f alse_conjunction (B, Clause)
false_goal((AB) ,(ABl)) - extract_body(B,Bl).
false_conjunction(((AB) ,Bs) ,Clause)
query_goal (A,false),
false_goal((AB) ,Clause).
false_conjunction((AB) ,Clause)
query_goal (A,false),
false_goal((AB) ,Clause).
false_conjunction((A,As) ,Clause)
false_conjunction(As ,Clause).
extract_body(Tree,Body) - See Program 17.14.
query_goal(A,Answer) - See Program 17.14.
Program 17.15 Top-down diagnosis of a false solution
Compare the behavior of the bottom-up algorithm with the following trace of the interactive behavior of Program 17.15:
false_solution(isort([3,2,1] ,X),C)?
Is the goal isort([2, 11,12,1]) true? false. Is the goal isort ([1] , [11) true? true. Is the goal insert (2, [1] [2,1]) true? false.
X = [3,2,1], C = insert(2, [11,12,1]) - 2 1.
There is a diagnosis algorithm for false solutions with an even better query complexity, called divide-and-query. The algorithm progresses by splitting the proof tree into two approximately equal parts and querying
339 Interpreters
the node at the splitting point. 1f the node is false, the algorithm is applied recursively to the subtree rooted by this node. If the node is true, its subtree is removed from the tree and replaced by true, and a new middle point is computed. The algorithm can be shown to require a number of queries logarithmic in the size of the proof tree. In case of close-to-linear proof trees, this constitutes an exponential improvement over both the top-down and the bottom-up diagnosis algorithms.
The third possible type of bug is a missing solution. Diagnosing a missing solution is more difficult than fixing the previous bugs. We say that a clause covers a goal A with respect to an intended meaning M if it has an instance whose head is an instance of A and whose body is in M.
For example, consider the goal insert (2, [1,3] ,Xs). lt is covered by the clause
insert(X, [YIYs] , [X,YJYs]) X Y.
of Program 17.13 with respect to the intended meaning M of the pro- gram, since in the following instance of the clause
insert(2,[1,3],[1,2,3]) - 2 1.
the head is ari instance of A and the body is in M. It can be shown that if a program P has a missing solution with respect
to an intended meaning M, then there is a goal A in M that is not covered by any clause in P. The proof of this claim is beyond the scope of the book. lt is embedded in the diagnosis algorithm that follows.
Diagnosing a missing solution imposes a heavier burden on the oracle. Not only does it have to know whether a goal has a solution but it must also provide a solution, if it exists. Using such an oracle, an uncovered goal can be found as follows.
The algorithm is given a missing solution, i.e., a goal in the intended meaning M of the program P, for which P fails. The algorithm starts with the initial missing solution. For every clause that unifies with it, it checks, using the oracle, if the body of the clause has an instance in M. If there is no such clause, the goal is uncovered, and the algorithm terminates. Otherwise the algorithm finds a goal in the body that fails. At least one of them should fail, or else the program would have solved the body, and hence the goal, in contrast to our assumption. The algorithm is applied recursively to this goal.
340 Chapter 17
missing_solution (A,Goal) - If A is a nonprovable true ground goal, then Goal is a true ground goal that is uncovered by the program.
missing_solution((A,B),Goal) -
(not A, missing_solution(A,Goal)
A, missing_solution(B,Goal))
inissing_solution(A,Goal) -
clause (A, B)
query_clause((AB)),
missing_solution (B , Goal)
missing_solution(A,A) -
not system(A)
query_clause(Clause) -
writeln(['Enter a true ground instance of ',Clause,
'if there is such, or no' otherwise']),
read (Answer)
check_answer (Answer , Clause)
check_answer(no,Clause) - !, fail.
check_answer(Clause,Clause)
check_answer (Answer ,Clause) -
write ('Illegal answer'),
query_clause (Clause)
Program 17.16 Diagnosing missing solution
An implementation of this algorithm is shown in Program 17.16. The program attempts to trace the failing path of the computation and to find a true goal which is uncovered. Following is a session with the program:
missing_solution(isort([2,1,3],[1,2,3]),C)?
Enter a true ground instance of (isort([2,1,3],[1,2,3]) -
isort([1,3],Xs),insert(2,Xs,[1,2,3]))
if there is such, or "no" otherwise
(isort([2,1,3],[1,2,3]) -
isort([1,3] ,[1,3]) ,insert(2, [1,3] [1,2,3])).
Enter a true ground instance of (isort([1,3] [1,3]) - isort([3] ,Ys) ,insert(1,Ys, [1,3]))
if there is such, or 'no' otherwise
341 Interpreters
(isort([1,3],[1,3]) - isort([3],[3]),insert(1,[3],[1,3])).
Enter a true ground instance of (insert(1, [3] ,[1,3]) - 1 3) if there is such, or no' otherwise
no.
C insert(1, [3] [1,3])
The reader can verify that the goal insert(1, [3] [1,3]) is not covered by Program 17.13.
The three algorithms shown can be incorporated in a high-quality in- teractive program development environment for Prolog.
17.4 An Explanation Shell for Rule-Based Systems
The final section of this chapter presents an application of interpreters to rule-based systems. An explanation shell is built that is capable of ex- plaining why goals succeed and fail and that allows interaction with the user during a computation. The shell is developed with the methodology of stepwise enhancement introduced in Section 13.3.
The skeleton interpreter in this section is written in the same style as the vanilla meta-interpreter and has the same granularity. It differs in two important respects. First, it interprets a rule language rather than Prolog clauses. Second, the interpreter has two levels to allow explana- tion of failed goals.
Before describing the interpreter, we give an example of a toy rule- based system written in the rule language. Program 17.17 contains some rules for placing a dish on the correct rack in an oven for baking. Facts have the form fact (Goal), For example, the first fact in Program 17.17 states that dishl is of type bread.
Rules have the form rule(Head,Body,Naine), where Head is a goal, Body is (possibly) a conjunction of goals, and Naine is the name of the rule. Individual goals in the body are placed inside a unary postfix func- tor is_true, for reasons to be explained shortly. Conjunctions in the body are denoted by the binary infix operator &, which differs from Pro- log syntax. Operator declarations for & and is_true are given in Program 17.17. To paraphrase a sample rule, rule place i in Program 17.17 states:
342 Chapter 17
Rule base for a simple expert system for placing dishes in an oven. The predicates used in the rules are place_in_oven (Dish,Rack) -
Dish should be placed in the oven at level Rack for baking. pastry(Dish) - Dish is a pastry. main_meal(Dish) Dish is a main meal. slow_cooker(Dish) - Dish is a slow cooker. type(Dish,Type) - Dish is best described as Type. size(Dish,Size) - The size of Dish is Size. The rules have the form rule(Head,Body,Name).
:- op(40,xfy,&).
op(30,xf,is_true).
rule (place_in_oven (Dish , top)
pastry(Dish) is_true & size(Dish,small) is_true,placei).
rule (place_in_oven (Dish ,middle)
pastry(Dish) is_true & size(Dish,big) is_true,place2).
rule (place_in_oven(Dish,middle) ,main_meal(Dish) is_true,place3).
rule(place_in_oven(Dish,bottom) ,slow_cooker(Dish) is_true ,place4)
rule(pastry(Dish) ,type(Dish, cake) is_true,pastryi).
rule (pastry(Dish) ,type(Dish,bread) is_true,pastry2).
rule(main_meal(Dish) ,type(Dish,meat) is_true,main_meal).
rule(slow_cooker(Dish) ,type(Dish,milk_pudding) is_true,slow_cooker).
fact (type (dishl ,bread)).
fact(size(dishi,big)).
Program 17.17 Oven placement rule-based system
"A dish should be placed on the top rack of the oven if it is a pastry and its size is small."
Why use a separate rule language when the syntax is so close to Prolog? The first rule, place 1, could be written as follows.
place_in_oven(Dish,top) - pastry(Dish), size(Dish,sxnall).
There are two main reasons for the rule language. The first is pedagog- ical. The rule interpreter is neater, avoiding complicated details associ- ated with Prolog's impurities such as the behavior of builtin predicates when called by clause. Avoiding Prolog's impurities also makes it easier to partially evaluate the interpreter, as described in Chapter 18.
343 Interpreters
monitor(Goal) - Succeeds if a result of yes is returned from solving Goal at the solve level, or when the end of the computation is reached.
monitor(Goal) - solve(Goal,Result) , filter(Result)
monitor (Goal)
fïlter(yes) 0/,
filter(no) .- fail.
solve(Goal,Result) - Given a set of rules of the form rule(A,B,Name), Goal has Result yes if it follows from the rules and no if it does not.
solve(A,yes) fact(A).
solve(A,Result) '- rule(A,B,Naxne), solve_body(B,Result).
solve (A na)
solve_body(A&B,Result)
solve(A,ResultA), solve_and(ItesultA,B,Result)
solve_body(A ïs_true,Result) - salve(A,Result).
solve_and(no,A,no).
solve_and(yes,B,Result) solve(B,Result).
Program 17.18 A skeleton two-level rule interpreter
The second reason is to show by example that the best way to develop a rule-based application in Prolog is to design a rule language on top of Prolog. Although the rule language is largely syntactic sugar, experience has shown that users of a rule-based system are happier working in a customized rule language than in Prolog. Rule languages are straightfor- ward to provide on top of Prolog.
We now start our presentation of the explanation shell. According to the method of stepwise enhancement, the skeleton constituting the basic control flow of the final program is presented first. Program 17.18 con- tains the skeleton of the rule ìnterpreter. The principal requirement that shaped the skeleton is the desire to handle both successful and failed computations in one interpreter.
The rule interpreter presented in Program 17.18 has two levels. The top level, or monitor level, consists of the predicates monitor and filter. The bottom level, or solve level, consists of the predicates solve, solve_ body, and solve_and. Two levels are needed to correctly handle failed computations.
344 Chapter 17
Let us consider the bottom level first. The three predicates consti- tute an interpreter at the same level of granularity as the vanilla meta- interpreter. There is one major difference. There is a result variable that says whether a goal succeeds or fails. A goal that succeeds, with the re- sult variable indicating failure, instead of failing gives rise to a different control flow, compensated for by the top level.
The predicate solve(Goal,Result) solves a single goal. There are three cases. The result is yes if the goal is a fact in the rule base. The result is no if no fact or head of a rule matches the goal. If there is a rule that matches the goal, the result will be returned by the predicate solve_body(Goal,Result). The order of the third clause is significant because we only want to report no for an individual goal if there is no suitable fact or rule. Effectively, solve succeeds for each branch of the search tree, the result being yes for successful branches and no for failed branches.
solve_body/2 has two clauses handling conjunctive goals and goals of the form A is_true. The functor is_true is a wrapper that allows uni- fication to distinguish between the two cases. A Prolog implementation with indexing would produce efficient code. The clause handling con- junctions calls a predicate solve_and/3, which uses the result of solving the first conjunct to decide whether to continue. The code for solve_ and results in behavior similar to the behavior of solve_conjunction in Program 17.11.
The monitor level is essentially a generate-and-test program. The solve level generates a branch of the search tree, and the test procedure fil- ter accepts successful branches of the search tree, indicated by the re- sult being yes. Failed branches, i.e., ones with result no, are rejected. Note that the second clause for filter could simply be omitted. We leave it in the program, albeit commented out, to make clear the later enhancement step for adding a proof tree.
The first enhancement of the rule interpreter makes it interactive. The interactive interpreter is given as Program 17.19. The user is given the opportunity to supply information at runtime for designated predicates. The designated predicates are given as a table of askable facts. For example, a fact askable(type(Dish,Type)). appearing in the table would indicate that the user could ask the type of the dish.
Interaction with the user is achieved by adding a new clause to the solve level:
345 Interpreters
solve(Goal,Result) - Given a set of rules of the form rule(A,B,Narne), Goal has Result yes if it follows from the rules and no if it does not. The user is prompted for missing information.
solve(A,yes) fact(A).
solve(A,Result) - rule(A,B,Name), solve_body(B,Result).
solve(A,Result) - askable(A), solve_askable(A,Result).
solve (Ano)
solve_body(A&B,Result)
solve_body(A,ResultA) , solve_and(ResultA,B,Result)
solve_body(A is_true,Result) solve(A,Result).
solve_and(no,A,no)
solve_and(yes,B,Result) - solve(B,Result)
solve_askable(A,Result) -
not known(A), ask(A,Response), respond(Response,A,Result).
The following predicates facilitate interaction with the user. ask(A,Response) - display_query(A) read(Response).
respond(yes,A,yes) - assert(known_to_be_true(A)).
respond(no,A,no) - assert(known_to_be_false(A)).
known(A) known_to_be_true(A).
known(A) known.to_be_false(A).
display_query(A) - write(A), write('? ').
Program 17.19 An interactive rule interpreter
solve(A,Result) - askable(A), solve_askable(A,Result).
An alternative method of making the rule interpreter interactive is to define a new class of goals in the body. An additional solve_body clause could be added, for example,
solve_body(A is_askable,Result) - solve_askable(A,Result).
We prefer adding a solve clause and having a table of askable facts to embedding in the rules the information about whether a predicate is askable. The rules become more uniform. Furthermore, the askable information is explicit meta-knowledge, which can be manipulated as needed.
To complete the interactive component of the rule interpreter, code for solve_askable needs to be specified. The essential components are
346 Chapter 17
displaying a query and accepting a response. Experience with users of rule-based systems shows that it is essential not to ask the same ques- tion twice. Users get very irritated telling the computer information they feel it should know. Thus answers to queries are recorded using assert. Program 17.19 contains appropriate code. Only the solve level is given. The monitor level would be identical to Program 17.18.
Program 17.19 queries the user. The interaction can be extended to allow the user also to query the program. The user may want to know why a particular question is being asked. A facility for giving a why ex- planation is common in rule-based systems, the answer being the rule containing the queried goal in its body. In order to give this why explana- tion, we need to extend the rule interpreter to carry the rules that have been used so far.
Program 17.20 is an enhancement of Program 17.18 that carries the list of rules that have been used in solving the query. All the predicates carry the rules as an extra argument. The rule list is initialized to be empty in the first monitor clause. The rule list is updated in the second solve clause when a new rule is invoked.
We now describe how the list of rules can be used to provide a why explanation. A new respond clause needs to be added to Program 17.19. The appropriate behavior is to display the rule, then prompt the user again for the answer to the query.
respond(why,A, [Rule Rules]) - display_rule(Rule),
ask(A,Answer), respond(Answer,A,Rules).
Repeated responses of why can be handled by giving the rule that invoked the current rule. The correct behavior is achieved by having the recursive respond goal use the rest of the rules. Finally, when there are no more rules to display, an appropriate response must be given. A suitable respond clause is
respond(cthy,A,[ ]) -
writeln(['No more explanation possible']), ask(A,Answer),
respond(Answer,A, E ]).
Now let us consider generating explanations of goals that have suc- ceeded or failed. The explanations will be based on the proof tree for successful goals and the search tree for failed goals. Note that a search
347 Interpreters
monitor(Goal) - Succeeds if a result of yes is returned from solving Goal at the solve level, or when the end of the computation is reached.
monitor(Goal) - solve(Goal,Result, [ 1), filter(Result). monitor (Goal)
filter(yes)
V0 filter(no) - fail.
so!ve(Goal,Result,Rules) - Given a set of rules of the form rule(A,B,Name), Goal has Result yes if it follows from the rules and no if it does not. Rules is the current list of rules that have been used.
solve(A,yes,Rules) - fact(A).
solve(A,Result,Rules) -
rule(A,B,Name), RulesB [Name Rules],
solve_body(B,Result ,RulesB).
solve(A,no,Rules).
solve_body(A&B,Result ,Rules) -
solve_body(A,ResultA,Rules),
solve_and(ResultA,B,Result ,Rules)
solve_body(A is_true,Result,Rules) - solve(A,Result,Rules)
solve_and(no,A,no,Rules)
solve_and(yes,B,Result,Rules) - solve(B,Result,Rules).
Program 17.20 A two-level rule interpreter carrying rules
tree is a sequence of branches. Each branch is either a proof tree or a fail- ure branch that is like a proof tree. Program 17.18 can be enhanced to in- corporate both cases. The enhanced program is given as Program 17.21. The solve level returns a branch of the search tree, and the monitor level keeps track of the failure branches since the last proof tree. The rela- tion between the predicate solve/3 in Program 17.21 and solve/2 in Program 17.18 is analogous to the relation between Programs 17.8 and 17.5.
Four predicates are added to the monitor level to record and remove branches of the search tree. The fact C search tree' (Proof) records the current sequence of branches of the search tree since the last suc cess. The predicate set_search_tree, called by the top-level monitor goal, initializes the sequence of branches to the empty list. Similarly,
348 Chapter 17
monitor(Goal,Proof) - Succeeds if a result of yes is returned from solving Goal at the solve level, in which case Proof is a proof tree representing the successful computation, or when the end of the computation is reached, in which case Proof is a list of failure branches since the last success.
monitor (Goal ,Proof) -
set_search_tree, solve(Goal,Result,Proof),
filter(Result ,Proof).
monitor (Goal ,Proof) -
collect_proof (P), reverse(P, E ] ,P1),
Proof = failed(Goal,P1).
filter(yes,Proof) - reset_search_tree.
filter(no ,Proof) - store_proof (Proof), fail.
solve( Goal,Result,Proof) - Given a set of rules of the form rule(A,B,Name), Goal has Result yes if it follows from the rules and no if it does not. Proof is a proof tree if the result is yes and a failure branch of the search tree if the result is no.
:- op(40,xfy,because). op(30,xfy,with).
solve(A,yes,Tree) - fact(A), Tree = fact(A).
solve(A,Result,Tree) -
rule(A,B,Name), solve_body(B,Result ,Proof),
Tree = A because B with Proof.
solve(A,no,Tree) -
not fact(A), not rule(A,B,Name), Tree = no_match(A).
solve_body(A&B,Result ,Proof) -
solve_body (A, ResultA, ProofA),
solve_and (ResultA, B ,Result, ProofB),
Proof = Proof A & ProofB.
solve_body(A is_true,Result ,Proof) - solve(A,Result ,Proof).
solve_and (no, A ,no, unsearched).
solve_and(yes,B,Result,Tree) - solve(B,Result,Tree).
Program 17.21 A two-level rule interpreter with proof trees
349 Interpreters
The following predicates use side effects to record and remove branches of the search tree. collect_proof(Proof) - retractY search tree' (Proof)).
store_proof (Proof) -
retract ('search tree'(Tree)),
assert('search tree'([ProofTreeJ))
set_search_tree assertYsearch tree'([ D). reset_search_tree
retract('search tree' (Proof)),
assert('search tree'([ 1)).
reverse(Xs,Ys) SeeProgram3.16.
Program 17.21 (Continued)
reset_search_tree initializes the search tree but first removes the cur- rent set of branches. It is invoked by filter when a successful compu- tation is detected. The predicate store_proof updates the search tree, while collect_proof removes the search tree. The failure branches are reordered in the second clause for monitor/2.
Having generated an explanation, we now consider how to print it. The proof tree is a recursive data structure that must be traversed to be explained. Traversing a recursive data structure is a straightforward exercise. Appropriate code is given in Program 17.22, and a trace of a computation given in Figure 17.4.
The explanation shell is obtained by combining the enhancements of Programs 17.19, 17.20, and 17.21. The final program is given as Pro- gram 17.23. Understanding the program is greatly facilitated by viewing it as a sum of the three components.
Exercises for Section 17.4
Add the ability to explain askable goals to the proof explainer in Program 17.22.
Add the ability to execute Prolog builtin predicates to the explana- tion shell.
Write a two-level meta-interpreter to find the maximum depth reached in any computation of a goal.
350 Chapter 17
explain(Goal) - Explains how the goal Goal was proved.
explain(Goal) - monitor(Goal ,Proof), interpret(Proof).
monitor(Goal,Proof) - See Program 17.21.
interpret (Proof A&ProofB)
interpret(ProofA), interpret(ProofB).
interpret(failed(A,Branches))
nl, writeln([A,' has failed with the following failure
branches: ']),
interpret (Branches).
interpret ([Fail I Fails]) - interpret(Fail), nl, write('NEW BRANCH'), nl,
interpret (Fails).
interpret([ J).
interpret(fact(A)) -
nl, writeln([A,' is a fact in the database.']).
interpret(A because B with Proof)
nl, writeln([A,' is proved using the rule']),
display_rule(rule(A,B)), interpret(Proof).
interpret (no_match (A))
nl, writeln([A,' has no matching fact or rule in the rule base.']).
interpret (unsearched) -
nl, writeln(['The rest of the conjunct is unsearched.']).
display_rule (rule (A, B)) -
write('IF '), write_conjunction(B), writeln(['THEN ',A ]).
write_conjunction(A&B)
write_conjunction(A), write(' AND '),
write_conjunction(B).
write_conjunction(A is_true) - write(A).
writeln(Xs) - See Program 12.1.
Program 17.22 Explaining a proof
35J Interpreters
place_in_oven(dishl ,middle) is proved using the rule IF pastry(dishl) AND size(dishl,big) THEN place_in_oven(dishl ,rniddle)
pastry(dishl) is proved using the rule IF type(dishl ,bread)
THEN pastry(dïshl)
type (dishi ,bread) is a fact in the database. size(dìshl,big) is a fact in the database.
Xmidd1e
place_in_oven(dishl ,x) has failed with the following failure branches: pltace_in_oven(dishl ,middle) is proved using the rule IF maiu_rneal(dishl)
THEN place_iu_oven(dishl,middle)
main_meal(dishl) is proved using the rule IF type(dishl,meat) THEN maïn_meal(dishl)
type(dishl,meat) has no matching fact or rule in the rule base. NEW BRANCH
place_in_oven(dïshl,lov) is proved using the rule 1F slow_cooker(djshl)
THEN place_in_oven(dishl ,low)
slow_cooker(dishl) is proved using the rule IF type(dishl ,milk_pudding)
THEN slow_cooker(dishl)
type(dishl,milk_pudding) has no matching fact or rule in the rule base.
Figure 17.4 Explaining a computation
352 Chapter 17
monitor ( Goal,Proof) - Succeeds if a result of yes is returned from solving Goal at the solve level, in which case Proof is a proof tree representing the successful computation, or when the end of the computation is reached, in which case Proof is a list of failure branches since the last success.
monitor (Goal ,Proof) - set_search_tree, solve(Goal,Result, E ] ,Proof), filter(Result,Proof).
monitor (Goal ,Proof) - collect_proof (P), reverse(P, E ] ,P1) Proof = failed(Goal,P1).
filter(yes,Proof) - reset_search_tree. filter(no,Proof) store_proof (Proof), fail. solve( Goal,Result,Rules,Proof) -
Given a set of rules of the form rule(A,B,Name), Goal has Result yes if it follows from the rules and no if it does not. Rules is the current list of rules that have been used. Proof is a proof tree if the result is yes and a failure branch of the search tree if the result is no.
:- op(40,xfy,because). : op(30,xfy,with).
solve(A,yes,Rules,Tree) - fact(A), Tree = fact(A). solve(A,Result,Rules,Tree) -
rule(A,B,Name), RulesB = [Naine IRules] solve_body(B,Result,RulesB,Proof), Tree = A because B with Proof.
solve(A,Result,Rules,Tree) - askable(A), solve_askable(A,Result,Rules), Tree = user(A).
solve(A,no,Rules,Tree) not fact(A), not rule(A,B,Narne), Tree = no_match(A).
solve_body(A&B,Result,Rules,Proof) - solve_body(A,ResultA,Rules,ProofA), solve_and (ResultA, B ,Result, Rules, ProofB), Proof = ProofA & ProofB.
solve_body(A is_true ,Result ,Rules ,Proof) - solve(A,Result,Rules,Proof).
solve_and (no, A ,no, Rules, unsearched). solve_and(yes,B,Result,Rules,Tree)
solve (B ,Result, Rules ,Tree).
Program 17.23 An explanation shell
353 Interpreters
The following predicates use side effects to record and remove branches of the search tree. collect_proof(Proof) - retract('search tree' (Proof)).
store_proof (Proof)
retract ('search tree'(Tree)),
assert('search tree' ([Proof ITree]))
set_search_tree assert('search tree'([ 1)).
reset_search_tree
retract('search tree'(Proof)) , assertYsearch tree' ([ 1))
reverse(Xs,Ys) SeeProgram3.16. The followrng predicates facilitate interaction with the user. ask(A,Response) - display_query(A), read(Response).
respond(yes,A,yes) '- assert(known_to_be_true(A))
respond(no,A,no) - assert(known_to_be_false(A)).
respond(why,A, [Rule Rules]) -
display_rule(Rule), ask(A,Answer), respond(Answer,A,Rules).
respond(why,A,[ J)
writeln( ['No more explanation possible']), ask(A,Answer)
respond(Answer,A, [ 1). known(A) - known_to_be_true(A).
known(A) known_to_be_f aise (A)
display_query(A) write(A), write('? ').
display_rule(rule(A,B)) -
writeYlF '), write_conjunction(B), nl, writeln(['THEN ' ,A]).
write_conjunction(A&B) -
write_conjunction(A), write(' AND '), write_conjunction(B).
write_conjunction(A is_true) - write(A).
writein(Xs) - See Program 12.1.
Program 1723 (Continued)
354 Chapter 17
17.5 Background
Our notation for automata follows Hoperoft and Uliman (1979). There is considerable confusion in the literature about the term meta-
interpreterwhether it differs from the term meta-level interpreter, for example. The lack of clarity extends further to the topic of meta- programming. A good discussion of meta-programming can be found in Yalçinalp (1991).
One dimension of the discussion is whether the interpreter is capable of interpreting itself. An ìnterpreter with that capability is also called meta-circular or self-applicable. An important early discussion of meta- circular interpreters can be found in Steele and Sussman (1978). That paper claims that the ability of a language to specify itself is a funda- mental criterion for language design.
The vanilla meta-interpreter is rooted in Prolog folklore. A version was in the suite of programs attached to the first Prolog interpreter developed by Colmerauer and colleagues, and was given in the early collection of Prolog programs (Coelho et al., 1980). Subsequently, meta- interpreters, and more generally meta-programs, have been written to affect the control flow of Prolog programs. References are Gallaire and Lasserre (1982), Pereira (1982), and Dincbas and Le Pape (1984). Using enhanced meta-interpreters for handling uncertainties is described by Shapiro (1983c).
There have been several papers on handling cuts in meta-interpreters. A variant of the vanilla meta-interpreter handling cuts correctly is de- scribed in Coelho et al. (1980) and attributed to Luis Pereira. One easy method to treat cuts is via ancestor cut, which is only present in a few Prologs like Waterloo Prolog on the IBM and Wisdom Prolog, described in the first edition of this book. There is a good discussion of meta- interpreters in general, and cuts in particular, in O'Keefe (1990).
Shapiro suggested that enhanced meta-interpreters should be the basis of a progranmiing environment. The argument, along with the debugging algorithms of Section 17.3, can be found in Shapiro (1983a). Shapiro's debugging work has been extended by Dershowitz and Lee (1987) and Drabent et al. (1989).
Prolog is a natural language for building rule-based systems. The basic statements are rules, and the Prolog interpreter can be viewed as a back-
355 Interpreters
ward chaining inference engine. Early advocates of Prolog for expert sys- tems were Clark and McCabe (1982), who discussed how explanation fa- cilities and uncertainty can be added to simple expert systems expressed as Prolog clauses by adding extra arguments to the predicates. Incorpo- rating interaction with the user in Prolog was proposed by Sergot (1983). An explanation facility incorporating Sergot's query_the_user was part of the APES expert system shell, described in Hammond (1984).
Using meta-interpreters as a basis for explanation facilities was pro- posed by Sterling (1984). Incorporating failure in a meta-interpreter has been discussed by several researchers, including Hammond (1984), Ster- ling and Latee (1986), and Bruffaerts and Henin (1989). The first descrip- tion of ari integrated meta-interpreter for both success and failure is in Yalçinalp and Sterling (1989). The rute interpreter given in Section 17.4 is an adaptation of the last paper. The layered approach can be used to explain cuts clearly, as in Sterling and Yalçinalp (1989), and also for uncertainty reasoning, as in Yalçinalp and Sterling (1991) and more com- pletely in Yalçinalp (1991).
18 Program Transformation
As stated in the introduction to Chapter 17, meta-prograniming, or the writing of programs that treat other programs as data, is particularly easy in Prolog. This chapter gives examples of programs that transform and manipulate Prolog programs. The first section looks at fold/unfold, the operation that underlies most applications of program transfor- mation for Prolog programs. The transformations given in Chapter 15 for using difference-lists to avoid explicit concatenation of lists can be understood as unfold operations, for example. The second section de- scribes a simple system for controlled unfolding and folding, which is especially good for removing layers of interpretation. The final section gives two examples of source-to-source transformation by code walking
18.1 Unfold/Fold Transformations
Logic programming arose from research on resolution theorem proving. The basic step in the logic programming computation model, goal re- duction, corresponds to a single resolution between a query and a pro- gram clause. Unfold/fold operations correspond to resolution between two Horn clauses. Loosely, unfolding corresponds to replacing a goal in the body of a clause by its definition, while folding corresponds to recog- nizing that goal(s) in the body of a clause are an instance of a definition. These two operations, being so similar, are often discussed together.
We demonstrate unfolding and folding with a running example in the first part of this chapter. The example is specializing the interpreter for nondeterministic pushdown automata (Program 17.3) for the particular
358 Chapter 18
pushdown automaton for recognizing palindromes (Program 17.4). In general, specializing interpreters is a good application for unfold/fold operations.
Definition Unfolding a goal B in a clause A - B1,.. .,B with respect to a clause B - C1,. . .,Cm where B and B unify with mgu O, produces a clause (A - B1,.. .,B1,C1,. . .,Cm,Bji.....B)O.
As an example of unfolding, we specialize the clause accept (Xs) - initial(Q), accept(Xs,Q, [ ]) to a particular initial state by unfold- ing the initial (Q) goal with respect to a particular initial fact. Specif- ically, unfolding with respect to the fact initial (push) produces the clause accept(Xs) - accept(Xs,push, E ]). (Note that in our running example we use the states push and pop for qQ and qi, respectively, from the NPDA of Program 17.4.)
The effect of the unfolding is to instantiate the initial state for the NPDA to push. In general, the effect of unfolding is to propagate variable bindings to the right, as in this example, and also to the left, to goals in the body of the clause and possibly also to the head.
There may be several clauses whose heads unify with a given goal in the body of a clause. We extend the definition of unfolding accordingly.
Definition Unfolding a goal B in a clause A - B1,. . .,B with respect to a procedure defining B is to unfold the goal with respect to each clause in the proce- dure whose head unifies with B. u
Unfolding the delta/5 goal in the clause accept([XIXs],Q,S) delta(Q,X,S,Q1,S1), accept(Xs,Q1,S1) withrespect to the following procedure for delta adapted from Program 17.4
delta(push,X,S,push, [XIS]). delta(push,X,S,pop, [XIS]).
delta(push,X,S,pop,S). delta(pop,X,[XIS],pop,S).
produces four clauses, one for each fact.
accept([XIXs],push,S) - accept(Xs,push, [XIS]).
accept([XIXs],push,S) - accept(Xs,pop, [XIS]).
accept([XIXs],push,S) - accept(Xs,pop,S).
accept([XIXs] ,pop, [XIS]) - accept(Xs,pop,S).
359 Program Transformation
palindrome(Xs) The string represented by the list Xs is a palindrome.
palindrome(Xs) - palindrome(Xs,push, E 1).
palindrome([XIXs],push,S) - palïndrome(Xs,push, [XIS]).
palindrome([XIXs] ,push,S) palindrome(Xs,pop, [XIS]).
palindrome([XIXs] ,push,S) - palindrome(Xs,pop,S).
paliridrome([XIXs] pop, [XIS]) - palindrome(Xs,pop,S).
palindrome([ ],pop,[ 1).
Program 18.1 A program accepting palindromes
This example shows variable bhidings being propagated both to the right, and to the head of the clause left of the goal being unfolded.
Folding is the reverse of unfolding. The occurrence of a body of a clause is replaced by its head. It is easiest to show with an example. Folding the goal accept (Xs,push, L 1) m the clause accept(Xs) - ac- cept(Xs,push, [ J) with respect to the clause palindrome(Xs,State, Stack) - accept(Xs,State,Stack) produces the clause accept(Xs)
palindrome(Xs,push,[ J). Note that if we now unfold the goal palindrome (Xs ,push, E ]) in ac-
cept(Xs) - paliridrome(Xs,push, L J) with respect to the clause just used for folding, palindrome(Xs,State,Stack) - accept(Xs,State, Stack), we arrive back at the original clause, accept (Xs) - accept (Xs, push, [ J). Ideally, fold/unfold are inverse operations.
Our example of folding used an iterative clause, i.e., one with a single goal in the body. Folding can be performed on a conjunction of goals, but there are technical difficulties arising from the scope of variables. Here we restrict ourselves to iterative clauses. The reader interested in the more general case should study the references given at the end of the chapter.
Specialization of the interpreter of Program 17.3 is completed by un- folding the f irial(L) goal in the third clause of Program 17.3, folding all occurrences of accept/3, and folding with respect to the clause palm- drome(Xs) - accept(Xs). Program 18.1 is then obtained.
Propagating bindings leftward in Prolog will not preserve correctness in general. For example, consider unfolding the goal r (X) with respect to the fact r(3) in the clause p(X) - var(X) , r(x). The resulting clause,
360 Chapter 18
p (3) - var(3), clearly always fails, in contrast with the original clause. Unfolding for Prolog can be performed correctly by not propagating bindings leftward, and replacing the unfolded goal by the unifier. For this example, the result would be p(X) - var(X), X=3. This will not be an issue in the examples we consider.
Exercise for Section 18.1
(i) Specialize the interpreter of Program 17.1 to the NDFA of Pro- gram 17.2, or any other NDFA, by unfold/fold operations.
18.2 Partial Reduction
In this section we develop a simple system for controlled unfold/fold operations according to prescribed user declarations. Systems for con- trolled unfolding are known in the logic programming literature as par- tial evaluators. This name reveals the influence of functional program- ming, where the basic computation model is evaluation. We prefer to refer to the system in terms of the computation model of logic program- ming, goal reduction. We thus, nonstandardly, say our system is doing partial reduction, and call it a partial reducer.
Considerable research on applying partial reduction has shown that partial reduction is especially useful for removing levels of interpreta- tion. The sequence of unfold/fold operations given in Section 18.1 typify what is possible. The general NPDA interpreter was specialized to a spe- cific NPDA, removing interpreter overhead. The resulting program, Pro- gram 18.1, only recognizes palindromes but does so far more efficiently than the combination of Programs 17.3 and 17.4.
Let us see how to build a system that can apply the unfold and fold operations that were needed to produce the palindrome program. The main idea is to recursively perform unfold/fold until no more "progress" can be achieved. A relation that replaces a goal by its equivalent under these operations is needed. The resulting equivalent goal is known as a residue. Let us call our basic relation preduce (Goal ,Residue), with in- tended meaning that Residue is a residue arising from partially reducing Goal by applying unfold and fold operations.
361 Program Transformation
preduce(Goal,Residue) - Partially reduce Goal to leave the residue Residue.
preduce(true,true) - L
preduce((A,B),(PA,PB)) , preduce(A,PA), preduce(B,PB).
preduce(A,B) - should_fold(A,B),
preduce(A,Residue)
should_unfold(A), ! , clause(A,B), preduce(B,Residue)
preduce(A,A).
Program 18.2 A meta-interpreter for determining a residue
Program 18.2 contains code for preduce. There are three possibilities for handling a single goal. It can be folded, unfolded, or left alone. The question immediately arises how to decide between the three possibil- ities. The easiest for a system is to rely on the user. Program 18.2 as- sumes that the user gives should_fold (Goal , FoldedGoal) declarations that say which goals should be folded and to what they should be folded, and also should_unfold(Goal) declarations that say which goals should be unfolded. Unification against the program clauses determines to what they should be unfolded. Goals not covered by either declaration are left alone. The remaining clauses in Program 18.2 handle the empty goal, true, and conjunctive goals, which are treated recursively in the obvious way.
Observe that Program 18.2 is essentially a meta-interpreter at the gran- ularity level of vanilla (Program 17.5). The meta-interpreter is enhanced to return the residue. Handling builtins is assigned to the exercises.
The query preduce((initial(Q), accept(Xs,Q,[ 1)), Residue)? assuming appropriate should_fold and should_unfold declarations (to be given shortly) has as solution Residue = (true,palindromo(Xs, push, E ])). lt would be preferable to remove the superfluous call to true. This can be done by modifying the clause handling conjunctive goals to be more careful in computing the conjunctive resolvent. A suit- able modification is
preduce((A,B) ,Res) -
!, preduce(A,PA), preduce(B,PB), combine(PA,PB,Res),
The code for combine, removing superfluous empty goals, is given in Program 18.3.
362 Chapter 18
process(Program, RedProgram) - Partially reduce each of the clauses in Program to produce RedProgram.
process(Prog,NewProg) -
findall(PC1, (member(C1 ,Prog) ,preduce(C1 ,PC1)) ,NewProg).
test (Name ,Program) -
program(Name,Clauses), process(Clauses,Prograxn).
preduce ( Goal,Residue) - Partially reduce Goal to leave the residue Residue.
preduce((A - B),(PA PB)) !, preduce(B,PB), preduce(A,PA).
preduce(true,true) -
preduce((A,B) ,Res) -
!, preduce(A,PA), preduce(B,PB), combine(PA,PB,Res).
preduce(A,B) - should_fold(A,B),
preduce(A,Residue) -
should_unf oid(A) , ! , ciause(A,B) preduce(B,Residue).
preduce(A,A).
combine(true,B,B) -
combine(A,true,A) -
combine(A,B,(A,B)).
Program 18.3 A simple partial reduction system
To extend Program 18.2 into a partial reducer, clauses must be han- dled as well as goals. We saw a need in the previous section to partially reduce the head and body of a clause. The only question is in which or- der. Typically, we will want to fold the head and unfold the body. Since unfolding propagates bindings, unfolding first will allow more specific folding. Thus our proposed rule for handling clauses is
preduce((A - B),(PA - PB)) - !, preduce(B,PB), preduce(A,PA).
This goal order is advantageous for the example of the rule interpreter to be presented later in this section.
To partially reduce a program, we need to partially reduce each of its clauses. For each clause, there may be several possibilities because of nondeterrninism. For example, the recursive accept/3 clause led to four rules because of the four possible ways of unfolding the delta goal. The
363 Program Transformation
prograin(npda,[(accept(Xsl) initial(Q1), accept(Xsl,Q1,[ 1), (accept([X2IXs2],Q2,S2) - delta(Q2,X2,S2,Q12,S12),
accept(Xs2,Q12,S12)), (accept([ ],Q3,[ J) true)]).
should_unf old C initial (Q))
should_unfold(final(Q))
should_unfold(delta(A,B,C,D,E)).
should_fold(accept(Q,Xs,Q1),palindrome(Q,Xs,Q1)).
should_fold (accept (Xs) ,palindrome (Xs)).
Program 18.4 Specializing an NPDA
cleanest way to get the whole collection of program clauses is to use the all-solutions predicate f indall. That gives
process (Prog, NewProg) -
f indall(PC1, (mernber(Cl,Prog), preduce(Cl,PC1)) ,NewProg).
Putting all the preceding actions together gives a simple system for partial reduction. The code is presented as Program 18.3. The program also contains a testing clause.
We now concentrate on how to specify should_fold and should_ unf old declarations. Consider the NPDA example for recognizing palin- dromes. The initial, final, and delta goals should all be unfolded. A declaration is needed for each. The accept/i and accept/3 goals should be folded into palindrome goals with the same argument. The declara- tion for accept/i is should_f old(accept(Xs),palindrome(Xs)). All the necessary declarations are given in Program 18.4. Program 18.4 also contains the test program as data. Note the need to make all the vari- ables in the program distinct. Applying Program 18.3 to Program 18.4 by posing the query test(npda,P)? produces Program 18.1, with the only difference being an explicit empty body for the last palindrome fact.
We now give a more complicated example of applying partial reduction to remove a level of overhead. We consider a simpler variant of the rule interpreter given in Section 17.4. The variant is at the bottom level of the layered interpreter. The interpreter, whose relation is solve(A,N), counts the number of reductions used in solving the goal A. The code for solve and related predicate solve_body is given in Program 18.5. The rules that we will consider constitute Program 17.17 for determining
364 Chapter 18
Rule interpreter for counting reductions solve(A,1) - fact(A).
solve(A,N) - rule(A,B,Name), solve_body(B,NB), N is NB+1.
solve_body (A&B , N)
solve_body(A,NA), solve_body(B,NB), N is NA+NB.
solve_body(A is_true,N) - solve(A,N).
Sample rule base
rule(oven(Dish,top) ,pastry(Dish) is_true
& size(Dish,small) is_true,placel)
rule(oven(Dish,middle) ,pastry(Dish) is_true
& size(Dish,big) is_true,place2).
rule (oven(Dish,middle) ,main_meal(Dish) is_true,place3).
rule (oven(Dish,bottom) , slow_cooker(Dish) is_true,place4)
rule(pastry(Dish),type(Dish,cake) is_true,pastryl).
rule (pastry(Dish) ,type(Dish,bread) is_true,pastry2).
rule(main_meal(Dish) ,type(Dish,meat) is_true ,main_meal).
rule(slow_cooker (Dish) ,type(Dish,milk_pudding)
is_true ,slow_cooker).
should_fold(solve(oven(D,P),N),oven(D,P,N)).
should_fold(solve(pastry(D),N),pastry(D,N)).
should_fold(solve(main_meal(D) ,N) ,main_meal(D,N)).
should_fold(solve(slow_cooker(D),N),slow_cooker(D,N)).
should_fold(solve(type(D,P),N),type(D,P,N)).
should_fold(solve(size(D,P),N),size(D,P,N)).
should_unfold(solve_body(G,N)).
should_unfold(rule(A,B,Name)).
program(rule_interpreter, [(solve(A1,l) - fact(A1)),
(solve(A2,N) - rule(A2,B,Name), solve_body(B,NB), N is NB+i)]).
Program 18.5 Specializing a rule interpreter
365 Program Transformation
where a dish should be placed in the oven. The rules are repeated in Program 18.5 for convenience.
The effect of partial reduction in this case will be to "compile" the rules into Prolog clauses where the arithmetic calculations are done. The resulting Prolog clauses can in turn be compiled, in contrast to the com- bination of interpreter plus rules. Rule place i will be transformed to
oven(Dish,top,N)
pastry(Dish,N1), size(Dish,small,N2),
N3 is N1+N2, N is N3+1.
The idea is to unfold the calls to rule so that each rule can be handled, and also to unfold the component of the interpreter that handles syn- tactic structure, specifically solvebody. What gets folded are the indi- vidual calls to solve, such as solve (oven (D,P) ,N), which gets replaced by a predicate oven (D , P, N) The necessary declarations are given in Pro- gram 18.5. Program 18.3 applied to Program 18.5 produces the desired effect.
Specifying what goals should be folded and unfolded is in general straightforward in cases similar to what we have shown. Nevertheless, making such declarations is a burden on the programmer. In many cases, the declarations can be derived automatically. Discussing how is beyond the scope of the chapter.
How useful partial reduction is for general Prolog programs is an open issue. As indicated, care must be taken when handling Prolog's impuri- ties not to change the meaning of the program. Further, interaction with Prolog implementations can actually mean that programs that have been partially reduced can perform worse than the original program. lt will be interesting to see how much partial reduction will be applied for Prolog compilation.
Exercises for Section 182
Extend Program 18.3 to handle builtins.
Apply Program 18.3 to the two-level rule interpreter with rules given as Program 17.20.
366 Chapter 18
18.3 Code Walking
The examples of meta-prograrnniing given so far in Chapters 17 and 18 are dynamic in the sense that they "execute" Prolog programs by per- forming reductions. Prolog is also a useful language for writing static meta-programs that perform syntactic transformations of Prolog pro- grams. In this section, we give two nontrivial examples in which pro- grams are explicitly manipulated syntactically.
The first example of explicit program manipulation is program com- position. In Section 13.3, stepwise enhancement for systematic construc- tion of Prolog programs was introduced. The third and final step in the method is composition of separate enhancements of a common skele- ton. We now present a program to achieve composition that is capable of composing Programs 13.1 and 13.2 to produce Program 13.3.
The running example we use to illustrate the program is a variant of the example in Chapter 13. The skeleton is the same, namely,
skel([XIXs],Ys) - member(X,Ys), skel(Xs,Ys). skel([XIXs],Ys) - nonmember(X,Ys), skel(Xs,Ys). skel([ ],Ys). The union program, Program 13.1, is also the same, namely,
union([XIXs],Ys,Us) - member(X,Ys), union(Xs,Ys,Us). union([XIXs] ,Ys, [XIUs]) - nonmember(X,Ys), union(Xs,Ys,Us). union([ ],Ys,Ys). The second program to be composed is different and represents when added goals are present. The relation to be used is common(Xs,Ys,N), which counts the number of common elements N in two lists Xs and Ys. The code is
coinmon([XIXs],Ys,N) - meinber(X,Ys), conimon(Xs,Ys,M), N is M+1.
coinmon([XIXs],Ys,N) - nonmember(X,Ys), coinmon(Xs,Ys,N). comnion([ J,Ys,O).
The program for composition makes some key assumptions that can be justified by theory underlying stepwise enhancement. Describing the theory is beyond the scope of this book. The most important assump- tion is that there is a one-to-one correspondence between the clauses of
367 Program Transformation
the two programs being composed, and one-to-one correspondences be- tween the clauses of each of the programs and the common skeleton.
Programs are represented as lists of clauses. The first clause in the first program corresponds to the first clause in the second program and to the first clause in the skeleton. Our assumption implies that the lists of clauses of programs being composed have the same length. The three programs have been written with corresponding clauses in the same or- der. (That the lists of clauses do have the same length is not checked explicitly.)
In order to perform composition, a composition specification is needed. It states how the arguments of the final program relate to the two extensions. The relation that we will assume is composition_ specification(Programl,Program2,Skeleton,FinalProgram). An ex-
ample of the specification for our running example is composition_ specification(union(Xs,Ys,Us), conunon(Xs,Ys,N), skel(Xs,Ys),
uc(Xs,Ys,Us,N). The composition specification is given as part of Pro- gram 18.6.
The program for composition is given as Program 18.6. The top-level relation is compose/4, which composes the first two programs assumed to be enhancements of the third argument to produce the composite program, which is the fourth argument.
The program proceeds clause by clause in the top ioop of Pro- gram 18.6, where compose_clause/4 does the clause composition. The arguments correspond exactly to the arguments for compose. To com- pose two clauses, we have to compose the heads and the bodies. Com- position of the heads of clauses happens through unification with the composition specification. The predicate compose_bodies/4 is used to compose the bodies. Note that the order of arguments has been changed so that we systematically traverse the skeleton. Each goal in the skeleton must be represented in each of the enhancements so that it can be used as a reference to align the goals in each of the enhancements.
The essence of compose_bodies is to traverse the body of the skeleton goal by goal and construct the appropriate output goal as we proceed. In order to produce tidy output and avoid superfluous empty goals, a difference-structure is used to build the output body. The first clause for compose_bodies covers the case when the body of the skeleton is nonempty. The predicates first and rest, which access the body of the skeleton, are a good example of data abstraction.
368 Chapter 18
compose (Program 1 ,Program2,Skeleton,FinalProgram) - FinaiProgram is the result of composing Program i and Program2, which are both enhancements of Skeleton.
compose([CllIClsl] , [Cl2ICis2] , [CiSkellCisSkei] ,[Cl Cisl) - compose_clause(Cll ,C12 ,ClSkel ,Cl),
compose (Cisl ,Cls2,CisSkei,Cls)
compose([ ]j ],[ ],[ 1). compose_clause((AlBl) , (A2B2) , (ASkelBSkel) ,(AB)) -
composition_specification(Al ,A2 ,ASkel ,A)
compose_bodies(BSkei,Bl ,B2,B\true).
compose_bodíes(SkeiBody,Bodyl ,Body2,B\BRest) -
first (SkelBody,G),
align(G,Bodyl Cl ,RestBodyl ,B\Bl),
align(G,Body2,G2,RestBody2,Bl\(Goal,B2)),
compose_goal(Gl ,G2,Goai),
rest(SkeiBody,Gs)
compose_bodies(Gs,RestBodyl ,RestBody2,B2\BRest).
compose_bodies (true, Bodyl , Body2 , B\BRest) -
rest_goals (Bodyl,B\Bl), rest_goals(Body2,Bl\BRest).
align(Goal,Body,G,RestBody,B\B) -
first(Body,G), correspond(G,Goai), , rest(Body,RestBody).
align(Goal,(G,Body),CorrespondingG,RestBody,(G,B)\Bl) -
align (Goal ,Body, CorrespondingG , RestBody , B\Bl).
first((G,Gs) ,G)
first(G,G) - G (A,B), G true.
rest((G,Gs) ,Gs)
rest(G,true) - G (A,B).
correspond (G , G)
correspond(G,B) map(G,B).
compose_goai(G,G,G) - !
compose_goai(Al ,A2,A) -
composition_specification(Al,A2,ASkei,A).
rest_goals(true,B\B) - !
rest_goals(Body, (G,B)\BRest) -
first(Body,G), , rest(Body,Bodyl), rest_goais(Bodyl,B\BRest).
Program 18.6 Composing two enhancements of a skeleton
369 Program Transformation
An important assumption made by Program 18.6 concerns finding the goals in the bodies of the program that correspond to the goals in the skeleton. The assumption made, embedded in the predicate correspond, is that a mapping will be given from goals in the enhancement to goals in the skeleton. In our running example, the predicates member and non- member map onto themselves, while both union and common map onto skel. This information, provided by the predicate map/2, is needed to correctly align goals from the skeleton with goals of the program being composed. The code for align as presented allows for additional goals to be present between goals in the skeleton. The only extra goal in our running example is the arithmetic calculation in common, which is after the goals corresponding to the skeleton goals.
The second clause for compose_bodies covers the case when the body is empty, either from dealing with a fact or because the skeleton has been traversed. In this case, any additional goals need to be included in the result. This is the function of rest_goals.
Program 18.7 contains a testing clause for Program 18.6, along with the specific data for our running example. As with Program 18.4, variables in the programs being composed must be named differently. Automatic generation of composition specifications for more complicated examples is possible.
The second example of explicit manipulation of programs is the con- version of context-free grammar rules to equivalent Prolog clauses. Context-free grammars are defined over a language of symbols, divided into nonterminal symbols and terminal symbols. A context-free grammar is a set of rules of the form
(head) - (body)
where head is a nonterminal symbol and body is a sequence of one or more items separated by commas. Each item can be a terminal or nonterminal symbol. Associated with each grammar is a starting symbol and a language that is the set of sequences of terminal symbols obtained by repeated (nondeterministic) application of the grammar rules starting from the starting symbol. For compatibility with Chapter 19, nonterminal symbols are denoted as Prolog atoms, terminal symbols are enclosed within lists, and II denotes the empty operation.
The language a(bc)* can be defined by the following context-free gram- mar consisting of four rules:
370 Chapter 18
test_compose (X, Prog) - programl(X,Progl), program2(X,Prog2), skeleton(X,Skeleton), compose(Progl ,Prog2 ,Skeleton,Prog).
programi (test, (union([X1IXs1] ,Ysl,Zsl) -
member(X1,Ysi), union(Xsl,Ysi,Zsl)), (union([X2IXs2] ,Ys2, [X2IZs2]) -
nonmember(X2,Ys2), union(Xs2,Ys2,Zs2)), (union([ ],Ys3,Ys3) - true)]).
program2(test, E (common([X1IXs1] ,Ysl,Ni)
member(Xi,Ysl), comxnon(Xsl,Ysl,M1), Nl is M1+l), (common([X2IXs2] ,Ys2,N2) -
nonmember(X2,Ys2), common(Xs2,Ys2,N2)), (cominon([ ],Ys3,O) - true)]).
skeleton(test, I (skel([X1IXs1],Ysl) - member(Xl,Ysl), skel(Xsl,Ysl)), (skel([X2IXs2] ,Ys2) - nonmember(X2,Ys2), skel(Xs2,Ys2)), (skel([ ],Ys3) - true)]).
composition_specification(union(Xs,Ys,Us), common(Xs,Ys,N) skel(Xs,Ys) ,uc(Xs,Ys,Us,N))
rnap(union(Xs,Ys,Zs), skel(Xs,Ys)) map(common(Xs,Ys,N), skel(Xs,Ys)).
Program 18.7 Testing program composition
s - [a], b.
b - [b], c.
b - [ ]. c - [c], b.
Another example of a context-free grammar is given in Figure 18.1. This grammar recognizes the language a*b*c*.
A context-free grammar can be immediately written as a Prolog pro- gram. Each nonterminal symbol becomes a unary predicate whose argu- ment is the sentence or phrase it identifies. The naive choice for repre- senting each phrase is as a list of terminal symbols. The first grammar rule in Figure 18.1 becomes
s(Xs) - a(As), b(Bs),
c(Cs), append(Bs,Cs,BsCs), append(As,BsCs,Xs).
371 Program Transformation
s -' a, b, c. a - [a], a. a -. [ ]. b [b], b. b [J. c -. [c], c. C £ J.
Figure 18.1 A context-free grammar for the language a*b*c*
s(As\Xs) '- a(As\Bs), b(Bs\Cs), c(Cs\Xs). a(Xs\Ys) - connect([a] ,Xs\Xsl), a(Xsl\Ys). a(Xs\Ys) - connect([ ] ,Xs\Ys) b(Xs\Ys) '- connect([b] ,Xs\Xsl), b(Xsl\Ys). b(Xs\Ys) - connect([ ],Xs\Ys). c(Xs\Ys) - cormect([c],Xs\Xsl), c(Xsl\Ys). c(Xs\Ys) - cormect([ ],Xs\Ys). connect([ ] ,Xs\Xs). connect([WIWs] , [WIXs]\Ys) connect(Ws,Xs\Ys).
Program 18.8 A Prolog program parsing the language a*b*c*
Completing the grammar of Figure 18.1 in the style of the previous rule leads to a correct program for parsing, albeit an inefficient one. The calls to append suggest, correctly, that a difference-list might be a more appropriate structure for representing the sequence of terminals in the context of parsing. Program 18.8 is a translation of Figure 18.1 to a Prolog program where difference-lists represent the phrases. The basic relation scheme is s(Xs), which is true if Xs is a sequence of symbols accepted by the grammar.
The predicate connect (Xs,Ws) is true if the list Xs represents the same sequence of elements as Ws. The predicate is used to make explicit the translation of terminal symbols to Prolog programs.
As a parsing program, Program 18.8 is a top-down, left-to-right re- cursive parser that backtracks when it needs an alternative solution. Al- though easy to construct, backtracking parsers are in general inefficient. However, the efficiency of the underlying Prolog implementation in gen- eral more than compensates.
372 Chapter 18
translate( Grammar,Program) - Program is the Prolog equivalent of the context-free grammar Grammar.
translate ([Rule Rules] , [Clause I Clauses]) -
translate_rule(Rule ,Clause),
translate (Rules , Clauses)
translate([ ],[ J). translate_rule ( GrammarRule,PrologClause) -
Prolog Clause is the Prolog equivalent of the grammar rule GrammarRule.
translate_rule((Lhs - Rhs),(Head - Body)) -
translate_head(Lhs,Head,Xs\Ys),
translate_body (Rhs ,Body, Xs\Ys).
translate_head(A,Al ,Xs) -
translate_goal(A,Al ,Xs).
translate_body(A,B) , (Al ,B1) ,Xs\Ys) -
translate_body(A,Al,Xs\Xsl), translate_body(B,Bl,Xsl\Ys).
translate_body (A,Al ,Xs)
translate_goal(A,Al ,Xs).
translate_goal (A,Al ,DList) -
nonterminal(A), functor(Al,A,l), arg(l,Al,DList).
translate_goal(Terms,coirnect(Terms,S) ,S)
terminals (Terms).
nonterminal(A) atom(A).
terminals(Xs) - list(Xs).
list(Xs) - seeProgram3.11.
Program 18.9 Translating grammar rules to Prolog clauses
We now present Program 18.9, which translates Figure 18.1 to Pro- gram 18.8. As for Program 18.6, the translation proceeds clause by clause. There is a one-to-one correspondence between grammar rules and Prolog clauses. The basic relation is translate(Rules,Clauses). Individual clauses are translated by translate.rule/2. To translate a rule, both the head and body must be translated, with the appropriate correspondence of difference-lists, which will be added as additional arguments.
Adding an argument is handled by the predicate translate_goal. If the goal to be translated is a nonterminal symbol, a unary predicate with
373 Program Transformation
the same functor is created. If the goal is a list of terminal symbols, the appropriate connect goal is created. When executed, the connect goal connects the two difference-lists. Code for connect is in Program 18.8.
Program 18.9 can be extended for automatic translation of definite clause grammar rules. Definite clause grammars are the subject of Chap- ter 19. Most versions of Edinburgh Prolog provide such a translator.
Exercise for Section 18.3
(i) Apply Program 18.6 to one of the exercises posed at the end of Section 13.3.
18.4 Background
Often research in logic programming has followed in the steps of related research in functional programming. This is true for unfold/fold and par- tial evaluation. Burstall and Darlington (1977) wrote the seminal paper on unfold/fold in the functional programming literature. Their work was adapted for logic programming by Tamaki and Sato (1984).
The term partial evaluation may have been used first in a paper by Lombardi and Raphael (1964), where a simple partial evaluator for Lisp was described. A seminal paper introducing partial evaluation to com- puter science is due to Futamura in 1971, who noted the possibility of compiling away levels of interpretation. Komorowski described the first partial evaluator for pure Prolog in his thesis in 1981. He has since preferred the term partial deduction. Gallagher in 1983 was the first to advocate using partial evaluation in Prolog for removing interpretation overhead (Gallagher, 1986). Venken (1984) was the first to list some of the problems of extending partial evaluation to full Prolog. The paper that sparked the most interest in partial evaluation in Prolog is due to Takeuchi and Furukawa (1986). They discussed using partial evalua- tion for removing runtime overhead and showed an order of magnitude speedup. Sterling and Beer (1989) particularize the work for expert sys- tems. Their paper introduces the issue of pushing down meta-arguments, which is subsumed in this chapter by should_fold declarations. Specific Prolog partial evaluation systems to read for more details are ProMiX
374 Chapter 18
(Lakhotia and Sterling, 1990) and Mixtus (Sahlin, 1991). An interesting application of partial evaluation is given by Smith (1991), where efficient string-matching programs were developed.
Composition was first discussed in the context of Prolog meta-inter- preters in Sterling and Beer (1989) and an informal algorithm was given in Sterling and Lakhotia (1988). A theory is found in Kirschenbaum, Ster- ling, and Jam (1993).
19 Logic Grammars
A very important application area of Prolog is parsing. In fact, Prolog originated from attempts to use logic to express grammar rules and to formalize the process of parsing. In this chapter, we present the most common logic grammar formalism, definite clause grammars. We show how grammar rules can be considered as a language on top of Prolog, and we apply granirnar rules to parse simple English sentences. In Chap- ter 24, definite clause grammars are used as the parsing component of a simple compiler for a Pascal-like language.
19.1 Definite Clause Grammars
Definite clause grammars arise from adding features of Prolog to context-free grammars. In Section 18.3, we briefly sketched how context- free grammars could be immediately converted to Prolog programs, which parsed the language specified by the context-free grammar. By adding the ability of Prolog to exploit the power of unification and the ability to call builtin predicates, a very powerful parsing formalism is indeed achieved, as we now show.
Consider the context-free grammar for recognizing the language a*b*c*, presented in Figure 18.1, with equivalent Prolog program Pro- gram 18.8. The Prolog program can be easily enhanced to count the number of symbols that appear in any recognized sequence of a's, b's, and c's. An argument would be added to each predicate constituting the number of symbols found. Arithmetic would be performed to add numbers together. The first clause would become
376 Chapter 19
s(As\Xs,N) - a(As\Bs,NA), b(Bs\Cs,NB), c(Cs\Xs,NC), N is NA+NB+NC.
The extra argument counting the number of a's, b's, and c's can be added to the grammar rule just as easily, yielding
s(N) - a(NA), b(NB), c(NC), N is NA+NB+NC.
Adding arguments to nonterminal symbols of context-free grammars, and the ability to call (arbitrary) Prolog predicates, increases their util- ity and expressive power. Grammars in this new class are called definite clause grammars, or DCGs. Definite clause grammars are a generaliza- tion of context-free grammars that are executable, augmented by the language features of Prolog.
Program 18.9, translating context-free grammars into Prolog programs, can be extended to translate DCGs into Prolog. The extension is posed as Exercise (i) at the end of this section. Throughout this chapter we write DCGs in grammar rule notation, being aware that they can be viewed as Prolog programs. Many Edinburgh Prolog implementations provide support for grammar rules. The operator used for - is -->. Grammar rules are expanded automatically into Prolog clauses with two extra ar- guments added as the last two arguments of the predicate to represent as a difference-list the sequence of tokens or words recognized by the predicate. Braces are used to delimit goals to be called by Prolog di- rectly, which should not have extra arguments added during translation. Grammar rules are not part of Standard Prolog but will probably be in- corporated in the future.
Program 19.1 gives a DCG that recognizes the language a*b*c* and also counts the number of letters in the recognized sequence. The en- hancement from Figure 18.1 is immediate. To query Program 19.1, con- sideration must be taken of the two extra arguments that will be added. Forexample,asuitablequeryiss(N,[a,a,b,b,b,c],1 ])?.
Counting the symbols could, of course, be accomplished by traversing the difference-list of words. However, counting is a simple enhancement to understand, which effectively displays the essence of definite clause grammars. Section 19.3 presents a wider variety of enhancements.
Our next example is a striking one of the increase in expressive power possible using extra arguments and unification. Consider recognizing the language a"b"c", which is not possible with a context-free grammar
377 Logic Grammars
s(N) a(NA), b(NB), c(NC), (N is NA+NB+NC}.
a(N) - [a], a(N1), (N is Ni+i}. a(0) [1. b(N) [b], b(N1), (N is N1+1}. b(0) [ ]. c(N) -' [c] , c(N1) , (N is N1+1} c(0) E].
Program 19.1 Enhancing the language a*b*c*
s - a(N), b(N), c(N). a(N) [a], a(N1), (N is N1+1}. a(0) - [ ]. b(N) -. [b] , b(N1) , (N is N11-1} b(0) - [ ]. c(N) -. [c] , c(N1) , {N is N1+1} c(0) -. E].
Program 19.2 Recognizing the language aN1cN
However, there is a straightforward modification to the grammar given as Program 191. All that is necessary is to change the first rule and make the number of a's, b's, and c's the same. The modified program is given as Program 19.2.
In Program 19.2, unification has added context sensitivity and in- creased the expressive power of DCGs over context-free grammars. DCGs should be regarded as Prolog programs. Indeed, parsing with DCGs is a perfect illustration of Prolog programming using nondeterrninistic pro- gramming and difference-lists. The top-down, left-to-right computation model of Prolog yields a top-down, left-to-right parser.
Definite clause grammars can be used to express general programs. For example, a version of Program 3.15 for append with its last two arguments swapped can be written as follows.
append([ 1) - E]. append([XIXs]) -. lix], append(Xs).
Using DCGs for tasks other than parsing is an acquired programming taste.
378 Chapter 19
The grammar for the declarative part of a Pascal program.
declarative_part -
const_declaration, type_declaration,
var_declaration, procedure_declaration.
Constant declarations
const_declaration - E J const_declaration
[const] , const_definition, E;] , const_definitions.
const_definitions E J
const_def initions
const_definition, E;] , const_definitions.
const_definition - identifier, [=1, constant.
identifier - [X] , {atom(X) }.
constant - [X], {constant(X)}.
Type declarations
type_declaration -. E ]. type_declaration
[type] , type_definition, [;] , type_definitions.
type_definitions E I type_definitions - type_definition, E;] , type_definitions.
type_definition identifier, [=1 , type.
type -. ['INTEGER']. type ['REAL'].
type ['BOOLEAN']. type ['CHAR'].
Variable declarations
var_declaration - [ I. var_declaration
[var] , var_definition, E;] , var_definitions.
var_definitions -' E J
var_definitions - var_definition, E;] , var_definitions. var_definition - identifiers, E:], type.
identifiers identifier.
identifiers - identifier, E,] , identifiers.
Program 19.3 Parsing the declarative part of a Pascal block
379 Logic Grammars
Procedure declarations procedure_declaration -. E procedure_declaration -. procedure_heading, E;] , block.
procedure_heading
[procedure] , identifier, formal_parameter_part.
formal_parameter_part E ]. formal_parameter_part [C] , f ormal_parameter_section, E)]
formal_parameter_section - formal_parameters.
formal_parameter_ection -.
formal_parameters, E;] , formal_parameter_section.
formal_parameters value_parameters.
formal_parameters -. variable_parameters.
value_parameters var_definition.
variable_parameters [var], var_definition.
Program 19.3 (Continued)
We conclude this section with a more substantial example. A DCG is given for parsing the declarative part of a block in a Pascal program. The code does not in fact cover all of Pascal - it is not complete in its defi- mtion of types or constants, for example. Extensions to the grammar are posed in the exercises at the end of this section. Parsing the statement part of a Pascal program is illustrated in Chapter 24.
The grammar for the declarative part of a Pascal block is given as Pro- gram 19.3. Each grammar rule corresponds closely to the syntax diagram for the corresponding Pascal statement. For example, the syntax diagram for constant declarations is as follows:
---> const -----> Constant Definition > >
+ < +
The second grammar rule for const_declaration in Program 19.3 says exactly the same. A constant declaration is the reserved word const followed by a constant definition, handled by the nontermrnal symbol const_definition; followed by a semicolon; followed by the rest of the constant definition, handled by the nonterminal symbol const_def initions. The first rule for const_declaratìon effectively states that the constant declaration is optional. A constant definition is
380 Chapter 19
an identifier followed by =, followed by a constant. The definition for const_def initions is recursive, being either empty or another constant definition; followed by a semicolon; followed by the rest of the constant definition.
The remainder of Program 19.3 is sirmiarly easy to understand. It clearly shows the style of writing grammars in Prolog.
Exercises for Section 19.1
Extend Program 18.9 so that it translates definite clause grammars to Prolog as well as context-free grammars.
Add to Program 19.3 the ability to correctly handle label declara- tions and function declarations.
Enhance Program 19.3 to return the list of variables declared in the declarative part.
Write a program to parse the language of your choice in the style of Program 19.3.
19.2 A Grammar Interpreter
Grammar rules are viewed in the previous section as syntactic sugar for Prolog clauses. This view is supported by Prolog systems with automatic grammar rule translation. There is a second way of viewing grammar rules, namely as a rule language.
This section takes the second view and considers grammar rules as an embedded language on top of Prolog. We consider applying the in- terpreter techniques of Chapter 17 to grammar rules.
Program 19.4 is an interpreter for grammar rules. The basic relation is parse (Symbol,Tokens), which is true ifa sequence of grammar rules can be applied to Symbol to reach Tokens. The tokens are represented as a difference-list.
The granularity of the DCG interpreter is at the clause reduction level, the same as for the vanilla meta-interpreter, Program 17.5, and the expert system rule interpreter, Program 17.18. Indeed, the code ni Program 19.4 is similar to those interpreters. There are four cases, handled by the
381 Logic Grammars
parse(Start,Tokens) The sequence of tokens Tokens represented as a difference-list can be reached by applying the grammar rules defined by /2, starting from Start.
parse(A,Tokens) - nonterminal(A), A -. B, parse(B,Tokens).
parse((A,B) ,Tokens\Xs) parse(A,Tokens\Tokensl), parse(B,Tokensl\Xs).
parse(A,Tokens) terminais(A), cormect(A,Tokens). parse({A},Xs\Xs) - A. terminals(Xs) See Program 18.9. connect(Xs,Tokens) - See Program 18.8.
Program 19.4 A definite clause grammar (DCG) interpreter
four clauses for parse in Program 19.4. The first rule handles the basic operation of reducing a nonterminal symbol, and the second rule handles conjunctions of symbols. The third rule handles terminal symbols, and the fourth rule covers the ability to handle Prolog predicates by calling them directly using the meta-variable facility.
Observe that the last argument in parse/2, the DCG interpreter, is a difference-list. This difference-list can be handled implicitly using gram- mar rule notation. In other words, Program 19.4 could itself be written as a DCG. This task is posed as Exercise 19.2(i).
Recall that the interpreters of Chapter 17 were enhanced. Similarly, the DCG interpreter, Program 19.4, can be enhanced. Program 19.5 gives a simple enhancement that counts the number of tokens used in pars- ing. As mentioned before, this particular enhancement could be accom- plished directly, but it illustrates how an interpreter can be enhanced.
Comparing Programs 19.1 and 19.5 raises an important issue. Is it better to enhance a grammar by modifying the rules, as in Program 19.1, or to add the extra functionality at the level of the interpreter? The second approach is more modular, but suffers from a lack of efficiency,
Exercises for Section 19.2
(i) Write Program 19.4 as a DCG.
382 Chapter 19
parse(Start, Tokens,N) The sequence of tokens Tokens, represented as a difference-list, can be reached by applying the grammar rules defined by -./2, starting from Start, and N tokens are found.
parse(A,Tokens,N) -
nonterminal(A), A -. B, parse(B,Tokens,N).
parse((A,B) ,Tokens\Xs,N)
parse(A,Tokens\Tokensl ,NA), parse(B,Tokensl\Xs ,NB),
N is NA+NB.
parse(A,Tokens,N) -
terminais(A), connect(A,Tokens), length(A,N).
parse({A},Xs\Xs,O) - A.
terninals(Xs) - See Program 18.9.
connect(A,Tokens) - See Program 18.8.
length(Xs,N) - SeeProgram8.11.
Program 19.5 A DCG interpreter that counts words
Use the partial reducer, Program 18.3, to specialize the interpreter of Program 19.4 to a particular grammar For example, Figure 18.1 should be transformed to Program 19.1.
Enhance Program 19.4 to build a parse tree.
19.3 Application to Natural Language Understanding
An important application area of logic programming has been under- standing natural languages. Indeed, the origins of Prolog lie within this application. In this section, it is shown how Prolog, through definite clause grammars, can be applied to natural language processing.
A simple context-free grammar for a small subset of English is given in Program 19.6. The nonterminal symbols are grammatical categories, parts of speech and phrases, and the terminal symbols are English words that can be thought of as the vocabulary. The first rule in Program 19.6 says that a sentence is a noun phrase followed by a verb phrase. The last rule says that surprise is a noun. A sample sentence recognized by the grammar is: "The decorated pieplate contains a surprise."
383 Logic Grammars
Grammar Rules
sentence - noun_phrase, verb_phrase.
noun_phrase determiner, noun_phrase2.
noun_phrase - noun_phrase2.
noun_phrase2 adjective, noun_phrase2.
noun_phrase2 -, noun.
verb_phrase -. verb.
verb_phrase -' verb, noun_phrase.
Vocabulary
determiner [the] . adjective [decorated]
determiner - [a]
noun - [pieplate] . verb -' [contains]
noun -' [surprise]
Program 19.6 A DCG context-free grammar
Using the terminology of stepwise enhancement introduced in Chap- ter 13, we can view a grammar as a skeleton. We proceed to show how useful grammatical features can be added by enhancement. The next two programs are enhancements of Program 19.6. The enhancements, although simple, typify how DCGs can be used for natural language ap- plications. Both programs exploit the power of the logical variable.
The first enhancement is constructing a parse tree for the sentence as it is being parsed. The program is given as Program 19.7. Arguments rep- resenting (subparts of) the parse tree must be added to Program 19.6. The enhancement is similar to adding structured arguments to logic pro- grams, as discussed in Section 2.2. The program builds the parse tree top-down, exploiting the power of the logic variable.
The rules in Program 19.7 can be given a declarative reading. For exam- ple, consider the rule
sentence(sentence(NP,VP)) - noun_phrase (NP), verb_phrase(VP).
This states that the parse tree built in recognizing the sentence is a struc- ture sentence (NP,VP), where NP is the structure built while recognizing the noun phrase and VP is the structure built while recognizing the verb phrase.
384 Chapter 19
sentence(sentence(NP,VP)) - noun_phrase(NP), verb_phrase(VP).
noun_phrase(np(D,N)) determiner(D), noun_phrase2(N).
noun_phrase(np(N)) noun_phrase2(N).
noun_phrase2(np2(A,N)) -. adjective(A), noun_phrase2(N).
noun_phrase2(np2(N)) noun(N).
verb_phrase(vp(V)) - verb(V).
verb_phrase(vp(V,N)) verb(V), noun_phrase (N).
Vocabulary
determiner(det(the)) - [the]
determiner(det(a)) [a]
noun(noun(pieplate)) -. [pieplate]
noun(noun(surprise)) -. [surprise]
adjective(adj(decorated)) - [decorated].
verb(verb(contains)) - [contains]
Program 19.7 A DCG computing a parse tree
The next enhancement concerns subject/object number agreement. Suppose we wanted our grammar also to parse the sentence "The dec- orated pieplates contain a surprise." A simplistic way of handling plural forms of nouns and verbs, sufficient for the purposes of this book, is to treat different forms as separate words. We augment the vocabulary by adding the facts
noun(noun(pieplates)) -P [pieplates]
verb(verb(contain)) - [contain].
The new program would parse "The decorated pieplates contain a sur- prise" but unfortunately would also parse "The decorated pieplates con- tains a surprise." There is no insistence that noun and verb must both be singular, or both be plural.
Number agreement can be enforced by adding an argument to the parts of speech that must be the same. The argument indicates whether the part of speech is singular or plural. Consider the grammar rule
sentence (sentence (NP ,VP))
noun_phrase(NP,Num), verb_phrase(VP,Nuin).
The rule insists that both the noun phrase, which is the subject of the sentence, and the verb phrase, which is the object of the sentence, have
385 Logic Grammars
sent once ( sentence (NP VP) noun_phrase(NP,Num), verb_phrase(VP,Num).
noun_phrase(np(D,N) Nun) determiner(D,Nurn), noun_phrase2(N,Num).
noun_phrase(np(N) Nun) -. noun_phrase2(N,Nuxn).
noun_phrase2(np2(A,N) ,Nuzn) -. adjective(A), noun_phrase2(N,Nuzn).
noun_phrase2(np2(N) ,Nuin) -. noun(N,Num).
verb_phrase (vp(V) ,Nuis) - verb(V,Num) vorb_phrase(vp(V,N) ,Num)
verb(V,Nuni), nouri_phrase(N,Nunìl).
Vocabulary
determiner(det(the) Nun) [the] detertniner(det(a),singular) - [a]. noun(noun(pieplate),singular) - [pieplate] noun(noun(pieplates) plural) -. [pieplates] noun(noun(surprïse) singular) -. [surprise] noun(noun(surprises) plural) [surprises] adjective(adj(decorated)) -. [decorated]. verb(verb(contains) ,singular) [contains]. verb(verb(contain) ,plural) [contain].
Program 19.8 A DCG with subject/object number agreement
the same number, singular or plural. The agreement is indicated by the sharing of the variable Nun. Expressing subject/object number agreement is context-dependent information, which is clearly beyond the scope of context-free grammars.
Program 19.8 is an extension of Program 19.7 that handles number agreement correctly. Noun phrases and verb phrases must have the same number, singular or plural. Similarly, the determiners and nouns in a noun phrase must agree in number. The vocabulary is extended to indi- cate which words are singular and which plural. Where number is unim- portant, for example, with adjectives, it can be ignored, and no extra argument is given. The determiner the can be either singular or plural. This is handled by leaving the argument indicating number uninstanti- ated.
386 Chapter 19
The next example of a DCG uses another Prolog feature, the ability to refer to arbitrary Prolog goals in the body of a rule. Program 19.9 is a grammar for recognizing numbers written in English up to, but not includmg, 1,000. The value of the number recognized is calculated using the arithmetic facilities of Prolog.
The basic relation is number (N), where N is the numerical value of the number being recognized. According to the grammar specified by the program, a number is zero or a number N of at most three digits, the rela- tion xxx (N) Similarly xx (N) represents a number N of at most two digits, and the predicates rest _xxx and rest_xx denote the rest of a number of three or two digits, respectively, after the leading digit has been removed. The predicates digit, teen, and tens recognize, respectively, single dig- its, the numbers 10 to 19 inclusive, and the multiples of ten from 20 to 90 inclusive.
A sample rule from the grammar is
xxx(N) -
digit(D), [hundred], rest_xxx(N1), {N is D*100+N1}.
This says that a three-digit number N must first be a digit with value D, followed by the word hundred followed by the rest of the number, which will have value Ni. The value for the whole number N is obtained by multiplying D by 100 and adding Nl.
DCGs inherit another feature from logic programming, the ability to be used backward. Program 19.9 can be used to generate the written representation of a given number up to, but not including, 1,000. In technical terms, the grammar generates as well as accepts. The behavior in so doing is classic generate-and-test. All the legal numbers of the grammar are generated one by one and tested to see whether they have the correct value, until the actual number posed is reached. This feature is a curiosity rather than an efficient means of writing numbers.
The generative feature of DCGs is not generally useful. Many grammars have recursive rules. For example, the rule in Program 19.6 defining a noun_phrase2 as an adjective followed by a noun_phrase2 is recursive. Using recursively defined grammars for generation results in a nonter- minating computation. In the grammar of Program 19.7, noun phrases with arbitrarily many adjectives are produced before the verb phrase is considered.
387 Logic Grammars
nuinber(0) -. [zero]
number(N) -. xxx(N).
xxx(N)
digit(D), [hundred], rest_xxx(N1), {N is D*100+N1}.
xxx(N) xx(N).
rest_xxx(0) -. E J. rest_xxx(N) [and] , xx(N).
xx(N) -, digit(N).
xx(N) -. teen(N).
xx(N) - tens(T), rest_xx(N1), {N is T+N1}.
rest_xx(0) -. E ]
rest_xx(N) digit(N).
digit(1) -. [one] . teen(1O) -. [ten]
digit(2) -. [two] . teen(11) - [eleven] digit(3) -' [three]. teen(12) -. [twelve].
digit(4) -. [four]. teeri(13) -. [thirteen]
digit(5) -, [five] teen(14) -' [fourteen]
digit(6) [six] teen(15) [fifteen]
digit(7) [seven]. teen(16) - [sixteen]
digit(8) - [eight]. teen(17) - [seventeen]
digit(9) -. [nine] teen(18) [eighteen]
teen(19) [nineteen]
tens(20) -. [twenty]
tens(30) - [thirty] tens(40) -. [forty] tens(50) [fifty]
tens(60) -' [sixty].
tens(70) - [seventy] tens(80) - [eighty]. tens(90) - [ninety].
Program 19.9 A DCG for recogmzing numbers
388 Chapter 19
Exercises for Section 19.3
Write a simple grammar for French that illustrates gender agree- ment.
Extend and modify Program 19.9 for parsing numbers so that it cov- ers all numbers less than i million. Don't forget to include things like "thirty-five hundred" and to not include "thirty hundred."
19.4 Background
Prolog was connected to parsing right from its very beginning. As men- tioned before, the Prolog language grew out of Colmerauer's interest in parsing, and his experience with developing Q-systems (Colmerauer, 1973). The implementors of Edinburgh Prolog were also keen on natu- ral language processing and wrote one of the more detailed accounts of definite clause grammars (Pereira and Warren, 1980). This paper gives a good discussion of the advantages of DCGs as a parsing formalism in comparison with augmented transition networks (ATN5).
The examples of using DCGs for parsing languages in Section 19.1 were adapted from notes from a tutorial on natural language analysis given by Lynette Hirschman at the Symposium on Logic Progranuning in San Francisco in 1987. The DCG interpreter of Section 19.2 is adapted from Pereira and Shieber (1987).
Even though the control structure of Prolog matches directly that of recursive-descent, top-down parsers, other parsing algorithms can also be implemented in it quite easily. For example, Matsumoto et al. (1986) describes a bottom-up parser in Prolog.
The grammar in Program 19.3 is taken from Appendix 1 of Findlay and Watt (1985). The grammar in Program 19.6 is taken from Winograd's (1983) book on computational linguistics.
For further reading on logic grammars, refer to Pereira and Shieber (1987) and Abramson and Daffi (1989).
20 Search Techniques
In this chapter, we show programs encapsulating classic AI search tech- niques. The first section discusses state-transition frameworks for solv- ing problems formulated in terms of a state-space graph. The second discusses the minimax algorithm with alpha-beta pruning for searching game trees.
20.1 Searching State-Space Graphs
State-space graphs are used to represent problems. Nodes of the graph are states of the problem. An edge exists between nodes if there is a transition rule, also called a move, transforming one state into the next. Solving the problem means finding a path from a given initial state to a desired solution state by applying a sequence of transition rules.
Program 20.1 is a framework for solving problems by searching their state-space graphs, using depth-first search as described in Section 14.2.
No cornniitment has been made to the representation of states. The moves are specified by a binary predicate move(State,Move), where Move is a move applicable to State. The predicate update(State,Move, Statel) finds the state Statel reached by applying the move Move to state State. It is often easier to combine the move and update proce- dures. We keep them separate here to make knowledge more explicit and to retain flexibility and modularity, possibly at the expense of perfor- mance.
The validity of possible moves is checked by the predicate legal (State), which checks if the problem state State satisfies the con- straints of the problem. The program keeps a history of the states visited
390 Chapter 20
solve_dfs (State,History,Moves) - Moves is a sequence of moves to reach a desired final state from the current State, where History contains the states visited previously.
solve_dfs(State,History,[ J) -
final_state (State).
solve_dfs(State,History, [Move Moves])
move (State,Move),
update (State,Move,Statei),
legal(Statel),
not member(Statel ,History),
solve_dfs(Statei, [Statelillistory] ,Moves).
Testing the framework
test_dfs(Problem,Moves) -
initial_state(Problem,State), solve_dfs(State,[State] Moves).
Program 20.1 A depth-first state-transition framework for problem solving
to prevent looping. Checking that looping does not occur is done by see- ing if the new state appears in the history of states. The sequence of moves leading from the initial state to the final state is built incremen- tally in the third argument of salve_df s/3.
To solve a problem using the framework, the programmer must decide how states are to be represented, and axiomatize the move, update, and legal procedures. A suitable representation has profound effect on the success of this framework.
Let us use the framework to solve the wolf, goat, and cabbage problem. We state the problem informally. A farmer has a wolf, goat, and cabbage on the left side of a river. The farmer has a boat that can carry at most one of the three, and he must transport this trio to the right bank. The problem is that he dare not leave the wolf with the goat (wolves love to eat goats) or the goat with the cabbage (goats love to eat cabbages). He takes all his jobs very seriously and does not want to disturb the ecological balance by losing a passenger.
States are represented by a triple, wgc(B,L,R), where B is the po- sition of the boat (left or right), L is the list of occupants of the left bank, and R the list of occupants of the right bank. The ini- tial and final states are wgc(lef t, [wolf,goat,cabbage] , E ]) and wgc (right, [ I , [wolf ,goat , cabbage]), respectively. In fact, it is not strictly necessary to note the occupants of both the left and right banks.
391 Search Techniques
The occupants of the left bank can be deduced from the occupants of the right bank, and vice versa. But having both makes specifying moves clearer.
lt is convenient for checking for loops to keep the lists of occupants sorted. Thus wolf will always be listed before goat, both of whom will be before cabbage if they are on the same bank.
Moves transport an occupant to the opposite bank and can thus be specified by the particular occupant who is the Cargo. The case when nothing is taken is specified by the cargo alone. The nondeterministic behavior of member allows a concise description of all the possible moves in three clauses as shown in Program 20.2: moving something from the left bank, moving something from the right bank, or the farmer's rowing in either direction by himself.
For each of these moves, the updating procedure must be specified, namely, changing the position of the boat (by update_boat/2) and up- dating the banks (by update_banks). Using the predicate select allows a compact description of the updating process. The insert procedure is necessary to keep the occupant list sorted, facilitating the check if a state has been visited before. It contains all the possible cases of adding an occupant to a bank.
Finally, the test for legality must be specified. The constraints are sim- ple. The wolf and goat cannot be on the same bank without the farmer, nor can the goat and cabbage.
Program 20.2, together with Program 20.1, solves the wolf, goat, and cabbage problem. The clarity of the program speaks for itself.
We use the state-transition framework for solving another classic search problem from recreational mathematicsthe water jugs prob- lem. There are two jugs of capacity 8 and 5 liters with no markings, and the problem is to measure out exactly 4 liters from a vat containing 20 liters (or some other large number). The possible operations are filling up a jug from the vat, emptying a jug into the vat, and transferring the contents of one jug to another until either the pouring jug is emptied completely, or the other jug is filled to capacity. The problem is depicted in Figure 20.1.
The problem can be generalized to N jugs of capacity C1.....CN. The problem is to measure a volume V, different from all the C but less than the largest. There is a solution if V is a multiple of the greatest common divisor of the C. Our particular example is solvable because 4 is a multiple of the greatest common divisor of 8 and 5.
392 Chapter 20
States for the wolf, goat and cabbage problem are a structure wgc(Boat,Left,Right), where Boat is the bank on which the boat currently is, Left is the list of occupants on the left bank of the river, and Right is the list of occupants on the right bank.
initial_state(wgc,wgc(left, [wolf ,goat,cabbagej 1 1)). final_state (wgc(right, E i , [wolf ,goat ,cabbage])).
move(wgc(lef t ,L,R) ,Cargo) - member(Cargo,L).
move(wgc(right ,L,R) ,Cargo) - member(Cargo,R).
move(wgc(B,L,R) ,alone).
update(wgc(B,L,R),Cargo,wgc(B1,L1,R1)) -
update_boat(B,B1), update_banks(Cargo,B,L,R,L1,R1).
update_boat (left ,right).
update_boat (right ,left).
update_banks(alone,B,L,R,L,R).
update_banks(Cargo,lef t ,L,R,Ll,R1)
select(Cargo,L,L1), insert(Cargo,R,R1).
update_banks(Cargo,right,L,R,L1,R1) -
select(Cargo,R,R1), insert(Cargo,L,L1).
insert(X,[YIYs],[X,YIYs]) -
precedes (X , Y)
insert(X,[YIYs],[YIZs]) -
precedes(Y,X) , insert(X,Ys,Zs)
insert(X,[ ],[X]).
precedes(wolf ,X).
precedes (X , cabbage)
legal(wgc(left,L,R)) - not illegal(R).
legal(wgc(right,L,R)) - not illegal(L).
illegal(Bank) - member(wolf Bank), menber(goat ,Bank).
illegal(Bank) - member(goat ,Bank), member(cabbage,Bank).
select(X,Xs,Ys) See Program 3.19.
Program 20.2 Solving the wolf, goat, and cabbage problem
393 Search Techniques
Jb 8 litres 5 litres
Figure 20.1 The water jugs problem
The particular problem we solve is for two jugs of arbitrary capacity, but the approach is immediately generalizable to any number of jugs. The program assumes two facts in the database, capacity(I CI), for I equals i and 2. The natural representation of the state is a structure jugs(V1 ,V2), where Vi and V2 represent the volumes of liquid currently in the two jugs. The initial state is jugs (0,0) and the desired final state either jugs(0,X) or jugs(X,0), where X is the desired volume. In fact, the only final state that needs to be specified is that the desired volume be in the larger jug. The volume can be transferred from the smaller volume, if it fits, by emptying the larger jug and pouring the contents of the smaller jug into the larger one.
Data for solving the jugs problem in conjunction with Program 20.1 are given in Program 20.3. There are six moves filling each jug, emptying each jug, and transferring the contents of one jug to another. A sam- pie fact for filling the first jug is ¡nove (jugs (Vi, V2) ,fili(i)). The jugs' state is given explicitly to allow the data to coexist with other problem solving data such as in Program 20.2. The emptying moves are optimized to prevent emptying an already empty jug. The updating procedure asso- ciated with the first four moves is simple, while the transferring opera- tion has two cases. If the total volume in the jugs is less than the capacity of the jug being filled, the pouring jug will be emptied and the other jug will have the entire volume. Otherwise the other jug will be filled to capacity and the difference between the total liquid volume and the ca- pacity of the filled jug will be left in the pouring jug. This is achieved by the predicate adjust/4. Note that the test for legality is trivial because all reachable states are legal.
Most interesting problems have too large a search space to be searched exhaustively by a program like 20.1. One possibility for improvement is
-u 4 litres
394 Chapter 20
iriitial_state(jugs,jugs(O,O)).
final_state(jugs(4,V)).
final_state(jugs(V,4).
move(jugs(V1,V2) ,fill(1)).
niove(jugs(Vt,V2) ,fill(2)).
move(jugs(V1,V2),empty(1)) - Vi > O.
move(jugs(V1,V2),empty(2)) - V2 > O.
move(jugs(Vi,V2),transfer(2,1)).
move(jugs(V1,V2) ,transfer(1,2)).
update(jugs(Vi,V2),fill(1),jugs(C1,V2)) - capacity(i,C1).
update(jugs(V1,V2),fill(2),jugs(V1,C2)) capacity(2,C2).
update(jugs(V1,V2) ,empty(1) ,jugs(O,V2)).
update(jugs(V1,V2),empty(2),jugs(Vi3O)).
update(jugs(Vi,V2),transfer(2,i),jugs(W1,W2)) -
capacity(i,Ci),
Liquid is Vi + V2,
Excess is Liquid - Cl,
adjust (Liquid,Excess,Wl ,W2).
update(jugs(Vl,V2),transfer(i,2),jugs(Wl,W2)) -
capacity(2,C2),
Liquid is Vi + V2,
Excess is Liquid - C2,
adjust (Liquid,Excess,W2,Wi).
adjust(Liquid,Excess,Liquid,O) - Excess O.
adjust(Liquid,Excess,V,Excess) -
Excess > O, V is Liquid - Excess.
iegai(jugs(Vi,V2)).
capacity(i,8).
capacity(2,5).
Program 20.3 Solving the water jugs problem
395 Search Techniques
to put more knowledge into the moves allowed. Solutions to the jug prob- lem can be found by filling one of the jugs whenever possible, emptying the other whenever possible, and otherwise transferring the contents of the jug being filled to the jug being emptied. Thus instead of six moves only three need be specified, and the search will be more direct, because only one move will be applicable to any given state. This may not give an optimal solution if the wrong jug to be constantly filled is chosen.
Developing this point further, the three moves can be coalesced into a higher-level move, fill_and_transfer. This tactic fills one jug and transfers all its contents to the other jug, emptying the other jug as necessary. The code for transferring from the bigger to the smaller jug is
move(jugs(V1,V2) ,fill_and_transfer(1)).
update(jugs(V1,V2),fill_and_transfer(1),jugs(0,V))
capacity(1,C1),
capacity(2,C2),
Cl > C2,
V is (Cl+V2) mod C2.
Using this program, we need only three fill and transfer operations to solve the problem in Figure 20.1.
Adding such domain knowledge means changing the problem descrip- tion entirely and constitutes programming, although at a different level.
Another possibility for improvement of the search performance, inves- tigated by early research in Al, is heuristic guidance. A general frame- work, based on a more explicit choice of the next state to search in the state-space graph, is used. The choice depends on numeric scores assigned to positions. The score, computed by an evaluation function, is a measure of the goodness of the position. Depth-first search can be considered a special case of searching using an evaluation function whose value is the distance of the current state to the initial state, while breadth-first search uses an evaluation function which is the inverse of that distance.
We show two search techniques that use an evaluation function explic- itly: hill climbing and best-first search. In the following, the predicate value (State,Value) is an evaluation function. The techniques are de- scribed abstractly.
396 Chapter 20
Hill climbing is a generalization of depth-first search where the suc- cessor position with the highest score is chosen rather than the leftmost one chosen by Prolog. The problem-solving framework of Program 20.1 is easily adapted. The hill climbing move generates all the states that can be reached from the current state in a single move, and then orders them in decreasing order with respect to the values computed by the evalu- ation function. The predicate evaluate_and_order(Moves ,State, MVs) determines the relation that MVs is an ordered list of move-value tuples corresponding to the list of moves Moves from a state State. The overall program is given as Program 20.4.
To demonstrate the behavior of the program we use the example tree of Program 14.8 augmented with a value for each move. This is given as Program 20.5. Program 20.4, combined with Program 20.Sand appropri- ate definitions of update and legal searches the tree in the order a, d, j. The program is easily tested on the wolf, goat, and cabbage problem using as the evaluation function the number of occupants on the right bank.
Program 20.4 contains a repeated computation. The state reached by Move is calculated in order to reach a value for the move and then re- calculated by update. This recalculation can be avoided by adding an extra argument to move and keeping the state along with the move and the value as the moves are ordered. Another possibility if there will be many calculations of the same move is using a memo-function. What is the most efficient method depends on the particular problem. For prob- lems where the update procedure is simple, the program as presented will be best.
Hill climbing is a good technique when there is only one hill and the evaluation function is a good indication of progress. Essentially, it takes a local look at the state-space graph, making the decision on where next to search on the basis of the current state alone.
An alternative search method, called best-first search, takes a global look at the complete state-space. The best state from all those currently unsearched is chosen.
Program 20.6 for best-first search is a generalization of breadth-first search given in Section 16.2. A frontier is kept as for breadth-first search, which is updated as the search progresses. At each stage, the next best available move is made. We make the code as similar as possible to Program 20.4 for hill climbing to allow comparison.
397 Search Techniques
solve_hill_climb (Sta te,History,Moves) - Moves is the sequence of moves to reach a desired final state from the current State, where History is a list of the states visited previously.
solve_hill_climb(State,History, [ ])
final_state (State).
solve_hill_climb(Sate,History,[MovePMoves]) -
hill_climb (State , Move)
update (State,Move,Statel),
legal(Statel)
not member(Statel,History),
solve_hill_climb(Statel, [Statel IHistory] Moves)
hill_climb(State,Move) -
findall(M,move(State,M) ,Moves)
evaluate_and_order(Moves,State, E ] ,MVs),
member((Move,Value) ,MVs).
evaluate_and_order (Moves,State,SoFar, Orde redMvs) All the Moves from the current State are evaluated and ordered as Orde redMvs. SoFar is an accumulator for partial computations.
evaluate_and_order([Move Moves] ,State ,MVs ,DrderedMVs)
update(State ,Move,Statel),
value (Statel, Value)
insert((Move,Value) ,MVs,MVs1),
evaluate_and_order (Moves ,State,MVs1 ,OrderedMVs).
evaluate_and_order([ I ,State,MVs,MVs).
insert(MV,[ ],[MV]).
insert((M,V),[(M1,V1)IMVs],[(M,V),(M1,V1)IMVs]) -
V Vi.
insert((M,V),[(Ml,Vi)JMVs]j(M1,V1)IMVs1]) - V < Vi, ïnsert((M,V),MVs,MVsi).
Testing the framework
test_hill_climb(Problem,Moves) -
initial_state (Problem,State),
solve_hill_climb(State, [State] ,Moves)
Program 20.4 Hill climbing framework for problem solving
398 Chapter 20
Program 20.5 Test data
At each stage of the search, there is a set of moves to consider rather than a single one. The plural predicate names, for example, updates and legals, indicate this. Thus legals(States,Statesl) filters a set of successor states, checking which ones are allowed by the constraints of the problem. One disadvantage of breadth-first search (and hence best- first search) is that the path to take is not as conveniently calculated. Each state must store explicitly with it the path used to reach it. This is reflected in the code.
Program 20.6 tested on the data of Program 20.5 searches the tree in the same order as for hill climbing.
Program 20.6 makes each step of the process explicit. In practice, it may be more efficient to combine some of the steps. When filtering the generated states, for example, we can test that a state is new and also le- gal at the same time. This saves generating intermediate data structures. Program 20.7 illustrates the idea by combining all the checks into one procedure, update_frontier.
Exercises for Section 20.1
Redo the water jugs program based on the two fill-and-transfer operations.
Write a program to solve the missionaries and cannibals problem:
Three missionaries and three cannibals are standing on the left bank of a river. There is a small boat to ferry them across with enough room for only one or two people. They wish to cross the river. If ever there are more missionaries than cannibals on a particular bank of the river, the missionaries will convert the cannibals. Find a series of ferryings to transport safely all the missionaries and cannibals across the river without exposing any of the cannibals to conversion.
initial_state(tree,a). value(a,O). final_state(j).
move(a,b). value(b,1). move(c,g). value(g,6).
move(a,c). value(c,5). move(d,j). value(j,9).
move(a,d). value(d,7). move(e,k). value(k,1).
move(a,e). value(e,2). move(f,h). value(h,3).
move(c,f). value(f,4). move(f,i). value(i,2).
399 Search Techniques
solve_best (Frontier,History,Moves) Moves is a sequence of moves to reach a desired final state from the initial state, where Frontier contains the current states under consideration, and History contains the states visited previously.
solve_best([state(State,Path,Value)IFrontior] ,History,Moves)'- final_state(State), reverse(Path,Moves).
solve_best([state(State,Path,Value)IFrontier] ,History,FinalPath)- findall(M,rnove(State,M) Moves), updates (Moves , Path, State , States) legals(States,Statesl) news(Statesl,History,States2), evaluates(States2,Values), inserts(Values,Frontier,Frontierl), solve_best (Frontierl, [State IHistory] ,FinalPath)
upda tes (Moves,Path,State,Sta tes) States is the list of possible states accessible from the current State, according to the list of possible Moves, where Path is a path from the initial node to State.
updates([MIMs],Path,S,[(S1,[MIPath]flSs]) - update(S,M,S1), updates(Ms,Path,S,Ss).
updates([ ],Patb,State,[ 1). le gals (States,Statesl) -
Statesi is the subset of the list of States that are legal. legals( E (S , P) IStates] , [ (S , P) IStatesi] )
legal(S), legals(States,Statesl). legals( E (S, P) States] , St at esi)
not legal(s), legals(States,Statesl) legals(E ],[ 1). news (States,History,Statesl) -
Statesi is the list of states in States but not in History. news([(S,P)IStates] ,History,Statesl)
member(S,History), news(States,Hïstory,Statesl). news([(S,P)IStates],History,[(S,P)IStatesl])
not member(S,History), news(States,History,Statesl). news([ ],History,[ 1). evaluates(States,Values) -
Values is the list of tuples of States augmented by their value.
evaluates([(S,PflStates],[state(S,P,VflValues]) - value(S,V), evaluates(States,Values).
evaluates([ ],[ 1).
Program 20.6 Best-first framework for problem solving
400 Chapter 20
inserts (States,Frontier,Frontierl) - Fron tien is the result of inserting States into the current Frontier.
inserts([ValuelValues] ,Frontier,Frontierl) -
insert (Value ,Frontier, FrontierO),
inserts (Values, FrontierO , Frontierl).
inserts([ I ,Frontier,Frontier).
insert (State, C j , [State]). insert(State,[StatellStates] , [State,StatellStates] -
lesseq_value(State , Statel).
insert (State, [Statel IStates] , [State IStates]) -
equals(State,Statel).
insert(State, [StateliStates] , [StateliStatesi]) -
greater_value(State,Statel), insert(State,States,Statesl)
equals(state(S,P,V) ,state(S,P1,V)).
lesseq_value(state(Si,Pi,V1),state(52,P2,V2)) - Si S2, Vi V2.
greater_value(state(S1,P1,V1),state(S2,P2,V2)) - Vi > V2.
Program 20.6 (Continued)
solve_best (Frontier,History,Moves) Moves is a sequence of moves to reach a desired final state from the initial state. Frontier contains the current states under consideration. History contains the states visited previously.
solve_best([state(State,Path,Value) Frontier] ,History,Moves) -
final_state(State), reverse(Path, [ j ,Moves). solve_best([state(State,Path,Value) Frontier] ,History,FinalPath)
findall(M,move(State,M) ,Moves),
update_frontier(Moves,State,Path,History,Frontier,Frontierl),
solve_best (Frontierl, [State I History] , FinalPath).
update_frontier([MIMs],State,Path,Hïstory,F,F1) -
update (State , M, Statel)
legal(Statel),
value (Statel,Value),
not member(Statel,History),
insert((Statel, [MiPath] ,Value) ,F,FO),
update_Írontier(Ms,State,Path,History,FO,F1).
update_frontier([ ],S,P,H,F,F).
insert(State,Frontier,Frontierl) - See Program 20.6.
Program 20.7 Concise best-first framework for problem solving
401 Search Techniques
Write a program to solve the five jealous husbands problem (Du- deney, 1917):
During a certain flood five married couples found themselves sur- rounded by water and had to escape from their unpleasant position in a boat that would only hold three persons at a time. Every husband was so jealous that he would not allow his wife to be in the boat or on either bank with another man (or with other men) unless he himself was present. Find a way of getting these five men and their wives across to safety.
Compose a general problem-solving framework built around breadth-first search analogous to Program 20.1, based on programs in Section 16.2.
(y) Express the 8-queens puzzle within the framework. Find an evalua- tion function.
20.2 Searching Game Trees
What happens when we play a game? Starting the game means setting up the chess pieces, dealing out the cards, or setting out the matches, for example. Once it is decided who plays first, the players take turns making a move. After each move the game position is updated accordingly.
We develop the vague specification in the previous paragraph into a simple framework for playing games. The top-level statement is
play(Game)
initialize (Gaine, Position ,Player),
display_game (Position,Player),
play (Position, Player , Result)
The predicate initialize (Gaine ,Posit on,Player) determines the ini- tial game position Position for Game, and Player, the player to start.
A game is a sequence of turns, where each turn consists of a player choosing a move, the move being executed, and the next player being determined. The neatest way of expressing this is as a tail recursive procedure, play, with three arguments: a game position, a player to move, and the final result. It is convenient to separate the choice of the move by choose_move/3 from its execution by move/3. The remaining
402 Chapter 20
play(Game) - Play game with name Game.
play(Game) -
initialize(Game,Position,Player),
display_game (Position , Player)
play(Position,Player,Result).
play(Position,Player,Result) -
game_over (Position,Player,Result), ! , announce(Result).
play(Position,Player,Result) -
choose_move (Position, Player , Move)
move(Move,Position,Positionl),
display_game (Positioni ,Player),
next_player(Player,Playerl),
play(Positionl,Playeri,Result).
Program 20.8 Framework for playing games
predicates m the clause for play/3 display the state of the game and determine the next player:
play(Position,Player,Result) -
choose_move(Position,Player,Move),
move(Move,Position,Positionl),
display_game (Positioni ,Player),
next_player(Player ,Player 1),
!, play(Positionl,Playerl,Result)
Program 20.8 provides a logical framework for game-playing programs. Using it for writing a program for a particular game focuses attention on the important issues for game playing: what data structures should be used to represent the game position, and how strategies for the game should be expressed. We demonstrate the process in Chapter 21 by writ- ing programs to play Nim and Kalah.
The problem-solving frameworks of Section 20.1 are readily adapted to playing games. Given a particular game state, the problem is to find a path of moves to a winning position.
A game tree is similar to a state-space graph. It is the tree obtained by identifying states with nodes and edges with players' moves. We do not, however, identify nodes on the tree, obtained by different sequences of moves, even if they repeat the same state. In a game tree, each layer is called a ply.
403 Search Techniques
evahiate_and_choose (Moves,Position,Record,BestMove) - Chooses the BestMove from the set of Moves from the current Position. Record records the current best move.
evaluate_and_choose( [MovelMoves] ,Positïon,Record,BestMove)
inove(Move,Position,Positionl)
value(Positionl ,Value),
update (Move , Value , Record , Recordi)
evaluate_and_choose(Moves,Position,Recordl ,BestMove).
evaluate_and_choose([ j ,Position, (Move,Value) ,Move).
update(Move,Valu,(Move1,Value1),(Move1,Value1))
Value Valuel.
update(Move ,Value, (Movel ,Valuel) ,(Move ,Value)) -
Value > Valuel.
Program 20.9 Choosing the best move
Most game trees are far too large to be searched exhaustively. This sec- tion discusses the techniques that have been developed to cope with the large search space for two-person games. In particular, we concentrate on the minimax algorithm augmented by alpha-beta pruning. This strat- egy is used as the basis of a program we present for playing Kalah in Chapter 21.
We describe the basic approach of searching game trees using evalua- tion functions. Again, in this section value(Position,Value) denotes an evaluation function computing the Value of Position, the current state of the game. Here is a simple algorithm for choosing the next move:
Find all possible game states that can be reached in one move. Compute the values of the states using the evaluation function. Choose the move that leads to the position with the highest score.
This algorithm is encoded as Program 20.9. It assumes a predicate move (Move,Position,Positionl) that applies a Move to the current Po- sition to reach Positioni. The interface to the game framework of Program 20.8 is provided by the clause
choose_move (Position, computer,Move) -
firidall(M,move(Position,M) ,Moves),
evaluate_and_choose (Moves,Position, (nil,-1000) ,Move).
The predicate move(Position,Move) is true if Move is a possible move from the current position.
404 Chapter 20
The basic relation is evaluate_and_choose (Noves ,Position,Record, BestMove) which chooses the best move BestMove in the possible Moves from a given Position. For each of the possible moves, the correspond- ing position is determined, its value is calculated, and the move with the highest value is chosen. Record is a record of the current best move so far. In Program 20.9, it is represented as a tuple (Move,Value). The structure of Record has been partially abstracted in the procedure up date/4. How much data abstraction to use is a matter of style and a trade-off among readability, conciseness, and performance.
Looking ahead one move, the approach of Program 20.9, would be sufficient if the evaluation function were perfect, that is, if the score reflected which positions led to a win and which to a loss. Games become interesting when a perfect evaluation function is not known. Choosing a move on the basis of looking ahead one move is generally not a good strategy. It is better to look several moves ahead and to infer from what is found the best move to make.
The minimax algorithm is the standard method for determining the value of a position based on searching the game tree several ply ahead.
The algorithm assumes that, when confronted with several choices, the opponent would make the best choice for her, i.e., the worst choice for me. My goal then is to make the move that maximizes for me the value of the position after the opponent has made her best move, i.e., that minimizes the value for her. Hence the name minimax This reasoning proceeds several ply ahead, depending on the resources that can be allocated to the search. At the last ply the evaluation function is used.
Assuming a reasonable evaluation function, the algorithm will produce better results the more ply are searched. It will produce the best move if the entire tree is searched.
The minimax algorithm is justified by a zero-sum assumption, which says, informally, that what is good for me must be bad for my opponent, and vice versa.
Figure 20.2 depicts a simple game tree of depth 2 ply. The player has two moves in the current position, and the opponent has two replies. The values of the leaf nodes are the values for the player. The oppo- nent wants to minimize the score, so will choose the minimum values, making the positions be worth +1 and 1 at one level higher in the tree. The player wants to maximize the value and will choose the node with value +1.
405 Search Techniques
Figure 20.2 A simple game tree
Program 20.10 encodes the rninimax algorithm. The basic relation is minimax(D,Position,MaxMin,Move,Value), which is true if Move is the move with the highest Value from Position obtained by searching D ply in the game tree. MaxMin is a flag that indicates if we are maximizing or minimizing. lt is i for maximizing and - i for minimizing, the particular values being chosen for ease of manipulation by simple arithmetic opera- tions. A generalization of Program 20.9 is used to choose from the set of moves. Two extra arguments must be added to evaluate_and_choose: the number of ply D and the flag MaxMin. The last argument is general- ized to reflirn a record including both a move and a value rather than just a move. The minimax procedure does the bookkeeping, changing the number of moves being looked ahead and also the minimax flag. The ini- tial record is (nil,-1000), where nil represents an arbitrary move and
i000 is a score intended to be less than any possible score of the evalu- ation function.
The observation about efficiency that was made about combining the move generation and update procedures in the context of searching state-space graphs has an analogue when searching game trees. Whether it is better to compute the set of positions rather than the set of moves (with the corresponding change in algorithm) will depend on the particu- lar application.
The minimax algorithm can be improved by keeping track of the re- sults of the search so far, using a technique known as alpha-beta pruning. The idea is to keep for each node the estimated minimum value found so far, the alpha value, along with the estimated maximum value, beta. If, on evaluating a node, beta is exceeded, no more search on that branch is necessary. In good cases, more than half the positions in the game tree need not be evaluated.
406 Chapter 20
evaluate_and_choose (Moves,Position,Depth,Flag,Record,BestMove) - Choose the BestMove from the set of Moves from the current Position using the minimax algorithm searching Depth piy ahead. Flag indicates if we are currently minimizing or maximizing. Record records the current best move.
evaluate_and_choose( [MovelMoves] ,Position,D,MaxMin,Record,Best)
move (Move,Position,Positionl),
minimax(D,Positionl ,MaxMin,MoveX,Value),
update (Nove ,Value, Record, Recordi),
evaluate_and_choose(Moves,Position,D,MaxMin,Recordl Best).
evaluate_and_choose([ I ,Position,D,MaxMin,Record,Record).
minimax(O,Position,MaxMin,Move,Value) -
value (Position,V),
Value is V*MaxMin.
minimax (D Position, MaxMin Move, Value)
D > O,
findall(M,move(Position,M) ,Moves),
Dl is D 1,
MinMax is -MaxMin,
evaluate_and_choose(Moves,Position,Dl,MinMax, (nil,-1000),
(Nove,Value)).
update (Move,Value ,Record,Recordl) - See Program 20.9.
Program 20.10 Choosing the best move with the minimax algorithm
Program 20.11 is a modified version of Program 20.10 that incor- porates alpha-beta pruning. The new relation scheme is alpha_beta (Depth,Position,Alpha,Beta,Move ,Value), which extends mirumax by replacing the minimax flag with alpha and beta. The same relation holds with respect to evaluate_and_choose.
Unlike the one in Program 20.10, the version of evaluate_arid_choose in Program 20.11 does not need to search all possibilities. This is achieved by introducing a predicate cutoff, which either stops searching the current branch or continues the search, updating the value of alpha and the current best move as appropriate.
For example, the last node in the game tree in Figure 20.2 does not need to be searched. Once a move with value 1 is found, which is less than the value of + 1 the player is guaranteed, no other nodes can contribute to the final score.
The program can be generalized by replacing the base case of alpha_ beta by a test of whether the position is terminal. This is necessary in chess programs, for example, for handling incomplete piece exchanges.
407 Search Techniques
evaluate_and_choose (Moves,Position,Depth,Alpha,Beta,Record,BestMove) Chooses the BestMove from the set of Moves from the current Position using the minimax algorithm with alpha-beta cutoff searching Depth piy ahead. Alpha and Beta are the parameters of the algorithm. Record records the current best move.
evaluate_and_choose([MovelMovesl ,Position,D,Alpha,Beta,Movel,
BestMove)
move(Move,Position,Positionl)
alpha_beta(D ,Positionl ,Alpha,Beta,MoveX,Value),
Valuel is -Value,
cutoff (Move,Valuel,D,Alpha,Beta,Moves,Positïon,Novel,BestMove)
evaluate_and_choose([ j ,Position,D,Alpha,Beta,Move,(Move,Alpha)).
alpha_beta(O ,Positïon,Alpha,Beta,Move ,Value) -
value (Position, Value)
alphabeta(D,Position,Alpha,Beta,Move ,Value)
findall(M,move(Position,M) ,Moves),
Alphal is -Beta,
Betal is -Alpha,
Dl is D-1,
evaluateand_choose(Moves,Position,D1,Alphal,Betal,nil,
i (Move,Value)).
cutoff (Move,Value,D,Alpha,Beta,Moves,Position,Movel, (Move,Value))
Value Beta.
cutoff (Move,Value,D,Alpha,Beta,Moves,Position,Movel,BestMove)
Alpha < Value, Value < Beta,
evaluateandchoose(Moves,Position,D,Value,Beta,Move,BestMove) cutoff (Move,Value,D,Alpha,Beta,Moves,Position,Movel,BestMove)
Value Alpha,
evaluateandchoose(Moves,Positiou,D,Alpha,Beta,Movel,BestMove).
Program 20.11 Choosing a move using minimax with alpha-beta pruning
20.3 Background
Search techniques for both planning and game playing are discussed in Al textbooks. For further details of search strategies or the minimax algo- rithm and its extension to alpha-beta pruning, see, for example, Nilsson (1971) or Winston (1977).
Walter Wilson originally showed us the alpha-beta algorithm in Prolog.
V. 4M 1 ' lr
r'--4- ..-i..-; 4r Ti+ T T 4 4. 1''-
L.J.U....._A_-_1-___¡ -
- r ___ -
r --' r'" .,. . 4_r. .Jws -.. -'1 ''r-f ,st.f,a44.w4.M -qe. ....» ..-''.1 . i .a .. L ' 'rà'. .... 1A4
Leonardo Da Vinci, The Proportions of the Human Figure, after Vitruvius. Pen and ink. About 1492. Venice Academy.
IV Applications
Prolog has been used for a wide range of applications: expert systems, natural language understanding, symbolic algebra, compiler writing, building embedded languages, and architectural design, to name a few. In this part, we give a flavor of writing application programs in Prolog.
The first chapter looks at programs for playing three games: master- mind, Nim, and Kalah. The next chapter presents an expert system for evaluating requests for credit. The third chapter presents a program for solving sI'mbolic equations, and the final chapter looks at a compiler for a Pascal-like language.
The emphasis in presentation in these chapters is on writing clear programs. Knowledge embedded in the programs is made explicit. Minor efficiency gains are ignored if they obscure the declarative reading of the program.
21 Game-Playing Programs
21.1 Mastermind
Our first program guesses the secret code in the game of mastermind lt is a good example of what can be programmed in Prolog easily with just a little thought.
The version of mastermind we describe is what we played as kids. It is a variant on the commercial version and needs less hardware (only pencil and paper). Player A chooses a sequence of distinct decimal digits as a secret codeusually four digits for beginners and five for advanced players. Player B makes guesses and queries player A for the number of bulls (number of digits that appear in identical positions in the guess and in the code) and cows (number of digits that appear in both the guess and the code, but in different positions).
There is a very simple algorithm for playing the game: Impose some order on the set of legal guesses; then iterate, making the next guess that is consistent with all the information you have so far until you find the secret code.
Learning how to play a game is fun. As well as understanding the rules of the game, we must constantly learn new strategies and tactics until the gaine is mastered. Writing a program to play games is also fun, and a good vehicle for showing bow to use Prolog for writing nontrivial pro- grams.
412 Chapter 21
Rather than defining the notion of consistency formally, we appeal to the reader's intuition: A guess is consistent with a set of answers to queries if the answers to the queries would have remained the same if the guess were the secret code.
The algorithm performs quite well compared with experienced players: an average of four to six guesses for a code with four digits with an observed maximum of eight guesses. However, it is not an easy strategy for humans to apply, because of the amount of bookkeeping needed. On the other hand, the control structure of Prolognondeterministic choice, simulated by backtrackingis ideal for implementing the algorithm.
We describe the program top-down. The entire program is given as Program 21.1. The top-level procedure for playing the game is
mastermind(Code) -
cleanup, guess(Code), check(Code), announce.
The heart of the top level is a generate-and-test loop. The guessing pro- cedure guess (Code), which acts as a generator, uses the procedure se- lects(Xs,Ys) (Program 7.7) to select nondeterrninistically a list Xs of elements from a list Ys. According to the rules of the game, Xs is con- strained to contain four distinct elements, while Ys is the list of the ten decimal digits:
guess(Code) -
Code = [X1,X2,X3,X4],
selects(Code,[1,2,3,4,5,6,7,8,9,O]).
The procedure check(Guess) tests the proposed code Guess. It first verifies that Guess is consistent with all (i.e., not inconsistent with any) of the answers to queries already made; then it asks the user for the number of bulls and cows in Guess. The ask(Guess) procedure also controls the generate-and-test loop, succeeding only when the number of bulls is four, indicating the correct code is found:
check(Guess) -
not inconsistent(Guess), ask(Guess).
Ask stores previous answers to queries in the relation query(X,B,C), where X is the guess, B is the number of bulls in it, and C the number of cows. A guess is inconsistent with a previous query if the number of bulls and cows do not match:
413 Game-Playing Programs
mastermind(Code) -
cleanup, guess(Code), check(Code), announce.
guess(Code) - Code = [X1,X2,X3,X4J, selects(Code,[1,2,3,4,5,6,7,8,9,O]).
Verify the proposed guess
check (Guess)
not inconsistent(Guess) , ask(Guess)
inconsistent (Guess)
query(OldGuess,Bulls,Cows),
not bulls_and_cows_rnatch(DldGuess,Guess,Bulls,Cows).
bulls_and_cows_match(OldGuess,Guess,Bulls,Cows) -
exact_matches(OldGuess,Guess,N1),
Bulls =: Nl, 'f, Correct number of bulls common_members (OldGuess Guess ,N2),
Cows =:= N2-Bulls. '/ Correct number of cows
exact_matches(Xs,Ys,N) -
size_of (A,saine_place(A,Xs,Ys) ,N)
common_members(Xs ,Ys ,N)
size_of (A, (member(A,Xs) ,member(A,Ys)) ,N).
sanie_place(X,[XIXs] , [XIYs]).
sane_place(A, [XIXs] , [YIYs]) - same_place(A,Xs,Ys)
Asking a guess ask (Guess)
repeat,
writeln(['How many bulls and cows in ',Guess,'?']), read((Bulls,Cows))
sensible (Bulls,Cows)
assert (query(Guess,Bulls,Cows)),
Bulls 4.
sensible(Bulls,Cows) -
integer(Bulls), integer(Cows), Bulls+Cows 4.
Bookkeeping
cleanup - abolish(query,3).
announce -
size_of (X,query(X,A,B) ,N)
writeln(['Found the answer after ',N,' queries']).
size_of(X,G,N) - findall(X,G,Xs), length(Xs,N).
length(Xs,N) - See Program 8.11.
selects(X,Xs) - See Program 7.7. abolish(F,N) - See Exercise 12.5(i).
Program 21.1 Playing mastermind
414 Chapter 21
inconsistent(Guess) - query(Úld,Bulls,Cows), not bulls_and_cows_match(Old,Guess,Bulls,Cows).
The bulls match between a previous guess OldGuess and a conjectured guess Guess if the number of digits in the same position in the two guesses equals the number of Bulls in DidGuess. It is computed by the predicate exact_matches(DldGuess,Guess,Bulls). The cows match if the number of common digits without respect to order corresponds to the sum of Bulls and Cows; it is computed by the procedure bulls_ and_cows_match. It is easy to count the number of matchtug digits and common digits in two queries, using an all-solutions predicate size_ of/3.
The ask(Guess) procedure is a memo-function that records the answer to the query. It performs some limited consistency checks on the input with the procedure sensible/2 and succeeds only if four bulls are indi- cated. The expected syntax for the user's reply is a tuple (Bulls,Cows).
The remaining (top-level) predicates are for bookkeeping. The first, cleanup, removes unwanted information from previous games. The predicate announce tells how many guesses were needed, which is de- termined using size_of /3.
A more efficient implementation of the exact_matches and common_ members procedures can be obtained by writing iterative versions:
exact_matches(Xs,Ys,N) - exact_matches(Xs,Ys,O,N). exact_matches([XIXs] , [XIYsI ,K,N) -
Kl is K+l, exact_matches(Xs,Ys,Kl,N). exact_matches([XIXs] , [YIYsI ,K,N) -
X Y, exact_matches(Xs,Ys,K,N). exact_matches([ ],[ ],N,N). common_members(Xs,Ys,N) - common_members(Xs,Ys,O,N).
comnaon_members([XIXs] ,Ys,K,N) - member(X,Ys), Kl is K+l, comnion_members(Xs,Ys,Kl,N).
comrnon_members([XIXs] ,Ys,K,N) - common_members (Xs , Ys , K, N).
comxnon_members([ I ,Ys,N,N).
Using the more efficient versions of exact_matches and common_ members saves about 10%-30% of the execution time.
415 Game-Playing Programs
21.2 Ni
We turn our attention now from mastermind to Nim, also a game for two players. There are several piles of matches, and the players take turns removing some of the matches (up to all) in a pile. The winner is the player who takes the last match. Figure 21.1 gives a common starting position, with piles of 1, 3, 5 and 7 matches.
To implement the Nim-playing program, we use the game-playing framework of Program 20.8.
The first decision is the representation of the game position and the moves. A natural choice for positions is a list of integers where elements of the list correspond to piles of matches. A move is a tu- pie (N,M) for taking M matches from pile N. Writing the procedure move(Move,Position,Positionl), where Position is updated to Posi- tioni by Move, is straightforward. The recursive rule counts down match piles until the desired pile is reached. The remaining piles of matches representing the new game position are computed routinely:
move((K,M) , [NINs] , [N1Ns1]) -
K > 1, Kl is K-1, move((K1,M),Ns,Nsl).
There are two possibilities for updating the specified pile of matches, the base case of the procedure. If all the matches are taken, the pile is removed from the list. Otherwise the new number of matches in the pile is computed and checked to be legal:
move((1,N), [NINs] ,Ns).
move((l,M), [NINs] , [Nl INs]) - N > M, Nl is N-M.
The mechanics of turns for two-person games is specified by two facts.
s
Figure 21.1 A starting position for Nim
s I I
416 Chapter 21
The initial piles of matches and who moves first must be decided by the two players. Assuming the computer moves second, the game of Figure 21.1 is specified as
initialize(niin, [1,3,5,7] ,opponent).
The game is over when the last match is taken. This corresponds to the game position being the empty list. The person having to move next is the loser, and the output messages of announce are formulated accord- ingly. The details are in Program 21.2.
It remains to specify how to choose the moves. The opponent's moves are accepted from the keyboard; how much flexibility is allowed in input is the responsibility of the programmer
choose_move (Pos ition,opponent, Move)
writeln(['Please make move']),
read (Move),
legal (Move,Position).
Choosing a move for the computer requires a strategy. A simple strat- egy to implement is taking all of the first pile of matches. It is recom- mended only for use against extremely poor players:
choose_move([NINs] ,computer, (1,N)).
A wirming strategy is known for Nim. It involves dividing game states, or positions, into two classes, safe and unsafe. To determine if a position is safe or unsafe, the binary representation of the number of matches in each pile is computed. The fErn-sum of these binary numbers is then calculated as follows. Each column is summed independently modulo 2. If the total in each colunm is zero, the position is safe. Otherwise the position is unsafe.
Figure 21.2 illustrates the process for the four piles of matches in Figure 21.1. The binary representations of 1, 3, 5, and 7 are 1, 11, 101, and 111 respectively. Calculating the mm-sum: there are four l's in the units column, two l's in the 2's colunm and two l's in the 4's colunm; an even number of l's in each. The mm-sum is zero, making the position [1,3,5,7] safe. On the other hand the position [2,61 is unsafe. The binary representations are 10 and 110 Summing them gives one i in the 4's column and two l's in the 2's colunm. The single i in the 4's column makes the position unsafe.
417 Game-Playing Programs
play(Game) '- See Program 20.8. Filling in the game-playing framework
initialize(nim, [1,3,5,7'] opponent). display_gaine(Position,X) write(Position), nl. ganie_over([ I ,Player,Player). anuounce(computer) - write('You won! Congratulations.'), nl. annol.mco(opponent) - write('I won.'), nl. Choosing moves
choose_move (Position,opponent Move) writeln(['Please make move']), read(Move), legal(Move,Position).
legal((K,N) Position) - nth_member(K,Position,M), N M. nth_member(1, [XIXs] ,X). nth_rnember(N,[XIXs],Y) N > 1, Nl is N-1, nth_member(N1,Xs,Y).
choose_move (Position, computer,Move) evaluate (Position,Saf ety,Suxri) decide_move(Safety,Position,Sum,Move).
evaluate(Positïon,Safety,Suin) - nim_sum(Position, [ I ,Suxn) , sa±ety(Sum,Salety)
safety(Sum,safe) - zero(Sum), !. safety(Suxn,unsafe) - not zero(Suin), decide_move(safe Position,Sum, (1,1))
'h The computer's ''arbitrary nove'' decide_move(unsaíe,Position,Sum,Move) -
saíe_move(Position,Sum,Move).
move(Move,Position,Positionl) Position i is the resi.ilt of executing the move Move from the current Position.
move((K,M),[NIN5],[NINs1]) - K > 1, Kl is K-1, move((K1,M),Ns,Nsl).
move1,N) [N INs] Ns). move((1,M),[NINs],[N1INs]) -
N > M, Nl is N-M.
next_player(computer,opponent). next_player(opponent computer).
Program 21.2 A program for playing a winning game of Nim
418 Chapter 21
nim_sum(Position,SoFar,Sum) - Sum is the nim-sum of the current Position, and SoFar is an accumulated value.
nim_sum([NINs] ,Bs,Sum) -
binary(N,Ds), nim_add(Ds,Bs,Bsl), nim_sum(Ns,Bsl,Sum).
nim_sum([ ] ,Sum,Sum).
nim_add(Bs, E I ,Bs). nim_add([ ],Bs,Bs).
nim_add([BIBs] , [CICs] , [DIDs]) -
D is (Bi-C) mod 2, nim_add(Bs,Cs,Ds).
binary(l, [1]).
binary(N,[DIDs]) -
N > i, D is N mod 2, Nl is N/2, binary(Ni,Ds).
decimal(Ds,N) - decimal(Ds,O,l,N).
decimal([ ],N,T,N).
decimal([DIDs],A,T,N) - Al is A+D*T, Ti is T*2, decimal(Ds,A1,Ti,N).
zero([ J).
zero([OIZs]) - zero(Zs).
safe_move (Position,NimSum,Move) - Move is a move from the current Position with the value NimSum that leaves a safe position.
safe_move(Piles,NimSum,Move) -
safe_move(Piles,NimSum, i ,Move).
safe_move([PileIPiles] ,NimSum,K,(K,M)) -
binary(Pile,Bs), can_zero(Bs,NimSum,Ds,O), decimal(Ds,M).
safe_move([PileIPiles] ,NimSum,K,Move)
Kl is K+l, safe_move(Piles,NimSum,Kl,Move).
can_zero([ ],NimSum,[ ],O) -
zero(NimSum).
can_zero([BIB5L[OlNimSuna],[CIDs],C) - can_zero (Es, NimSum , Ds , C).
can_zero([BIBs] , [ilNimSum] , [DIDs] ,C) -
D is i-B*C, Cl is l-B, can_zero(Bs,NimSuxn,Ds,Ci).
Program 21.2 (Continued)
419 Game-Playing Programs
000 Figure 21.2 Computmg mm-sums
The winning strategy is to always leave the position safe. Any unsafe position can be converted to a safe position (though not all moves do), while any move from a safe position creates an unsafe one. The best strategy is to make an arbitrary move when confronted with a safe posi- tion, hoping the opponent will blunder, and to convert unsafe positions to safe ones.
The current position is evaluated by the predicate evaluate/3, which determines the safety of the current position. An algorithm is needed to compute the nim-sum of a position. The nim-sum is checked by the predicate safety(Sum,Safety), which labels the position safe or unsafe depending on the value of Sum.
choose_move (Position,computer, Move) -
evaluate (Poition,Safety, Sum),
decide_move (Saf ety, Position , Sum , Move)
The move made by the computer computed by decide_move/4 de- pends on the safety of the position. If the position is safe, the computer makes the "arbitrary" move of one match from the first pile. If the posi- tion is unsafe, an algorithm is needed to compute a move that converts an unsafe position into a safe one. This is done by safe_move/3.
In a prior version of the program evaluate did not return Sum. In the writing of safe_move it transpired that the mm-sum was helpful, and it was sensible to pass the already computed value rather than recomput- ing it.
The nim-sum is computed by nim_sum(Ns,SoFar,Suzn). The relation computed is that Sum is the mm-sum of the numbers Ns added to what has been accumulated in SoFar. To perform the additions, the numbers must first be converted to binary, done by biriary/2:
nìm_sum([N INs] ,Bs,Sum) -
binary(N,Ds), nim_add(Ds,Bs,Bsl), nim_sum(Ns,Bsl,Sum).
420 Chapter 21
The binary form of a number is represented here as a list of digits. To overcome the difficulty of adding lists of unequal length, the least signif- icant digits are earliest in the list. Thus 2 (in binary 10) is represented as [0,1], while 6 is represented as [0,1,1]. The two numbers can then be added from least significant digit to most significant digit, as is usual for addition. This is done by nim_add/3 and is slightly simpler than regu- lar addition, since no carry needs to be propagated. The code for both binary and nim_add appears in Program 21.2.
The nim-sum Suni is used by the predicate safe_move (Ns ,Sum ,Move) to find a winning move Move from the position described by Ns. The piles of matches are checked in turn by safe_move/4 to see if there is a number of matches that can be taken from the pile to leave a safe position. The interesting clause is
safe_move([PileIPiles] ,NimSuni,K,(K,M)) -
binary(Pile,Bs), can_zero(Bs,NimSum,Ds,O), decimal(Ds,M).
The heart of the program is can_zero(Bs,NimSum,Ds,Carry). This re- lation is true if replacing the binary number Bs by the binary number Ds would make NimSum zero. The number Ds is computed digit by digit. Each digit is determined by the corresponding digit of Bs, NimSum, and a carry digit Carry initially set to 0. The number is converted to its decimal equivalent by decimal/2 in order to get the correct move.
Program 21.2 is a complete program for playing Nim interactively in- corporating the winning strategy. As well as being a program for playing the game, it is also an axiomatization of what constitutes a winning strat- egy.
21.3 Kalah
We now present a program for playing the game of Kalah that uses alpha- beta pruning. Kalah fits well into the paradigm of game trees for two reasons. First, the game has a simple, reasonably reliable evaluation func- tion, and second, its game tree is tractable, which is not true for games such as chess and go. It has been claimed that some Kalah programs are unbeatable by human players. Certainly, the one presented here beats us.
Kalah is played on a board with two rows of six holes facing each other. Each player owns a row of six holes, plus a kalah to the right of the holes.
421 Game-Playing Programs:: Ô Ô
Ô Ô
Figure 21.3 Board positions for Kalah
In the initial state there are six stones in each hole and the two kalahs are empty. This is pictured in the top half of Figure 21.3.
A player begins his move by picking up all the stones in one of his holes. Proceeding counterclockwise around the board, he puts one of the picked-up stones in each hole and in his own kalah, skipping the opponent's kalah, until no stones remain to be distributed. There are three possible outcomes. If the last stone lands on the kalah, the player has another move. If the last stone lands on an empty hole owned by the player, and the opponent's hole directly across the board contains at least one stone, the player takes all the stones in the hole plus his last landed stone and puts them all in his kalah. Otherwise the player's turn ends, and his opponent moves.
The bottom kalah board in Figure 21.3 represents the following move from the top board by the owner of the top holes. He took the six stones in the rightmost hole and distributed them, the last one ending in the kalah, allowing another move. The stones in the fourth hole from the right were then distributed.
If all the holes of a player become empty (even if it is not his turn to play), the stones remaining in the holes of the opponent are put in the opponent's kalah and the game ends. The winner of the game is the first player to get more than half the stones in his kalah.
422 Chapter 21
The difficulty for programming the game in Prolog is finding an effi- cient data structure to represent the board, to facilitate the calculation of moves. We use a four-argument structure board (Holes , Kalah, Opp- Holes, OppKalah), where Holes is a list of the numbers of stones in your six holes, Kalah is the number of stones in your kalah, and OppHoles and OppKalah are, respectively, the lists of the numbers of stones in the opponent's holes and the number of stones in his kalah. Lists were cho- sen rather than six-place structures to facilitate the writing of recursive programs for distributing the stones in the holes.
A move consists of choosing a hole and distributing the stones therein. A move is specified as a list of integers with values between 1 and 6 inclusive, where the numbers refer to the holes. Hole i is farthest from the player's kalah, while hole 6 is closest. A list is necessary rather than a single integer because a move may continue. The move depicted in Figure 21.3 is [1,4].
The code gives all moves on backtracking. The predicate stones_in_ hole(M,Board,N) returns the number of stones N in hole M of the Board if N is greater than O, failing if there are no stones in the hole. The predicate ext end_move (M ,Board, N, Ms) returns the continuation of the move Ms. The second clause for move handles the special case when all the player's holes become empty during a move.
Testing whether the move continues is nontrivial, since it may involve all the procedures for making a move. If the last stone is not placed in the kalah, which can be determined by simple arithmetic, the move will end, and there is no need to distribute all the stones. Otherwise the stones are distributed, and the move continues recursively.
The basic predicate for making a move is distribute_stones (Stones, N ,Board, Board 1), which computes the relation that Board i is obtained from Board by distributing the number of stones in Stones starting from hole number N. There are two stages to the distribution, putting the stones in the player's holes, distribute_my_holes, and putting the stones in the opponent's holes, distribute_your_holes.
The simpler case is distributing the stones in the opponent's holes. The holes are updated by distribute, and the distribution of stones continues recursively if there is an excess of stones. A check is made to see if the player's board has become empty during the course of the move, and if so, the opponent's stones are added to his kalah.
Distributing the player's stones must take into account two possibili- ties, distributing from any particular hole, and continuing the distribu-
423 Game-Playing Programs
tion for a large number of stones. The pick_up_and_distribute pred- icate is the generalization of distribute to handle these cases. The predicate check_capture checks if a capture has occurred and updates the holes accordingly; update_kalah updates the number of stones in the player's kalah. Some other necessary utilities such as ri_substitute are also included in the program.
The evaluation function is the difference between the number of stones in the two kalahs:
value(board(H,K,Y,L),Value) - Value is K-L.
The central predicates have been described. A running program is now obtained by filling in the details for I/O, for initializing and terminating the game, etc. Simple suggestions can be found in the complete program for the game, given as Program 21.3.
[n order to optimize the performance of the program, cuts can be added. Another tip is to rewrite the main loop of the program as a failure- driven ioop rather than a tail recursive program. This is sometimes nec- essary in implementations that do not incorporate tail recursion opti- mization and a good garbage collector.
21.4 Background
The mastermind program, slightly modified, originally appeared in SIGART (Shapiro, 1983d) in response to a program for playing master- mind in Pascal. The SIGART article provoked several reactions, both of theoretical improvements to algorithms for playing mastermind and practical improvements to the program. Most interesting was an analy- sis and discussion by Powers (1984) of how a Prolog program could b rewritten to good benefit using the mastermind code as a case study. Eventually, speedup by a factor of 50 was achieved.
A proof of the correctness of the algorithm for playing Nim can be found in any textbook discussing games on graphs, for example, Berge (1962).
Kalah was an early AI target for game-playing programs (Slagle and Dixon, 1969).
424 Chapter 21
Play framework
play(Game) See Program 20.8.
Choosing a move by minimax with alpha-beta cutoff
choose_move (Position,computer, Move)
lookahead(Depth),
alpha_beta(Depth,Position,-40,40,Move,Value),
nl, write(Move), nl.
choose_move (Position,opponent, Move)
nl, writeln(['please nake move']), read(Move), legal(Move).
alpha_beta(Depth,Position,Alpha,Beta,Move,Value)
See Program 20.11.
move(Board, [MIMs])
member(M, [1,2,3,4,5,6]),
stones_in_hole (M,Board,N),
extend_move (N ,M, Board, Ms)
nove(board([O,O,O,O,O,O],K,Ys,L),[]).
stones_in_hole(M,board(Hs,K,Ys,L),Stones)
nth_nember(M,Hs,Stones), Stones > O.
extend_move(Stones,M,Board,[ J)
Stones =\ (7-M) nod 13,
extend_move(Stones ,M,Board,Ms)
Stones (7-M) mod 13, !,
distribute_stones(Stones ,M,Board,Boardl),
move(Boardl,Ms).
Executing a move
move([N Nsj ,Board,FinalBoard)
stones_in_hole (N, Bod,Stones),
distribute_stones (Stones, N ,Board, Boardi),
move (Ns, Boardi , FinalBoard).
move([ ],Boardl,Board2)
swap(Eoardl ,Board2).
distribute_stones (Stones,Hole,Board,Boardl) Boardi is the result of distributing the number of stones Stones from Hole from the current Board. It consists of two stages: distributmg the stones in the player's holes, distribute_my_holes, and distributing the stones in the opponent's holes, distribute_your_holes.
Program 21.3 A complete program for playing Kalah
425 Garne-PlQying Progrcims
distribute_stones(Stones,Hole,Board,FinalBoard) distribute_my_holes(Stones,Hole,Board,Boardl,Stonesl), distribute_your_holes (Stonesi ,Boardl ,FinalBoard).
distribute_my_holes(Stones,N,board(Hs,K,Ys,L), board(Hsl,Kl,Ys,L),Stonesl) -
Stones > 7-N, pick_up_axid_distribute(N,Stones,Hs,Hsl), Kl is K+1, Stonesi is Stones-fN-7.
dïstribute_my_holes(Stones,N,board(Hs,K,Ys,L),Board,O) - Stones 7-N, pïckup_ancLdistribute(N,Stones,Hs,Hsl), check_capture(N,Stones,Hsl,Hs2,Ys,Ysi,Pieces), update_kalah(Pieces,N,Stones,K,Kl) checkïffinished(board(Hs2,Kl,Ysl,L),Board).
checkcapture(N,Stones,Hs,Hsl,Ys,Ysl,Pieces) Finishingllole is N+Stones, nth_rnernber(FinishingHole,Hs, 1), OppositeHole is 7-FinishingHole, ath_member (OppositeHole , Ys , Y), Y> O, I, n_substitute(OppositeHole,Ys ,O,Ysl) n_substitute (FinishingHole,Hs ,O,Hsl), Pieces is Y+l.
check_capture(N,Stones,Hs,Hs,Ys,Ys,O)
check.if_finished(board(Hs,K,Ys,L),board(Hs,K,Hs,Ll)) - zero(Hs) , ! , suinlist(Ys,YsSum) , Li is L-i-YsSum.
check_iLfinished(board(Hs,K,Ys,L),board(Ys,Kl,Ys,L)) - zero(Ys), !, sunilist(Hs,HsSun), Kl is K+HsSum.
check_if_f inished(Soard,Board) update_kalah(O,Stones,N,K,K) '- Stones < 7-N, !. update_kalah(O,Stones,N,K,Kl) - Stones 7-N, !, Kl is K+l. update_kalah(Pieces,Stones,N,K,Kl) - Pieces > O, , Kl is K+Pieces. distributeyour_holes(O,Board,Board) - ! distributeyour_holes(Stones,board(Hs,K,Ys,L),bosrd(Hs,K,Ysl,L))
i Stones, Stones < 6, non_zero(Hs), ! distribute (Stones,Ys,Ysl)
Program 21.3 (Continued)
426 Chapter 21
distribute_your_holes (Stones ,board(Hs,K,Ys,L) ,Board) -
Stones > 6, !,
distribute(6,Ys,Ysl),
Stonesi is Stones-6,
distribute_stones(Stonesl ,O,board(Hs ,K,Ysi ,L) ,Board).
distribute_your_holes(Stones,board(Hs,K,Ys,L),board(Hs,K,Hs,Ll)) -
zero(Hs), !, sumlist(Ys,YsSuni), Li is Stones+YsSum+L.
Lower-level stone distribution
pick_up_and_distribute(O,N,Hs,Hsi) -
!, distribute(N,Hs,Hsl).
pick_up_and_distribute(l,N, [HIHs] ,[OIHsi]) -
distribute(N,Hs,Hsi).
pick_up_and_distribute(K,N,[HIHs] ,[HIHs1]) -
K > i, !, Kl is K-1, pick_up_and_distribute(Kl,N,Hs,Hsi).
distribute(O,Hs,Hs) -
distribute(N, [HIHs] , [HhIHsl]) -
N > O, !, Nl is N-i, Hl is H+l, distribute(Nl,Hs,Hsl).
distribute(N,[ ],[ ]) - L
Evaluation function
value(board(H,K,Y,L),Value) - Value is K-L.
Testing for the end of the game game_over(board(O,N,O,N) ,Player,draw) -
pieces(K), N 6*K,
game_over(board(H,K,Y,L) ,Player,Player) -
pieces(N), K > 6*N, !.
gameover(board(H,K,Y,L) ,Player,Opponent) -
pieces(N), L > 6*N, next_player(Player,Opponent).
announce(opponent) - writeln( ['You won! Congratulations.']).
announce(computer) - writeln(['I won.']).
announce(draw) - writeln(['The game is a draw']).
Miscellaneous game utilities
nthjnember(N, [HIHs] ,K) -
N > 1, !, Ni is N-1, nthmember(Nl,Hs,K).
nthmember(l, [HIHs] ,H).
n_substitute(l, [XIXs] ,Y, [YIXs]) - !
n_substitute(N, [XXs] ,Y, [XIXs1]) -
N > 1, !, Ni is N-1, n_substitute(Nl,Xs,Y,Xsi).
Program 21.3 (Continued)
427 Game-Playing Programs
next_player(computer opponent).
next_player (opponent ,computer).
legal([NINsJ) - O < N, N < 7, legal(Ns).
legal([ ]).
swap(board(Hs,K,Ys,L) ,board(Ys,L,Hs ,K)).
display_game (Position, computer)
show (Posit ion)
display_game (Position, opponent)
swap(Position,Positionl) , show(Positionl)
show(board(H,K,Y,L)) -
reverse(H,HFt), write_stones(HR),
write_kalahs(K,L) , write_stones(Y)
write_stones (H)
nl, tab(5), display_holes(H).
display_holes( [H IHs]) -
write_pile (H) , display_holes(Hs)
display_holes([ J) - nl. write_pile(N) N < 10, write(N), tab(4)
write_pile(N) - N 10, write(N), tab(3)
write_kalahs(K,L)
write(K), tab(34), write(L), nl.
zero([O,O,0,0,O,O]).
non_zero(Hs) - Hs [0,0,0,0,0,0]
Initializing lookahead(2).
initialize(kalali,board([N,N,N,N,N,N],O,[N,N,N,N,N,N],O),opponent) -
pieces (N)
pieces (6).
Program 21.3 (Continued)
22 A Credit Evaluation Expert System
22.1 Developing the System
This chapter presents an account of developing a prototype expert sys- tem. The example comes from the world of banking: evaluating requests for credit from small business ventures. We give a fictionalized account of the development of a simple expert system for evaluating client re- quests for credit from a bank. The account is from the point of view of Prolog programmers, or knowledge engineers, commissioned by the bank to write the system. It begins after the most difficult stage of building an expert system, extracting the expert knowledge, has been under way for some time. In accordance with received wisdom, the programmers have been consulting with a single bank expert, Chas E. Manhattan. Chas has
When the first edition of this book was published, there was a surge of activity in the application of artificial intelligence to industry. Of partic- ular interest were expert systemsprograms designed to perform tasks previously allocated to highly paid human experts. One important fea- ture of expert systems is the explicit representation of knowledge.
This entire book is relevant for programming expert systems. The ex- ample programs typify code that might be written. For instance, the equation-solving program of Chapter 23 can be, and has been, viewed as an expert system. The knowledge of expert systems is often expressed as rules. Prolog whose basic statements are rules is thus a natural language for implementing expert systems.
430 Chapter 22
told us that three factors are of the utmost importance in considering a request for credit from a client (a small business venture).
The most important factor is the collateral that can be offered by the client in case the venture folds. The various types of collateral are di- vided into categories. Currency deposits, whether local or foreign, are first-class collateral. Stocks are examples of second-class collateral, and the collateral provided by mortgages and the like is illiquid.
Also very important is the client's financial record. Experience in the bank has shown that the two most important factors are the client's net worth per assets and the current gross profits on sales. The client's short-term debt per annual sales should be considered in evaluating the record, and slightly less significant is last year's sales growth. For knowl- edge engineers with some understanding of banking, no further expla- nation of such concepts is necessary. In general, a knowledge engineer must understand the domain sufficiently to be able to communicate with the domain expert.
The remaining factor to be considered is the expected yield to the bank. This is a problem that the bank has been working on for a while. Programs exist to give the yield of a particular client profile. The knowl- edge engineer can thus assume that the information will be available in the desired form.
Chas uses qualitative terms in speaking about these three factors: "The client had an excellent financial rating, or a good form of collateral. His venture would provide a reasonable yield," and so on. Even concepts that could be determined quantitatively are discussed in qualitative terms. The financial world is too complicated to be expressed only with the numbers and ratios constantly being calculated. In order to make judg- ments, experts in the financial domain tend to think in qualitative terms with which they are more comfortable. To echo expert reasoning and to be able to interact with Chas further, qualitative reasoning must be mod- eled.
On talking to Chas, it became clear that a significant amount of the expert knowledge he described could be naturally expressed as a mixture of procedures and rules. On being pressed a little in the second and third interviews, Chas gave rules for determining ratings for collateral and financial records. These involved considerable calculations, and in fact, Chas admitted that to save himself work in the long term, he did a quick initial screening to see if the client was at all suitable.
431 A Credit Evaluation Expert System
This information is sufficient to build a prototype. We show how these comments and observations are translated into a system. The top-level basic relation is credit(Client,Answer), where Answer is the reply given to the request by Client for credit. The code has three modules- collateral, financial_rating, and bank_yieldcorrespondirig to the three factors the expert said were important. The initial screening to determine that the client is worth considering in the first place is per- formed by the predicate ok_prof ile(Client). The answer Answer is then determined with the predicate evaluate (Prof ile,Answer), which evaluates the Profile built by the three modules.
Being proud knowledge engineers, we stress the features of the top- level formulation in credit/2. The modularity is apparent. Each of the modules can be developed independently without affecting the rest of the system. Further, there is no commitment to any particular data struc- ture, i.e., data abstraction is used. For this example, a structure pro f ile(C,F,Y) represents the profile of collateral rating C, the financial rating F, and the yield Y of a client. However, nothing central depends on this decision, and it would be easy to change it. Let us consider some of the modular pieces.
Let us look at the essential features of the collateral evaluation module. The relation collateral_ratirig/2 determines a rating for a particu- lar client's collateral. The first step is to determine an appropriate pro- file. This is done with the predicate collateral_profile, which classi- fies the client's collateral as first_class, second_class, or illiquid and gives the percentage each covers of the amount of credit the client requested. The relation uses facts in the database concerning both the bank and the client. In practice, there may be separate data- bases for the bank and the client. Sample facts shown in Program 22.1 indicate, for example, that local currency deposits are first-class collateral.
The profile is evaluated to give a rating by collateral_evaluation. lt uses rules of thumb to give a qualitative rating of the collateral: excellent, good, etc. The first collateral_evaluation rule, for example, reads: "The rating is excellent if the coverage of the requested credit amount by first-class collateral is greater than or equal to loo percent."
Two features of the code bear comment. First, the terminology used in the program is the terminology of Chas. This makes the program (al- most) self-documenting to the experts and means they can modify it with
432 Chapter 22
Credit Evaluation
credit (Client,Answer) Answer is the reply to a request by Client for credit.
credit (Client ,Answer) -
ok_profile (Client),
collateral_rating (Client ,Collateralhating),
financial_rating(Client ,FinancialRating),
bank_yield (Client ,Yield),
evaluate(profile(CollateralRating,FinancialRating,Yield) Answer).
The collateral rating module
collateraL rating ( Client,Rating) - Rating is a qualitative description assessing the collateral offered by Client to cover the request for credit.
collateral_rating (Client Rating)
collateral_profile (Client, FirstClass , SecondClass , Illiquid),
collateral_evaluation (FirstClass ,SecondClass , Illiquid,Rating).
collateral_prof ile(Client ,FirstClass,SecondClass,Illiquid) -
requested_credit (Client ,Credit),
collateral_percent (f irst_class,Client ,Credit ,FirstClass),
collateral_percent(second_class,Clïent ,Credit,SecondClass),
collateral_percent(illiquid,Client,Credit ,Illiquid).
collateral_percent (Type ,Client Total ,Value) -
findall(X, (collateral(Collateral Type),
amount (Collateral,Client,X)) ,Xs) sumlist(Xs,Suni),
Value is Sum*100/Total.
Evaluation rules
collateral_evaluation(FirstClass , SecondClass ,Illiquid ,excellent) - FirstClass loo.
collateral_evaluation(FirstClass,SecondClass ,Illiquid,excellent) -
FirstClass > 70, FirstClass + SecondClass 100.
collateral_evaluation(FirstClass ,SecondClass,Illiquid,good) -
FirstClass + SecondClass > 60,
FirstClass + SecondClass < 70,
FirstClass + SecondClass + Illiquid 100.
Bank data classification of collateral
collateral (local_currency_deposits ,first_class)
collateral (foreign_currency_deposits ,first_class)
collateral (negotiate_instruments ,second_class).
collateral (mortgage, illiquid).
Program 22.1 A credit evaluation system
433 A Credit Evaluation Expert System
Financial rating financial_rating C Client,Rating)
Rating is a qualitative description assessing the financial record offered by Client to support the request for credit.
financial_rating(Client Rat ing)
financial_f actors (Factors)
score (Factors ,Client, 0, Score),
calibrate (Score , Rating)
Financial evaluation rules
calibrate(Score,bad) Score -500. calibrate(Score,mediuin) - -500 < Score, Score < 150. calibrate(Score,good) 150 Score, Score < 1000.
calibrate(Score,excellent) - Score 1000.
Bank data - weighting factors
financial_factors(E(net_worth_per_assets,5), (last_year_sales_growth, 1), (gross_prof its_on_sales,5), (short_term_debt_per_annual_sales, 2) 1).
score([(Factor,Weight)IFactorsl ,Client,Acc,Score) -
value (Factor,Client ,Value)
Acci is Acc + Weight*Value,
score (Factors,Client,Accl ,Score)
score([ I ,Client,Score,Score)
Final evaluation
evaluate (Pro file,Outcome) Outcome is the reply to the client's Profile.
evaluate(Profile Answer)
rule(Conditions,Answer), verify(Conditions,Profile).
verify([condition(Type,Test,Rating)IConditions] ,Profile) -
scale(Type,Scale),
select_value (Type Profile ,Fact)
compare(Test ,Scale Fact Rating),
verify(Conditions Profile).
verify([ I Profile)
compare('=',Scale,Rating,Rating).
compare('>' Scale,Ratingl,Rating2) -
precedes(Scale,Ratingl ,Rating2).
compare('' Scale,Ratingl,Rating2) -
precedes(Scale,Ratingl,Rating2) Ratingi Rating2.
Program 22.1 (Continued)
434 Chapter 22
compare('<' ,Scale,Ratingi,Rating2) -
precedes(Scale,Rating2,Ratingl).
compare('' ,Scale,Ratingl,Rating2) -
precedes(Scale,Rating2,Ratingl) ; Ratingi = Rating2.
precedes([R1IRs] ,R1,R2).
precedes([RIRs],Ri,R2) - R R2, precedes(Rs,Rl,R2).
select_value(collateral,profile(C,F,Y) ,C)
select_value(finances,profile(C,F,Y) ,F).
select_value(yield,profile(C,F,Y) ,Y).
Utilities
sumlist(Xs,Sum) - See Program 8.6b.
Bank data and rules
rule([condition(collateral, '' ,excellent), condition(finances, '' ,good), condition(yield, '' ,reasonable)] ,give_credit).
rule([condition(collateral, ,good) ,condition(finances, C ,good),
condition(yield, '' ,reasonable)] ,consult_superior). rule ( [conditíon(collateral, '' ,moderate),
condition(finances, '' ,mediuin)] refuse_credit)
scale(collateral, [excellent,good,moderate]).
scale(finances, [excellent ,good,medium,bad]).
scale (yield, [excellent ,reasonable, poor]).
Program 22.1 (Continued)
little help from the knowledge engineer. Allowing people to think in do- main concepts also facilitates debugging and assists in using a domain- independent explanation facility as discussed in Section 17.4. Second, the apparent naivete of the evaluation rules is deceptive. A lot of knowledge and experience are hidden behind these simple numbers. Choosing poor values for these numbers may mean suffering severe losses.
The financial evaluation module evaluates the financial stability of the client. It uses items taken mainly from the balance and profit/loss sheets. The financial rating is also qualitative. A weighted sum of financial fac- tors is calculated by score and used by calibrate to determine the qualitative class.
It should be noted that the modules giving the collateral rating and the financial rating both reflect the point of view and style of a particular
435 A Credit Evaluation Expert System
expert, Chas Manhattan, rather than a universal truth. Within the bank there is no consensus about the subject. Some people tend to be conser- vative and some are prepared to take considered risks.
Programming the code for determining the collateral and financial rat- ings proceeded easily. The knowledge provided by the expert was more or less directly translated into the program. The module for the overall evaluation of the client, however, was more challenging.
The major difficulty was formulating the relevant expert knowledge. Our expert was less forthcoming with general rules for overall evaluation than for rating the financial record, for example. He happily discussed the profiles of particular clients, and the outcome of their credit requests and loans, but was reluctant to generalize. He preferred to react to sug- gestions rather than volunteer rules.
This forced a close reevaluation of the exact problem we were solving. There were three possible answers the system could give: approve the request for credit, refuse the request, or ask for advice. There were three factors to be considered. Each factor had a qualitative value that was one of a small set of possibilities. For example, the financial rating could be bad, medium, good, or excellent. Further, the possible values were ranked on an ordinal scale.
Our system clearly faced an instance of a general problem: Find an outcome from some ordinal scale based on the qualitative results of several ordinal scales. Rules to solve the problem were thus to give a conclusion based on the outcome of the factors. We pressed Chas with this formulation, and he rewarded us with several rules. Here is a typical one: "If the client's collateral rating is excellent (or better), her financial rating good (or better), and her yield at least reasonable, then grant the credit request."
An inirnediate translation of the rule is
evaluate (profile (excellent,good,reasonable) ,give_credit).
But this misses many cases covered by the rule, for example, when the client's profile is (excellent,good,excellent). All the cases for a given rule can be listed. lt seemed more sensible, however, to build a more general tool to evaluate rules expressed in terms of qualitative values from ordinal scales.
436 Chapter 22
There is potentially a problem with using ordinal scales because of the large number of individual cases that may need to be speci- fied. If each of the N modules have M possible outcomes, there are NM cases to be considered. In general, it is infeasible to have a sep- arate rule for each possibility. Not only is space a problem for so many rules but the search involved in finding the correct rule may be prohibitive. So instead we defined a small ad hoc set of rules. We hoped the rules defined, which covered many possibilities at once, would be sufficient to cover the clients the bank usually dealt with. We chose the structure rule(Conditions,Conclusion) for our rules, where Conditions is a list of conditions under which the rule applies and Conclusion is the rule's conclusion. A condition has the form con- dition(Factor,Relation,Rating), insisting that the rating from the factor named by Factor bears the relation named by Relation to the rating given by Rating.
The relation is represented by the standard relational operators: <, =, >, etc. The previously mentioned rule is represented as
rule([condition(collateral, '' ,excellent), condition(finances, '' ,good), condition(yield, '' ,reasonable)] ,give_credit).
Another rule given by Chas reads: "If both the collateral rating and fi- nancial rating are good, and the yield is at least reasonable, then consult your superior." This is translated to
rule( [condition(collateral, ' ,good),
condition(finances, '=' ,good), condition(yield, '' ,reasonable)] ,consult_superior).
Factors can be mentioned twice to indicate they lie in a certain range or might not be mentioned at all. For example, the rule
rule( [condition(collateral, '' ,moderate), condition(finances, '' ,medium)]
refuse_credit).
states that a client should be refused credit if the collateral rating is no better than moderate and the financial rating is at best medium. The yield is not relevant and so is not mentioned.
437 A Credit Evaluation Expert System
Client Data
bank_yield(clientl ,excellent) requested_credit (clienti 50000). amount (local_currency_deposits client i 30000). axnount(foreign_currencydeposits ,clienti ,20000). amount (bank_guarantees, clienti 3000).
axnount(negotiate_instruxnents,clientl,5000). amount(stocks,clientl,9000) amount (mortgage ,clienti , i2000). amount (do cuments client i, 14000).
value (net_worth_per_assets,clienti ,40). value(last_year_sales_growth,clientl 20). value (gross_prof its_on_sales,clienti 45). value(short_term_debt_per_annual_sales,clienti ,9). ok_profile(clienti).
Program 22.2 Test data for the credit evaluation system
The interpreter for the rules is written nondeterministically. The pro- cedure is: "Find a rule and verify that its conditions apply," as defined by evaluate. The predicate verify(Conditions,Profile) checks that the relation between the corresponding symbols in the rule and the ones that are associated with the Profile of the client is as specified by Con- ditions. For each Type that can appear, a scale is necessary to give the order of values the scale can take. Examples of scale facts in the bank database are scale(collateral, [excellent,good,Inoderate]) and scale(finances, [excellent,good,mediuni,bad]). The predicate select_value returns the appropriate symbol of the factor under the or- dinality test that is performed by compare. lt is an access predicate, and consequently the only predicate dependent on the choice of data struc- ture for the profile.
At this stage, the prototype program is tested. Some data from real clients are necessary, and the answer the system gives on these individ- uals is tested against what the corresponding bank official would say. The data for clienti is given in Program 22.2. The reply to the query credit(clientl,X) isX = give_credit.
Our prototype expert system is a composite of styles and methods - not just a backward chaining system. Heuristic rules of thumb are used
438 Chapter 22
to deterrnme the collateral rating; an algorithm, albeit a simple one, is used to deterrmne the financial rating; and there is a rule language, with an interpreter, for expressing outcomes in terms of values from discrete ordinal scales. The rule interpreter proceeds forward from conditions to conclusion rather than backward as in Prolog. Expert systems must become such composites in order to exploit the different forms of knowl- edge already extant.
The development of the prototype was not the only activity of the knowledge engineers. Various other features of the expert system were developed in parallel. An explanation facility was built as an extension of Program 17.22. A simulator for rules based on ordinal scales was built to settle the argument among the knowledge engineers as to whether a reasonable collection of rules would be sufficient to cover the range of outcomes in the general case.
Finally, a consistency checker for the rules was built. The following meta-rule is an obvious consistency principle: "If all of client A's factors are better than or equal to client B's, then the outcome of client A must be better than or equal to that of client B."
22.2 Background
More details on the credit evaluation system can be found in Ben-David and Sterling (1986).
23 An Equation Solver
A very natural area for Prolog applications is symbolic manipulation. For example, a Prolog program for symbolic differentiation, a typical symbol manipulation task, is just the rules of differentiation in different syntax, as shown in Program 3.30.
In this chapter, we present a program for solving symbolic equations. It is a simplification of PRESS (PRolog Equation Solving System), developed in the mathematical reasoning group of the Department of Artificial In- telligence at the University of Edinburgh. PRESS performs at the level of a mathematics student ni her final year of high school.
The first section gives an overview of equation solving with some exam- ple solutions. The remaining four sections cover the four major equation- solving methods implemented in the equation solver.
23.1 An Overview of Equation Solving
The task of equation solving can be described syntactically. Given an equation Lhs = Rhs in an unknown X, transform the equation into an equivalent equation X = Rhsl, where Rhsl does not contain X. This final equation is the solution. Two equations are equivalent if one is trans- formed into the other by a finite number of applications of the axioms and rules of algebra.
Successful mathematics students do not solve equations by blindly applying axioms of algebra. Instead they learn, develop, and use various methods and strategies. Our equation solver, modeling this behavior, is accordingly a collection of methods to be applied to an equation to be
440 Chapter 23
cos(x) (1 2 sin(x)) = O x2-3x+2=O
(lii) 22x - 5 2x+ + 16 = O
Figure 23.1 Test equations
solved. Each method transforms the equation by applying identities of algebra expressed as rewrite rules. The methods can and do take widely different forms. They can be a collection of rules for solving the class of equations to which the method is applicable, or algorithms implementing a decision procedure.
Abstractly, a method has two parts: a condition testing whether the method is applicable, and the application of the method itself.
The type of equations our program can handle are indicated by the three examples [n Figure 23.1. They consist of algebraic functions of the unknown, that is +, -, *, /, and exponentiation to an integer power, and also trigonometric and exponential functions. The unknown is x in all three equations.
We briefly show how each equation is solved. The first step in solving equation (i) in Figure 23.1 is factorization. The
problem to be solved is reduced to solving cos(x) = O and i - 2 sin(x) = O. A solution to either of these equations is a solution to the original equation.
Both the equations cos(x) = O and 1 - 2 . sin(x) = O are solved by mak- ing x the subject of the equation. This is possible because x occurs once in each equation.
The solution to cos(x) = O is arccos(0). The solution of 1 - 2 sin(x) = O takes the following steps:
i - 2 sin(x) = O, 2 . sín(x) = 1,
sin(x) = 1/2, x = arcsin(1/2).
In general, equations with a single occurrence of the unknown can be solved by an algorithmic method called isolation. The method repeatedly applies an appropriate inverse function to both sides of the equation
441 An Equation Solver
until the single occurrence of the unknown is isolated on the left-hand side of the equation. Isolation solves i - 2 sin(x) = O by producing the preceding sequence of equations.
Equation (ii) in Figure 23.1, x2 - 3 x + 2 = O, is a quadratic equation in x. We all learn in high school a formula for solving quadratic equations. The discriminant, b2 4 a c, is calculated, in this case (_3)2 4 . i 2, which equals 1, and two solutions are given: x = ((-3) + \/T)/2, which equals 2, and x = ((-3) - fÏ)/2, which equals 1.
The key to solving equation (iii) fri Figure 23.1 is to realize that the equation is really a quadratic equation in 2x. The equation 22'x - 5 2i+16=Ocanberewrittenas(2x)2_5.2.2x+16=O.Thiscanbe solved for 2, giving two solutions of the form 2X =Rhs, where Rhs is free of x. Each of these equations are solved for x to give solutions to equation (iii).
PRESS was tested on equations taken from British A-level examinations in mathematics. It seems that examiners liked posing questions such as equation (in), which involved the student's manipulating logarithmic, exponential, or other transcendental functions into fortiis where they could be solved as polynomials A method called homogenization evolved to solve equations of these types.
The aim of homogenization is to transform the equation into a poly- nomial in some term containing the unknown. (We simplify the more general homogenization of PRESS for didactic purposes.) The method consists of four steps, which we illustrate for equation (ili). The equa- tion is first parsed and all maximal nonpolynomial terms containing the unknown are collected with duplicates removed. This set is called the of- fenders set. In the example, it is 22), 2x+1}. The second step is finding a term, known as the reduced term. The result of homogenization is a poly- nomial equation in the reduced term. The reduced term in our example is 2V. The third step of homogenization is finding rewrite rules that express each of the elements of the offenders set as a polynomial in the reduced term. Finding such a set guarantees that homogenization will succeed. In our example the rewrite rules are 22x (2x)2 and 2x+1 = 2 . 2X Finally, the rewrite rules are applied to produce the polynomial equation.
We complete this section with a brief overview of the equation solver. The basic predicate is solve_equation(Equation,X,Solution). The re- lation is true if Solution is a solution to Equation in the unknown X. The complete code appears as Program 23.1.
442 Chapter 23
solve_equation (Equation, Unknown,Solution) - Solution is a solution to the equation Equation in the unknown Unknown.
solve_equation(A*B=O ,X ,Solution) -
factorize(A*B,X,Factors\[ J),
remove_duplicates (Factors ,Factorsl),
solve_factors(Factorsl ,X,Solution).
solve_equation(Equation,X,Solution)
single_occurrânce (X ,Equation),
position(X,Equation, [SidelPosition]),
maneuver_sides(Side ,Equation,Equationl),
isolate (Position,Equationl ,Solution).
solve_equation(Lhs=Rhs ,X Solution)
polynomial(Lhs,X),
polynomial (Rhs , X),
polynomial_normal_form (Lhs-Rhs , X, PolyForm),
solve_polynomial_equation(PolyForm,X,Solution).
solve_equation (Equation,X ,Solution) -
homogenize (Equation,X,Equationl ,X1),
solve_equation(Equationl Xl ,Solutionl),
solve_equation(Solutionl ,X,Solution).
The factorization method factorize (Expression,Subterm,Factors) -
Factors is a difference-list consisting of the factors of the multiplicative term Expression that contain the Subterm.
factorize(A*B,X,Factors\Rest) - ! , factorize(A,X,Factors\Factorsl),
factorize(B,X,Factorsl\Rest).
factorize(C,X, [ClFactors]\Factors) -
subterm(X,C), !.
factorize(C,X ,Factors\Factors).
solve_factors (Factors, Unknown,Solution) - Solution is a solution of the equation Factor = O in the Unknown for some Factor in the list of Factors.
solve_factors([FactorlFactors] ,X,Solution) -
solve_equation (Factor=O ,X,Solution).
solve_factors([FactortFactors] ,X,Solution)
solve_f actors (Factors ,X,Solution).
Program 23.1 A program for solving equations
443 An Equation Solver
The isolation method
maneuver_sides(i,Lhs = Rhs,Lhs Rhs) - maneuver_sides(2,Lhs Rhs,Rhs Lhs) - isolate([NlPosition],Equation,IsolatedEquation) -
isolax(N,Equation,Equationl), isolate(Positíon,Equationi,IsolatedEquatïon).
isolate([ 1 ,Equation,Equatiori). Axioms for isolation
isolax(1,-Lhs = Rhs,Lhs -Rhs). isolax(i,Terml+Term2 = Rhs,Termi = Rhs-Term2). isolax(2,Termi+Term2 Rhs,Term2 Rhs-Teruil).
isolax(1,Terml-Term2 Rhs,Tenni Rhs+Term2). isolax(2,Terml-Term2 Rhs,Term2 Terml-Rhs).
isolax(1,Terml*Term2 Rhs,Terml Rhs/Term2) Term2 O.
isolax(2,Termi'Term2 Rbs,Term2 Rhs/Terml) - Tenni O.
'h Unary minus
'I, Addition X Addition
X Subtraction X Subtraction 'I Multiplication
X Multiplication
isolax(1,TermllTerm2 = Rhs,Terml = RhsI(-Term2)). X Exponentiation
isolax(2,TerrniîTerm2 Rhs,Term2 log(base(Terml),Rhs)). 'h Exponentiation
isolax(l,sin(U) = V,U = arcsin(V)). X Sine isolax(1,sin(U) V,U Tr-arcsin(V)). X Sine isolax(1,cos(U) V,IJ = arccos(V)). 'h Cosine isolax(i,cos(U) V,tJ = -arccos(V)). X Cosine The polynomial method
polynomial(Term,X) See Program 11.4.
polynomiaL normal_form (Expression, Term ,PolyNormalForrn) PolyNormalForm is the polynomial normal form of Expression, which is a polynomial ¡n Term.
polynomial_normal_f orm(Polynomial ,X,NormalForm) - polynomial_f orm(Polynomial , X, PolyForm), remove_zero_terms(PolyForm,NormalForm)
polynomial_form(X,X, C(i,i)]). polynomial_form(XIN,X,[(i,N)]). polynomial_f orm(Terml+Term2 ,X,PolyForin) -
polynomial_f orm(Termi ,X ,PolyForml), polynomial_f ors (Term2,X,PolyForm2), add_polynomials (PolyFormi , PolyForm2 , PolyForm).
Program 23.1 (Continued)
444 Chapter 23
polynomial_f orm(Terml-Term2 ,X,PolyForm) - polynomial_f orm(Termt ,X,PolyForml), polynomial_form (Term2 , X, PolyForm2), subtract_polynomials (PolyFormi , PolyForm2 , PolyForm).
polynomial_form(Terml*Term2,X PolyForm) - polynomial_f orn (Termi ,X,PolyForml), polynomial_f orni(Term2 ,X,PolyForm2), multiply_polynomials(PolyForml ,PolyForm2,PolyForm).
polynomial_form(TermtN,X,PolyForm) - polynomial_f orm(Term,X ,PolyForml), binomial (PolyFormi , N, PolyForm).
polynomial_f orin(Tern,X, [(Term,O)]) - free_of (X,Term),
remove_zero_terms([(O,N)IPoly] Polyl) - !, remove_zero_terms(Poly,Polyi).
remove_zero_terms([(C,N)IPoly] , [(C,N)JPolyl]) - C O, , remove_zero_terms(Poly,Polyl).
remove_zero_terms([ il,[ ]). Polynomial manipulation routines add_polynomials (Polyl,Poly2,Poly)
Poly is the sum of Polyl and Po!y2, where Polyl, PoIy2, and Poly are all in polynomial form.
add_polynomials([ ],Poly,Poly) - add_polynoinials(Poly, [ ] ,Poly) - add_polynomials([(Ai,Ni)lPolyl] , [(Aj ,Nj)Poly2] ,[(Ai,Ni)IPoly]) -
Ni > Nj, !, add_polynomials(Polyl, [(Aj,Nj)IPoly2l,Poly). add_polynomials([(Ai,Ni)tPolyl],[(Aj,Nj)jPoly2] ,[(A,Ni)IPolyl) -
Ni =:= Nj, !, A is Ai+Aj, add_polynomials(Polyl,Poly2,Poly). add_polynomials([(Ai,Ni)IPolyl] ,f(Aj ,Nj)Poly2],[(Aj,Nj)IPoly])
Ni < Nj, ! , add_polynomials([(Ai,Ni)IPolyl] ,Poly2,Poly). subtract_polynomials (Polyl,Poly2,Poly) -
Po/y is the difference of Polyl and Poly2, where Polyl, Poly2, and Poly are all in polynomial form.
subtract_polynomials (Polyl , Poly2 , Poly) - multiply_single (Poly2, (1,O) ,Poly3), add_polynomials(Polyl,Poly3,Poly),
multiply_single(Polyl,Monomial,Poly) - Poly is the product of Po/yl and Monomial, where Polyl and Po/y are in polynomial form, and Monomial has the form (C,N) denoting the monomial C1XN.
multiply_single([(C1,N1)IPolyl],(C,N),[(C2,N2)IPoly]) - C2 is C1*C, N2 is N1+N, multiply_single(Polyl,(C,N),Poly).
multiply_single([ ],Factor,[ 1).
Program 23.1 (Continued)
445 Ari Equation Solver
multiply_polynomials (Polyl,Poly2,Poly) - Poly is the product of Polyl and Poly2, where Polyl, Poly2, and Poly are all in polynomial form.
multiply_polynomials([(C,N)lPolyl],Poly2,Poly) - multiply_single(Poly2, (C,N) ,Poly3), multiply_polynomials (Polyl , Poly2 , Poly4), add_polynomials (Poly3, Poly4 , Poly).
multiply_polynomials([ ],P,[ 1). binornial(Poly, i ,Poly).
Polynomial equation solver
solve_polynomial_equation (Equation, Unknown,Solution) - Solution is a solution to the polynomial Equation in the unknown Unknown.
solve_polynomial_equation(PolyEquation,X,X = - linear(PolyEquat ion), !, pad(PolyEquation, [(A,i) , (B4O)1).
solve_polyriomial_equation(PolyEquation,X, Solution) - quadratic(PolyEquation), !, pad(PolyEquation, [(A,2) , (B,1) , (C,O)]) discrimïnarit(A,B,C,Discriminant), root(X,A,B,C,Discriminant,Solution).
discriminant(A,fl,C,D) D is B*B - 4*A*C.
root(X,A,B,C,O,X= -B/(2*A)). root(X,A,B,C,D,X (-B+sqrt(D))/(2*A)) .- D > O. root(X,A,B,C,D,X (-B-sqrt(D))/(2*A)) - D > O. pad([(C,N) IPoly] , [(C,N) IPolyl])
pad(Poly,Polyl). pad(Poly,[(O,N)lPolylI)
pad(Poly,Polyl). pad([ J,[ 1). linear([(Coeff 1) IPoly]) quadratic( [(Coeff 2)1 Poly]) The homogenization method
homogenize (Equation,X,Equation 1,X1) - The Equation in X is transfoi iiied to the polynomial Equation i in Xl where Xi contains X.
homogenïze(Equation,X,Equationl Xl) - offenders (Equation,X,Dffenders), reduced_term(X,Dffenders,Type Xl), rewrite(Offenders,Type,Xi ,Substitutions), substitute(Equation,Substitutions ,Equationl)
Program 23.1 (Continued)
446 Chapter 23
offenders(Equation, Unknown,Offenders) Offenders is the set of offenders of the Equation in the Unknown.
offenders(Equation,X,Offenders) -
parse(Equation,X,Offendersl\[ 1), remove_duplicates(Offendersl ,Offenders),
multiple(Offenders).
reduced_term(X,Offenders,Type,X1) -
classify(Offenders,X,Type),
candidate(Type,Dffenders,X,X1).
Heuristics for exponential equations classify(Offenders,X,exponential) -
exponential_offenders(Offenders ,X).
exponential_offenders([AIBIOffs] ,X) -
free_of(X,A), subterm(X,B), exponential_offenders(Offs ,X).
exponential_offenders([ ],X).
candidate(exponential,Offenders,X,AIX) -
base (Offenders,A), polynomial_exponents(Offenders,X).
base([AIBlOffs] ,A) - base(Offs,A)
base([ ],A).
polynomial_exponents([AIBIOffs] ,X) -
polynomial(B,X), polynomial_exponents(Dffs,X).
polynomial_exponents( E ] ,X).
Parsing the equation and making substitutions parse (Expression, Term, Offenders) -
Expression is traversed to produce the set of Offenders in Term, that is, the nonalgebraic subterms of Expression containing Term.
parse(A+B,X,L1\L2) - 1, parse(A,X,L1\L3), parse(B,X,L3\L2).
parse(A*B,X,Li\L2)
!, parse(A,X,L1\L3), parse(B,X,L3\L2).
parse(A-B,X,Li\L2) -
!, parse(A,X,L1\L3), parse(B,X,L3\L2).
parse(A=B,X,L1\L2) -
parse(A,X,L1\L3), parse(B,X,L3\L2).
parse(AIB,X,L) -
integer(B), !, parse(A,X,L).
parse(A,X,L\L) -
freeof(X,A), !.
parse(A,X,EAIL]\L) -
subterm(X,A), !.
Program 23.1 (Continued)
447 An Equation Solver
substitute (Expression,Substitutions,Expressionl) - The list of Substitutions is applied to Expression to produce Expression 1.
substitute (A+B ,Subs, NewA+NewB) ', substitute(A,Subs,NewA), substitute(B,Subs,NewB).
substitute (A*B ,Subs, NewA*New) - substitute(A,Subs,NewA), substitute(B,Subs,NewB).
substitute (-B,Subs, NewA-NewB) - substitute(A,Subs,NewA), substitute(B,Subs,NewB).
substitute (A=B ,Subs, NewA=NewB) - substitute(A,Subs,NevA), substitute(B,Subs,NewB).
substituto(AIB,Subs,NewAIB) - iuteger(B), ! , substitute(A,Subs,NewA).
substitute(A,Subs,B) - member(A=B,Subs), L
substitute(A,Subs,A). Finding homogenization rewrite rules
rewrite([DffIOffs] ,Type,X1, [Off=TermlRewrites]) * homogenize_axiom(Type,Off ,X1 ,Terin), rewrite (Offs,Type,X1 ,Rewrites).
rewrite([ 1,Type,X,[ ]). Homogenization axioms
homogenize_axiom(exponential,A1(N*X) ,AIX, (AIX) TN) homogenize_axioin(exponeritial,At(-X),AIX,1/(AIX)). homogenize_axiom(exponential,A1(X+B),AIX,ATB*AIX).
Utilities
subterm(Sub,Term) See Program 9.2.
position(Term,Term,[ 1) - L position(Sub,Term,Path) -
compound(Term), functor(Term,F,N), position(N,Sub,Term,Path), positïon(N,Sub,Term, [NiPath]) -
arg(N,Terrn,Arg), position(Sub,Arg,Path) position(N,Sub,Terin,Path) -
N > 1, Nl is N-1, position(N1,Sub,Term,Path).
free_of(Subterm,Term) - occurrence(Subterm,Terrn,O). single_occurrerice(Subterm,Term) - occurrence(Subterm,Term,l). occurrence(Terin,Term,l) - occurrence(Sub,Term,N) -
cornpound(Terrn), ! , functor(Term,F,M), occurrence(M,Sub,Term,O,N). occurrence(Sub,Term,O) - Term Sub.
Program 23.1 (Continued)
448 Chapter 23
occurrence(M,Sub,Term,N1,N2) - M > O, !, arg(M,Term,Arg), occurrence(Sub,Arg,N), N3 is N+N1,
Ml is M-1, occurrence(M1,Sub,Term,N3,N2).
occurrence (O , Sub , Term ,N, N)
remove_duplicates(Xs,Ys) no_doubles(Xs,Ys).
no_doubles(Xs,Ys) - SeeProgram7.9. multiple([X1,X2IXs]).
Testing and data test_press(X,Y) - equation(X,E,U), solve_equation(E,U,Y).
equation(1,cos(x)*(l-2*sin(x))=O,x).
equation(2,x12-3*x+20,x).
equation(3,21(2*x)-5*21(x+1)+16=O,x).
Program 23.1 (Continued)
Program 23.1 has four clauses for solve_equation, one for each of the four methods needed to solve the equations in Figure 23.1. More generally, there is a clause for each equation-solving method. The full PRESS system has several more methods.
Our equation solver ignores several features that might be expected. There is no simplification of expressions, no rational arithmetic, no record of the last equation solved, no help facility, and so forth. PRESS does contain many of these facilities as discussed briefly in Section 23.6.
23.2 Factorization
Factorization is the first method attempted by the equation solver. Note that the test whether factorization is applicable is trivial, being unifica- tion with the equation A * B = O. If the test succeeds, the simpler equa- tions are recursively solved. The top-level clause implementing factoriza- tion is
solve_equation(A*B=O,X,Solution) -
factorize(A*B,X,Factors\[ 1), remove_duplicates (Factors ,Factorsl),
solve_factors (Factors 1, X, Solution).
449 An Equation Solver
The top-level clause in Program 23.1 has a cut as the first goal in the body. This is a green cut: none of the other methods depend on the success or failure of factorization. In general, we omit green cuts from clauses we describe in the text.
23.3 Isolation
A useful concept to locate and manipulate the single occurrence of the unknown is its position. The position of a subterm in a term is a list of argument numbers specifying where it appears. Consider the equation cos(x) = O. The term cos(x) containing x is the first argument of the equation, and x is the first (and only) argument of cos(x). The position of x in cos(x) = O is therefore [1,1]. This is indicated in the diagram in Figure 23.2. The figure also shows the position of x in i - 2 . sin(x) = O which is 11,2,2,1].
The clause defining the method of isolation is
solve_equation(Equation,X,Solution) -
single_occurrence (X, Equation)
position(X,Equation, [Side Position]),
maneuver_sides (Side , Equation, Equationi)
isolate (Position,Equationl ,Solution).
ZN cos O
Figure 23.2 Position of subterms in terms
450 Chapter 23
The condition characterizing when isolation is applicable is that there be a single occurrence of the unknown X in the equation, checked by single_occurrence. The method calculates the position of X with the predicate position. The isolation of X then proceeds in two stages. First, maneuver_sides ensures that X appears on the left-hand side of the equation, and second, isolate makes it the subject of the formula.
It is useful to define single_occurrence in terms of the more general predicate occurrence(Subterm,Term,N), which counts the number of times N that Subterm occurs in the term Term. Both occurrence and position are typical structure inspection predicates. Both are posed as exercises at the end of Section 9.2. Code for them appears in the utilities section of Program 23.1.
The predicate maneuver_sides (N ,Equation, Equationi) consists of two facts:
maneuver_sides(1,Lhs = Rhs,Lhs = Rhs).
maneuver_sides(2,Lhs = Rhs,Rhs = Lhs).
Its effect is to ensure that the unknown appears on the left-hand side of Equationi. The first argument N, the head of the position list, indicates the side of the equation in which the unknown appears. A i means the left-hand side, and the equation is left intact. A 2 means the right-hand side, and so the sides of the equation are swapped.
The transformation of the equation is done by isolate/3. It repeatedly applies rewrite rules until the position list is exhausted:
isolate([NlPosition] ,Equation,IsolatedEquation) -
isolax(N,Equation,Equationl),
isolate (Position,Equationl , IsolatedEquation).
isolate( [ J ,Equation,Equation).
The rewrite rules, or isolation axioms, are specified by the predicate isolax(N,Equation,Equationl). Let us consider an example used in solving i - 2 . sin(x) = O. An equivalence transformation on equations is adding the same quantity to both sides of an equation. We show its translation into an isolax axiom for manipulating equations of the form u - y = w. Note that rules need only simplify the left-hand side of equa- tions, since the unknown is guaranteed to be on that side.
Two rules are necessary to cover the two cases whether the first or second argument of u - y contains the unknown. The term u - y = w
451 Ari Equotion Solver
can be rewritten to either u = w + i' or y = u - w. The first argument of isolax specifies which argument of the sum contains the unknown. The Prolog equivalent of the two rewrite rules is then
isolax(1,Terinl-Term2 = Rhs,Terml Rhs+Term2). isolax(2,Terml-Term2 = Rhs,Term2 = Terml-Rhs).
Other isolation axioms are more complicated. Consider simplifying a product on the left-hand side of an equation. One of the expected rules would be
isolax(1,Terml*Term2 = Rhs,Terml = Rhs/Term2).
If Teiui2 equals zero, however, the rewriting is invalid. A test is therefore added that prevents the axioms for multiplication being applied if the term by which it divides is O. For example,
isolax(1,Terinl*Term2 = Rhs,Terml = Rhs/Term2) - Terin2 O.
Isolation axioms for trigonometric functions illustrate another possi- bility that must be catered for - multiple solutions. An equation such as sin(x) = 1/2 that is reached in our example has two solutions between O and 2 . Tr. The alternative solutions are handled by having separate iso- lax axioms:
isolax(1,sin(U) = V,U = arcsin(V)). isolax(1,sin(U) V,U = Tr - arcsin(V)).
In fact, the equation has a more general solution. Integers of the form 2 n iT can be added to either solution for arbitrary values of n. The decision whether a particular or general solution is desired depends on context and on semantic information independent of the equation solver.
Further examples of isolation axioms are given in the complete equa- tion solver, Program 23.1.
The code described so far is sufficient to solve the first equation in Fig- ure 23.1, cos(x) . (1 - 2 . sin(x)) = O. There are four answers arccos(0), - arccos(0), arcsin((1 - 0)/2), rr - arcsin((1 - 0)/2). Each can be simpii- fled, for example, arcsin((1 - O)/2) to rr/6, but will not be unless the expression is explicitly evaluated.
The usefulness of an equation solver depends on how well it can per- form such simplification, even though simplification is not strictly part
452 Chapter 23
of the equation-solving task. Writing an expression simplifier is nontriv- ial, however. It is undecidable whether two expressions are equivalent in general. Some simple identities of algebra can be easily incorporated, for example, rewriting O + u to u. Choosing between other preferred forms, e.g., (1 + x)3 and i + 3 x + 3 x2 + x3, depends on context.
23.4 Polynomial
Polynomial equations are solved by a polynomial equation solver, apply- ing various polynomial methods. Both sides of the equation are checked as to whether they are polynomials in the unknown. If the checks are successful, the equation is converted to a polynomial normal form by polynomial_normal_form, and the polynomial equation solver solve_ polynomial_equat ion is invoked:
solve_equation(Lhs=Rhs , X ,Solution)
polynomial(Lhs,X),
polynomial(Rhs,X),
polynomial_normal_form (Lhs-Rhs , X, PolyForm),
solve_polynomial_equation (PolyForm,X, Solution).
The polynomial normal form is a list of tuples of the form (A,N1), where A is the coefficient of which is necessarily nonzero. The tuples are sorted into strictly decreasing order of N; for each degree there is at most one tuple. For example, the list [(1,2), (-3, 1),(2,O)] is the normal form for x2 - 3 . x + 2. The leading term of the polynomial is the head of the list. The classic algorithms for handling polynomials are applicable to equations in normal form. Reduction to polynomial normal form occurs in two stages:
polynomial_normal_form (Polynomial , X, NormalForm) -
polynomial_f orm (Polynomial, X, PolyForm),
remove_zero_terms (PolyForm NormalForm).
The predicate polynornial_form(X,Polynomial PolyForm) decom- poses the polynomial. PolyForm is a sorted list of coefficient-degree tuples, where tuples with zero coefficients may occur.
It is convenient for many of the polynomial methods to assume that all the terms in the polynomial form have nonzero coefficients. There-
453 An Equation Solver
fore the final step of polynomial_normal_form is removing those ternis whose coefficients are zero. This is achieved by a simple recursive proce- dure remove_zero_terms.
The code for polynomial_form directly echoes the code for polyno- mial. For each clause used in the parsing process, there is a correspond- ing clause giving the resultant polynomial. For example, the polynomial form of a term x1 is [(1,n)], which is expressed in the clause
polynomial_form(XIN,X, [(1,N)J).
The recursive clauses for polynomial_form manipulate the polynomi- als in order to preserve the polynomial form. Consider the clause
polynomial_form(Polyl+Poly2,X,PolyForm) -
polynomial_f orm(Polyl ,X,PolyForml),
polynomial_f orm(Poly2 , X, PolyForm2),
add_polynomials (PolyFormi , PolyForm2 , PolyForm).
The procedure add_polynomials contains an algorithm for adding poly- nomials in normal form. The code is a straightforward list of the possi- bilities that can arise:
add_polynomials([ I ,Poly,Poly). add_polynomials (Poly, E I ,Poly).
add_polynomials([(Ai,Ni) IPolyl] , [(Aj,Nj)IPoly2], [(Ai,Ni)IPoly])
Ni > Nj, add_polynomials(Polyl, E(Aj , Nj) IPoly2] , Poly).
add_polynomials([(Ai,Ni)IPolyl] ,[(Aj,Nj)IPoly2] , [(A,Ni)Poly]) -
Ni =:= Nj, A is Ai+Aj, add_polynomials(Polyl,Poly2,Poly).
add_polynomials([(Ai,Ni)IPolyl],[(Aj,Nj)IPoly2],[(Aj,Nj)IPoly]) - Ni < Nj, add_polynomials([(Ai,Ni)IPolyl] ,Poly2,Poly).
Similarly, the procedures subtract_polynomials, mult iply_polyno- mials, and binomial are algorithms for subtracting, multiplying, and binomially expanding polynomials in normal form to produce results in normal form. The subsidiary predicate multiply_single(Polyl,Mono- mial,Poly2) multiplies a polynomial by a monomial (C,N) to produce a new polynomial.
Once the polynomial is in normal form, the polynomial equation solver is invoked. The structure of the polynomial solver follows the structure of the overall equation solver. The solver is a collection of methods that
454 Chapter 23
are tried in order to see which is applicable and can be used to solve the equation. The predicate solve_polynomial_equation is the analogous relation to solve_equation.
The second equation in Figure 23.1 is quadratic and can be solved with the standard formula. The equation solver mirrors the human method. The polynomial is identified as being suitable for the quadratic method by checking (with quadratic) if the leading term in the polynomial is of second degree. Since zero terms have been removed in putting the poly- nomial into its normal form, pad puts them back if necessary. The next two steps are familiar: calculating the discriminant, and returning the roots according to the value of the discriminant. Again multiple solutions are indicated by having multiple possibilities:
solve_polynomial_equation(Poly,X,Solution) -
quadratic(Poly),
pad(Poly, [(A,2) , (B,1) , (C,O)])
discriminant(A,B,C,Discriminant),
root(X,A,B,C,Discriminant,Solution).
discriminant(A,B,C,D) - D is (B*B - 4*A*C).
root(X,A,B,C,O,X= -B/(2*A)).
root(X,A,B,C,D,X= (-B+sqrt(D))/(2*A)) - D > O.
root(X,A,B,C,D,X= (-B-sqrt(D))/(2*A)) - D > O.
Other clauses for solve_polynomial_equation constitute separate methods for solving different polynomial equations. Linear equations are solved with a simple formula. In PRESS, cubic equations are handled by guessing a root and then factoring, reducing the equation to a qua- dratic. Other tricks recognize obvious factors, or that quartic equations missing a cubic and a linear term are really disguised quadratics.
23.5 Homogenization
The top-level clause for homogenization reflects the transformation of the original equation into a new equation in a new unknown, which is recursively solved; its solution is obtained for the original unknown:
455 An Equation Solver
solve_equation(Equation,X,Solution) - hornogenize(Equation,X,Equationl,X1), solve_equation(Equationl ,X1 ,Solutionl), solve_equation(Solutionl ,X, Solution).
The code for homogenize/4 implements the four stages of homoge- nization, described in Section 23.1. The offenders set is calculated by offenders/3, which checks that there are multiple offenders. If there is oily a single offender, homogenization will not be useful:
hornbgenize(Equation,X,Equationl ,X1) - offenders (Equation, X,Offenders), reduced_terni(X,Offenders Type ,X1) rewrite(Offenders,Type,X1,Substitutions), substitute (Equation,Substitutions, Equatìonl).
The predicate reduced_term/4 finds a reduced term, that ìs, a candi- date for the new unknown. In order to structure the search for the re- duced term, the equation is classified into a type. This type is used in the next stage to find rewrite rules expressing each element of the offenders set as an appropriate function of the reduced term. The type of the exam- ple equation is exponential. PRESS encodes a lot of heuristic knowledge about finding a suitable reduced term. The heuristics depend on the type of the terms appearing in the offenders set. To aid the structuring (and retrieval) of knowledge, finding a reduced term proceeds in two stages - classifying the type of the offenders set, and finding a reduced term of that type:
reduced_terni(X ,Offenders, Type ,Xl) - classify(Offenders,X,Type), candidate(Type,Offenders,X,X1).
We look at the set of rules appropriate to our particular equation. The offenders set is of exponential type because all the elements in the offenders set have the form A, where A does not contain the unknown but B does. Standard recursive procedures check that this is true.
The heuristic used to select the reduced term in this example is that if all the bases are the same, A, and each exponent is a polynomial in the unknown, X, then a suitable reduced term is Ax:
456 Chapter 23
candidate (exponential ,Offenders, X ,AIX) -
base(Offenders,A), polynomial_exponents(Offenders,X).
The straightforward code for base and polynomial_exponents is in the complete program. The heuristics in PRESS are better developed than the ones shown here. For example, the greatest common divisor of all the leading terms of the polynomials is calculated and used to choose the reduced term.
The next step is checking whether each member of the offenders set can be rewritten in terms of the reduced term candidate. This involves finding an appropriate rule. The collection of clauses for homogenize_ axiom constitute the possibly applicable rewrite rules. In other words, relevant rules must be specified in advance. The applicable rules in this case are
homogenize_axiom(exponential,AI (N*X) ,AIX, (AIX) IN).
homogenize_axiom(exponential,AI(X+B) ,AIX,AIB*AIX).
Substituting the term in the equation echoes the parsing process used by offenders as each part of the equation is checked to see whether it is the appropriate term to rewrite.
Exercises for Chapter 23
Add isolation axioms to Program 23.1 to handle quotients on the left-hand side of the equation. Solve the equation x/2 = 5.
Add to the polynomial equation solver the ability to solve disguised linear and disguised quadratic equations. Solve the equations 2 x3-8=x3, andx4 5 2 +6=0.
The equation cos(2 . x) - sin(x) = O can be solved as a quadratic equation in sin(x) by applying the rewrite rule cos(2 . X) = i - 2 sin2(x). Add clauses to Program 23.1 to solve this equation. You will need to add rules for identifying terms of type trigonometric, heuristics for finding trigonometric reduced terms, and appropriate homogenization axioms.
(iv) Rewrite the predicate free_of (Term,X) so that it fails as soon as it finds an occurrence of X in Term.
457 An Equation Solver
(V) Modify Program 23.1 so that it solves simple simultaneous equa- tions.
23.6 Background
Symbolic manipulation was an early application area for Prolog. Early examples are programs for symbolic integration (Bergman and Kanoui, 1973) and for proving theorems in geometry (Welham, 1976).
The PRESS program, from which Program 23.1 is adapted, owes a debt to many people. The original version was written by Bob Weiham Many of the researchers in the mathematical reasoning group working with Alan Bundy at the University of Edinburgh subsequently tinkered with the code. Published descriptions of the program appear in Bundy and Weiham (1981), Sterling et al. (1982), and Silver (1986). The last reference has a detailed discussion of homogenization.
PRESS includes various modules, not discussed in this chapter, that are interesting ¡n their own right: for example, a package for inter- val arithmetic (Bundy, 1984), an infinite precision rational arithmetic package developed by Richard O'Keefe, and an expression simplifier based on difference-structures as described in Section 15.2, developed by Lawrence Byrd. The successful integration of all these modules is strong evidence for the practicality of Prolog for large programming projects.
The development of PRESS showed up classic points of software engi- neering. For example, at one stage the program was being tuned prior to publishing some statistics. Profiling was done on the program, which showed that the predicate most commonly called was free_of. Rewriting it as suggested in Exercise 23(iv) resulted in a speedup of 35 percent in the performance of PRESS.
Program 23.1 is a considerably cleaned-up version of PRESS. Tidying the code enabled further research. Program 23.1 was easily translated to other logic programming laliguages, Concurrent Prolog and FCP (Sterling and Codish, 1986). Making the conditions when methods were used more explicit enabled the writing of a program to learn new equation-solving methods from examples (Silver, 1986).
24.1 Overview of the Compiler
The source language for the compiler is PL, a simplified version of Pascal designed solely for the purposes of this chapter. lt contains an assign- ment statement, an if-then-else statement, a while statement, and simple 1/O statements. The language is best illustrated with an example. Fig- ure 24.1 contains a program for computing factorials written in PL. A formal definition of the syntax of the language is implicit in the parser in Program 24.1.
The target language is a machine language typical for a one-accumu- lator computer. Its instructions are given in Figure 24.2. Each instruction has one (explicit) operand, which can be one of four things: an integer constant, the address of a storage location, the address of a program instruction, or a value to be ignored. Most of the instructions also have a second implicit operand, which is either the accumulator or its contents. In addition, there is a pseudoinstruction block that reserves a number of storage locations as specified by its integer operand.
The scope of the compiler is clear from its behavior on our example. Figure 24.3 is the translation of the PL program in Figure 24.1 into ma-
24 A Compiler
Our final application is a compiler. The program is presented top-down. The first section outlines the scope of the compiler and gives its defim- tion. The next three sections describe the three major components: the parser, the code generator, and the assembler.
460 Chapter 24
program factorial;
begin
read value;
count 1;
result : 1;
while count < value do
begin
count count+1;
result := result*count
end
write result
end
Figure 24.1 A PL program for computing factorials
Figure 24.2 Target language instructions
chine language. The compiler produces the colunms labeled Instruction and Operand.
The task of compiling can be broken down into the five stages given in Figure 24.4. The first stage transforms a source text into a list of tokens. The list of tokens is parsed in the second stage, syntax analysis, to give a source structure. The third and fourth stages transform the source structure into relocatable code and assemble the relocatable code into absolute object code, respectively. The final stage outputs the object program.
Our compiler implements the middle three stages. Both the first stage of lexical analysis and the final output stage are relatively uninteresting and are not considered here. The top level of the code handles syntax analysis, code generation, and assembly.
Arithmetic
Literals Menory Control
junpeq
I/O, etc.
read
addc add jumpne write
subc sub jumplt halt
mulc mul junpgt
divc div junple
loadc
store
load jumpge
jump
461
Figure 24.3 Assembiy code version of a factorial program
A Compiler
Symbol Address Instruction Operand Symbol
i READ 21 VALUE 2 LOADC i 3 STORE 19 COUNT 4 LOADC 20 5 STORE 20 RESULT
LABEL1 6 LOAD 19 COUNT
7 SUB 21 VALUE
8 JUMPGE 16 LABEL2 9 LOAD 19 COUNT
10 ADDC 1
11 STORE 19 COUNT 12 LOAD 20 RESULT 13 MUL 19 COUNT 14 STORE 20 RESULT 15 JUMP 6 LABEL1
LABEL2 16 LOAD 20 RESULT 17 WRITE O
18 HALT O
COUNT 19 BLOCK 3 RESULT 20 VALUE 21
Object Object
Source Text
Token List
Source Structure
Structure (relocat-
able)
Structu re (abso-
lute)
Object Program
Lexical Syntax Code Assembly Output Analysis Analysis Generation
Figure 24.4 The stages of compilation
The basic predicate compile (Tokens,ObjectCode) relates a list of to- kens Tokens to the ObjectCode of the program the tokens represent. The compiler compiles correctly any legal PL program but does not han- dle errors; that is outside the scope of this chapter. The list of tokens is assumed to be input from some previous stage of lexical analysis. The parser performing the syntax analysis, implemented by the predi- cate parse, produces from the Tokens an internal parse tree Structure.
462 Chapter 24
compile(Tokens,ObjectCode) ObjectCode is the result of compilation of a list of Tokens representing a PL program.
compile (Tokens ,ObjectCode)
parse (Tokens ,Structure),
encode (Structure ,Dictionary,Code),
assemble (Code ,Dictionary,ObjectCode).
The parser parse( Tokens,Structure) -
Structure represents the successfully parsed list of Tokens. parse (Source ,Structure)
pl_program(Structure,Source\[ j).
p1_program(S) - [program], identifier(X), E';'], statement(S).
statement((S;Ss)) -
[begin] , statement(S), rest_statements(Ss).
statement(assign(X,V))
identifier(X) , E' :'] , expression(V). statement(if(T,Si,S2)) -
[if] , test(T), [then] , statement(S1), [else] , statement(S2)
statement(while(T,S)) -
[while] , test(T) , [do] , statement(S)
statement(read(X)) -.
[read] , identifier(X)
statement(write(X)) -.
[write] , expression(X).
rest_statements((S;Ss)) - [';'] , statement(S), rest_statements(Ss).
rest_statements(void) [end]
expression(X) -e pl_constant(X).
expression(expr(Op,X,Y))
pl_constant(X), arithmetic_op(Op), expression(Y).
arithmetic_op('+') -. ['+'j
arithmetic_op('-') - E'-']. arithmetic_op('*') -. ['*'].
arithmetic_op('/') - E'!'].
pl_constant(name(X)) -. identifier(X).
pl_constant(number(X)) -' pl_integer(X).
identifier(X) -' [X], {atom(X)}.
pl_integer(X) -' EX], {integer(X)}.
test(compare(Op,X,Y)) -
expression(X), comparison_op(Op), expression(Y).
Program 24.1 A compiler from PL to machine language
463 A Compiler
comparison_opY=') - cornparison_op(') ['']. comparison_opY>') . ['>'1. comparison_opY<') ['<'J. comparison_op('') . ['>']
comparison_op('') [''].
The code generator
encode C Stru cture,Dictionary,RelocatableCode) - Relocatable Code is generated from the parsed Structure building a Dictioriry associating variables with addresses.
encode((X;Xs),D,(Y;Ys)) -
encode(X,D,Y), encode(Xs,D,Ys).
encode(void,D,no_op).
encode(assign(Name,E),D,(Code; instr(store,Address))) -
lookup(Name,D,Address) , encode_expression(E,D,Code)
encode(if(Test,Then,Else) ,D,
(TestCode; ThenCode; instr(jump,L2);
label (Li); ElseCode; label(L2))) -
encode_test (Test ,L1 ,D ,TestCode)
encode (Then,D ,ThenCode),
encode (Else,D,ElseCode)
encode (wliile(Test,Do) ,D,
(label(L1); TestCode; DoCode; instr(jump,L1); label(L2))) -
encode_test(Test,L2,D,TestCode), encode(Do,D,DoCode).
encode(read(X) ,D,instr(read,Address))
lookup(X,D,Address).
encode(write(E),D,(Code; instr(write,O)))
encode_expression(E,D,Code).
encode_expression (Expression,Dictionary, Code) - Code corresponds to an arithmetic Expression.
encode_expression(number(C) ,D,ïnstr(loadc,C)).
encode_expression(nanie(X),D,instr(load,Address))
lookup(X,D,Address)
encode_expression(expr(Op,E1,E2),D,(Load;Instruction))
single_instruction(Op,E2,D,Instruction),
encode_expression(E1 ,D,Load).
encode_expression(expr(Dp,Ei,E2) ,D,Code) -
not sïngle_instruction(Op,E2,D,Instructïon)
single_operation(Op,E1 ,D,E2Code,Code),
encode_expression(E2,D,E2Code).
Program 24.1 (Continued)
464 Chapter 24
single_instruction(Op,nuinber(C),D,instr(OpCode,C)) -
literal_operation(Op,OpCode).
single_instruction(Op,naine(X),D,instr(OpCode,A)) -
memory_operation(Op,OpCode), lookup(X,D,A).
single_operation(Op,E,D,Code, (Code; Instruction)) -
coinmutative(Op), single_instruction(Op,E,D,Instruction).
single_operation (Op. E, D ,Code,
(Code;instr(store,Address);Load;instr(OpCode,Addressfl) -
not commutative(Op),
lookupC$temp' ,D,Address),
encode_expression(E,D,Load),
op_code (E , Op , OpCode)
op_code(number(C) ,Op,OpCode) - literal_operation(Op,OpCode).
op_code(name(X) ,Op,OpCode) - niemory_operation(Op,OpCode).
literal_operation('+' ,addc). memory_operationY+' ,add).
literal_operation('-' ,subc). memory_operationY-' sub).
literal_operation('*' ,mulc). memory_operationY*' mul).
literal_operationY/' ,divc). memory_operationY/' div).
comlnutative('-*'). cominutative('*).
encode_test (compare(Op,E1 E2) Label,D,
(Code; instr(OpCode ,Label))) -
coniparison_opcode (Op OpCode)
encode_expression(exprY' ,Ei,E2) ,D,Code).
comparison_opcode('=' jumpne). comparison_opcode(' ' ,jumpeq).
comparison_opcodeY>' ,jumple). comparison_opcode('' ,jumplt).
comparison_opcodeY<' ,jumpge). comparison_opcode(' ' jumpgt). lookup(Naine,Dictionary,Address) - See Program 15.9.
The assembler
assemble ( Code,Dictionary, TidyCode) - TidyCode is the result of assembling Code removing no_Ups and labels, and filling in the Dictionary.
assemble(Code Dictionary,TidyCode) -
tidy_and_count (Code, 1 N,TidyCode\(instr(halt 0) ;block(L)))
Ni is N+i,
allocate (Dictionary,Nl,N2),
L is N2-N1, L
tidy_and_count((Codei ;Code2) M,N,TCodel\TCode2) -
tidy_and_count(Codel,M,M1,TCodel\Rest), tidy_and_count(Code2,Mi N,Rest\TCode2).
tidy_and_count(instr(X,Y),N,Ni,(instr(X,Y);Code)\Code) -
Nl is N+l.
tidy_and_count (label (N) N N, Code\Code).
tidy_and_count (no_op ,N, N, Code\Code)
Program 24.1 (Continued)
465 A Compiler
allocate(void,N,N).
allocate(dict(Name,N1,Before,After),N0,N)
allocate (Bef ore ,NO , Nl)
N2 is Ni1,
allocate (After,N2,N)
Program 24.1 (Continued)
test_compiler(X,Y)
proram(X,P) , compile(P,Y)
program(testl, [program,testl,' ; ,begïn,write,
x + ' , y , - ' , z , ' / ' , 2 , end] prograxn(test2, [prograni,test2,' ;',
begin,if,a,'>' ,b,then,max, :' ,a,else,max, :' ,b,end])
prograni(factorial,
[prograin,factorial, ;'
,begin
,read,value,'
,count , :=' , 1,';' ,result, :=' ,1, (;) ,while,count,'<' ,value,do
,begin
,count, ,count, '+' 1, ,result, := ,result, 't' ,count
end, '
,write, result
,end])
Program 24.2 Test data
The structure is used by the code generator encode to produce relocat- able code Code. A dictionary associating variable locations to memory addresses and keeping track of labels is needed to generate the code. This is the second argument of encode. Finally, the relocatable code is assembled into object code by assemble with the aid of the constructed Dictionary.
The testing data and instructions for the program are given as Pro- gram 24.2. The program factorial is the PL program of Figure 24.1 trans- lated into a list of tokens. The two small programs consist of a single statement each, and test features of the language not covered by the factorial example. The program testi tests compilation of a nontrivial arithmetic expression, and test2 checks the if-then-else statement.
466 Chapter 24
24.2 The Parser
The parser proper is written as a definite clause grammar, as described in Chapter 19. The predicate parse as given in Program 24.1 is just an interface to the DCG, whose top-level predicate is p1_program. The DCG has a single argument, the structure corresponding to the statements, as described later. A variant of Program 18.9 is assumed to translate the DCG into Prolog clauses. The convention of that program is that the last argument of the predicates defined by the DCG is a difference-list:
parse (Source ,Structure) -
pl_program(Structure,Source\[ 1).
The first statement of any PL program must be a program statement. A program statement consists of the word program followed by the name of the program. We call words that must appear for rules of the grammar to apply standard identifiers, the word program being an example. The name of the program is an identifier in the language. What constitutes identifiers, and more generally constants, is discussed in the context of arithmetic expressions. The program name is followed by a semicolon, another standard identifier, and then the program proper begins. The body of a PL program consists of statements or, more precisely, a sin- gle statement that may itself consist of several statements. All this is summed up in the top-level grammar rule:
p1_program(S) -
[program], identifier(X), [C;] , statement(S).
The structure returned as the output of the parsing is the statement constituting the body of the program. For the purpose of code genera- tion, the top-level program statement has no significance and is ignored in the structure built.
The first statement we describe is a compound statement. Its syntax is the standard identifier begin followed by the first statement, S, say, in the compound statement, and then the remaining statements Ss. The structure returned for a compound statement is (S; Ss), where ; is used as a two-place infix functor. Note that S, Ss, or both may be compound statements or contain them. The semicolon is chosen as functor to echo its use in PL for denoting sequencing of statements:
467 A Compiler
statement ( (S; Ss) )
[begin] , statement(S), rest_statements(Ss).
Statements in PL are delimited by semicolons. The rest of the state- ments are accordingly defined as a semicolon followed by a nonempty statement, and recursively the remaining statements:
rest_statements((S;Ss)) -.
E';'], tatement(S), rest_statements(Ss).
The end of a sequence of statements is indicated by the standard iden- tifier end. The atom void is used to mark the end of a statement in the internal structure. The base case of rest_statements is therefore
rest_statements(void) -. [end]
The above definition of statements precludes the possibility of empty statements. Programs and compound statements in PL cannot be empty.
The next statement to discuss is the assignment statement. It has a simple syntactic definition - a left-hand side, followed by the standard identifier is, followed by the right-hand side. The left-hand side is re- stricted to being a PL identifier, and the right-hand side is any arithmetic expression whose definition is to be given:
statement(assign(X,E))
identifier(X), [':='], expression(E).
The structure returned by the successful recognition of an assignment statement has the form assign(X,E). The (Prolog) variable E represents the structure of the arithmetic expression, and X is the name of the (PL) variable to be assigned the value of the expression. It is implicitly assumed that X will be a PL identifier.
For simplicity of both code and explanation, we restrict ourselves to a subclass of arithmetic expressions. Two rules define the subclass. An expression is either a constant or a constant followed by an arithmetic operator and recursively an arithmetic expression. Examples of expres- sions in the subclass are x, 3, 2 . t and x + y - z/2, the expression in the first test case in Program 24.2:
expression(X) -. pl_constant(X).
expression(expr(Op,X,Y)) -.
pl_constant(X), arithxnetic_op(Op), expression(Y).
468 Chapter 24
This subclass of expressions does not respect the standard precedence of arithmetic operators. The expression x . 2 + y is parsed as x (2 + y). On the other hand, the expression x + y - z/2 is interpreted unambigu- ously as x + (y - (z/2)).
For this example, we restrict ourselves to two types of constants in PL: identifiers and integers. The specification of p1_constant duly consists of two rules. Which of the two is found is reflected in the structure returned. For identifiers X, the structure name (X) is returned, whereas number (X) is returned for the integer X:
pl_constant(name(X)) - identifier(X).
pl_constant(number(X)) -. pl_integer(X).
For simplicity we assume that PL integers and PL identifiers are Prolog integers and atoms, respectively. This allows the use of Prolog system predicates to identify the PL identifiers and integers. Recall that the curly braces notation of DCGs is used to specify Prolog goals:
identifier(X) [X], {atom(X)}.
pl_integer(X) - [X], {integer(X)}.
In fact, all grammar rules that use PL identifiers and constants could be modified to call the Prolog predicates directly if greater efficiency is needed.
A list of arithmetic operators is necessary to complete the definition of arithmetic expressions. The form of the statement for addition, repre- sented by +, follows. The grammar rules for subtraction, multiplication, and division are analogous, and appear in the full parser in Program 24.1:
arithmetic_op(+') [C] The next statement to be discussed is the conditional statement, or
if-then-else. The syntax for conditionals is the standard identifier if fol- lowed by a test (to be defined). After the test, the standard identifier then is necessary, followed by a statement constituting the then part, the stan- dard identifier else and a statement constituting the else part, in that order. The structure built by the parser is if (T,Si , S2), where T is the test, Si is the then part, and S2 is the else part:
statement(if(T,Si,S2))
[if], test(T), [then], statement(Si),
[else], statement(S2).
469 A Compiler
Tests are defined to be an expression followed by a comparison oper- ator and another expression. The structure returned has the form com- pare (Op , X, Y), where Op is the comparison operator, and X and Y are the left-hand and right-hand expressions in the test, respectively:
test(compare(Op,X,Y)) -. expression(X), comparison_op(Op), expression(Y).
The definition of comparison operators using the predicate compari- son_op is analogous to the use of arithmetic_op to define arithmetic operators. Program 24.1 contains definitions for =, , >, <,, and .
While statements consist of a test and the action to take if the test is true. The structure returned is while(T,S), where T is the test and S is the action. The syntax is defined by the following rule:
statement(while(T,S)) -. [while], test(T), [do], statement(S).
I/O is handled in PL with a simple read statement and a simple write statement. The input statement consists of the standard identifier read followed by a PL identifier; it returns the structure read(X), where X is the identifier. Write statements are similar:
statement(read(X)) -. [read], identifier(X).
statement(write(X)) - [write], expression(X).
Collecting the various pieces of the DCG just described gives a parser
for the language. Note that ignoring the arguments in the DCG gives a formal BNF grammar for PL.
Let us consider the behavior of the parser on the test data in Pro- gram 24.2. The parsed structures produced for the two single statement programs have the form (structure) ;void, where (structure) repre- sents the parsed statement. The write statement is translated to
write(expr(+,name(x),expr(-,name(y),expr(/,name(z),
nuinber(2))))),
and the if-then-else statement is translated to
if (compare (> ,riame (a) ,name (b)) , assign(max name (a) ) assign(max,name(b))).
The factorial program is parsed into a sequence of five statements fol- lowed by void. The output after parsing for all three test programs is
470 Chapter 24
Program testi: write(expr(+,nanie(x) ,expr(-,name(y),
expr(/,name(z) ,nuniber(2)flfl;void Program test2:
if(compare(>,name(a) ,name(b)),assign(max,naine(a)), assign(max,name(bfl) ;void
Program test3:
read(value);assign(count,number(lfl;assign(result,number(lfl; while(compare(< ,name(count) ,name(value)), (assign(count,expr(+,name(count),number(1))); assign(result,expr(*,name(result),nanie(count)fl;void)); write(name(result)) ;void
Figure 24.5 Output from parsing
given in Figure 24.5. This is the input for the second stage of compila- tion, code generation.
24.3 The Code Generator
The basic relation of the code generator is encode (Structure,Dictio- nary ,Code), which generates Code from the Structure produced by the parser. This section echoes the previous one. The generated code is de- scribed for each of the structures produced by the parser representing the various PL statements. Dictionary relates PL variables to memory locations, and labels to
instruction addresses. The dictionary is used by the assembler to resolve locations of labels and identifiers. Throughout this section D refers to this dictionary. An incomplete ordered binary tree is used to implement it, as described in Section 15.3. The predicate lookup(Name,D,Value) (Program 15.9) is used for accessing the incomplete binary tree.
The structure corresponding to a compound statement is a sequence of its constituent structures. This is translated into a sequence of blocks of code, recursively defined by encode. The functor ; is used to denote sequencing. The empty statement denoted by void is translated into a null operation, denoted no_op. When the relocatable code is traversed during assembly this "pseudoinstruction" is removed.
471 A Compiler
The structure produced by the parser for the general PL assignment statement has the form assign(Name,Expression), where Expression is the expression to be evaluated and assigned to the PL variable Name. The correspondmg compiled form calculates the expression followed by a store instruction whose argument is the address corresponding to Name. The representation of individual instructions in the compiled code is the structure instr(X,Y), where X is the instruction and Y is the operand. The appropriate translation of the assign structure is there- fore (Code; instr(store,Address)), where Code is the compiled form of the expression, which, after execution, leaves the value of the ex- pression in the accumulator. It is generated by the predicate encode_ expression(Expression, D, ExpressionCode), Encoding the assignment statement is performed by the clause
ericode(assign(Name,Expression) ,D,(Code;instr(store,Address)))
- lookup(Name ,D,Address), encode....expression(Expression,
D,Code).
This clause is a good example of Prolog code that is easily understood declaratively but hides complicated procedural bookkeeping. Logically, relations have been specified between Name and Address, and between Expression and Code. From the programmer's point of view it is irrele- vant when the final structure is constructed, and in fact the order of the two goals in the body of this clause can be swapped without changing the behavior of the overall program. Furthermore, the lookup goal, in re- lating Name with Address, could be making a new entry or retrieving a previous one, where the final instantiation of the address happens in the assembly stage. None of this bookkeeping needs explicit mention by the programmer. It goes on correctly in the background.
There are several cases to be considered for compiling the expression. Constants are loaded directly; the appropriate machine instruction is loadc C, where C is the constant. Similarly identifiers are compiled into the instruction load A, where A is the address of the identifier. The two corresponding clauses of encode_expression are
encode_expression(nuniber(C) ,D,instr(loadc,C)).
encode_expression(name(X) ,D,instr(load,Address)) -
lookup(X,D,Address).
472 Chapter 24
The general expression is the structure expr (Op ,El , E2), where Op is the operator, El is a PL constant, and E2 is an expression. The form of the compiled code depends on E2. If it is a PL constant, then the final code consists of two statements: an appropriate load instruction determined recursively by encode_expression and the single instruction corresponding to Op. Again, it does not matter in which order the two instructions are determined. The clause of encode_expression is
encode_expression(expr(Op,El,E2),D, (Load;Instruction)) - single_instruction(Op,E2,D, Instruction), encode_expression(El ,D,Load).
The nature of the single instruction depends on the operator and whether the PL constant is a number or an identifier. Numbers refer to literal operations, and identifiers refer to memory operations:
single_instruction(Op,nuniber(C) ,D,instr(Opcode,C)) - literal_operation(Op , Opcode).
single_instruction(Op,name(X) ,D,instr(Opcode,A)) - meniory_operation(Op,Opcode), lookup(X,D,A).
A separate table of facts is needed for each sort of operation. The respective form of the facts is illustrated for +:
literal_operation(+ , addc). mernory_operation(+ ,add).
A separate calculation is necessary when the second expression is not a constant and caimot be encoded in a single instruction. The form of the compiled code is determined from the compiled code for calculating E2, and the single operation is determined by Op and El:
encode_expression(expr(Op,El,E2) ,D,Code) - not single_instruction(Op,E2,D,Instruction), single_operation(Op El, D, E2Code ,Code), encode_expression(E2,D,E2Code).
In general, the result of calculating E2 must be stored iii some tempo- rary location, called $telnp in the following code. The sequence of instruc- tions is then the code for E2, a store instruction, a load instruction for El, and the appropriate memory operation addressing the stored con- tents. The predicates shown previously are used to construct the final form of the code:
473 A Compiler
single_operation(Op, E, D ,Code, (Code; instr(store,Address); Load; instr (OpCode ,Address))
not cornmutative(Dp), lookup('$temp' ,D,Address), encode_expression (E,D,Load), op_code (E,Op,OpCode).
An optimization is possible if the operation is commutative, e.g., ad- dition or multiplication, which circumvents the need for a temporary variable. In this case, the memory or literal operation can be performed on El, assuming that the result of computing E2 is in the accumulator:
single_operation(Op,E,D,Code, (Code;Instruction)) - cominutative(Op), single_instruction(Op,E,D,Instruction).
The next statement is the conditional if-then-else parsed into the struc- ture if(Test,Then,Else). To compile the structure, we have to intro- duce labels to which instructions can jump. For the conditional we need two labels marking the beginning and end of the else part respectively. The labels have the form label (N), where N is the address of the instruc- tion. The value of N is filled in during the assembling stage, when the label statement itself is removed. The schematic of the code is given by the third argument of the following encode clause:
encode(if(Test,Then,Else) ,D, (TestCode; ThenCode; instr(junip,L2); label (Li); ElseCode; label(L2))
encode_test (Test ,Li , D, TestCode), encode (Then,D ,ThenCode), encode (Else, D, ElseCode).
474 Chapter 24
In order to compare two arithmetic expressions, we subtract the sec- ond from the first and make the jump operation appropriate to the par- ticular comparison operator. For example, if the test is whether two ex- pressions are equal, we circumvent the code if the result of subtracting the two is not equal to zero. Thus comparison_opcode(c= ,jumpne) is a fact. Note that the label that is the second argument of encode_test is the address of the code following the test.
encode_test(compare(Op,E1,E2) ,Label,D,
(Code; instr(OpCode,Label)))
comparison_opcode(Op,DpCode),
encode_expression(expr('-' ,E1,E2),D,Code).
The next statement to consider is the while statement. The statement is parsed into the structure while (Test, Statements). A label is necessary before the test, then the test code is given as for the if-then-else state- ment, then the body of code corresponding to Statements and a jump to reperform the test. A label is necessary after the jump instruction for when the test fails.
encode (while (Test,Do) ,D,
(label (Li);
TestCode;
DoCode;
instr(jump,L1);
label(L2))
)
encode_test (Test ,L2,D,TestCode),
encode (Do, D, DoCode).
The I/O statements are straightforward. The parsed structure for in- put, read (X), is compiled into a single read instruction, and the table is used to get the correct address:
encode (read(X) ,D,instr(read,Address)) -
lookup(X,D,Address).
The output statement is translated into encoding an expression and then a write instruction:
encode(write(E) ,D, (Code; instr(write,O))) -
encode_expression(E,D,Code).
475 A Compiler
Program testi
((instr(load,Z);instr(divc,2fl;instr(store,Teuip); instr(load,Y);instr(sub,Ternp));instr(add,X)); instr(write 0)); no_op
Program test2:
(((instr(load,A);instr(sub,B));ïnstr(jumple,Llfl; (instr(load,A);instr(store,Max));instr(juxnp,L2);label(L1); (instr(load,B);instr(store,Max));label(L2));no_op
Program factorial: instr(read,Value); (instr(loadc,1);ïnstr(store,Count));
(instr(loadc,i);instr(store,Result)); (label (Li); ((instr(load,Count);instr(sub,Value));instr(jumpge,L2)); (((instr(load,Count);instr(addc,i));instr(store,Count)); ((instr(load,Result);iritr(mul,Countfl;instr(store,Resultfl; no_op) ;instr(jwnp,Li);label(L2)) ;(instr(load,Result); instr(write,0));no_op
Figure 24.6 The generated code
Figure 24.6 contains the relocatable code after code generation and be- fore assembly for each of the three examples of Program 24.2. Mnemonic variable names have been used for easy reading.
24.4 The Assembler
The final stage performed by the compiler is assembling the relocatable code into absolute object code. The predicate assemble(Code,Dictio- nary, ObjectCode) takes the. Code and Dictionary generated in the pre- vious stage and produces the object code. There are two stages in the as- sembly. During the first stage, the instructions in the code are counted, at the same time computing the addresses of any labels created during code generation and removing unnecessary null operations. Tuis tidied code is further augmented by a halt instruction, denoted by instr(halt,O), and a block of L memory locations for the L PL variables and tempo- rary locations in the code. The space for memory locations is denoted by block (L). In the second stage, addresses are created for the PL and temporary variables used in the program:
476 Chapter 24
assemble(Code,Dictionary,TidyCode) -
tidy_and_count(Code,1,N,TidyCode\(instr(halt,O) ;block(L))),
Nl is N+1,
allocate (Dictionary ,Nl, N2),
L is N2-N1, !.
The predicate tidy_and_count(Code,M,N,TidyCode) tidies the Code into TidyCode, where the correct addresses of labels have been filled in and the null operations have been removed. Procedurally, executing tidy_and_count constitutes a second pass over the code. M is the ad- dress of the begimiing of the code, and N is i more than the address of the end of the original code. Thus the number of actual instructions in Code is N+1-M. TidyCode is represented as a difference-structure based on;
The recursive clause of tidy_and_count demonstrates both standard difference-structure technique and updating of numeric values:
tidy_and_count((Codel;Code2) ,M,N,TCodel\TCode2) -
tidy_and_count (Codel ,M,M1 ,TCodel\Rest),
tidy_and_count (Code2 ,Ml , N, Rest\TCode2).
Three types of primitives occur in the code: instructions, labels, and no_ops. Instructions are handled routinely. The address counter is incre- mented by 1, and the instruction is inserted into a difference-structure:
tidy_and_count(instr(X,Y),N,N1,(instr(X,Y);Code)\Code) -
Nl is N+1.
Both labels and no_ops are removed without updating the current ad- dress or adding an instruction to the tidied code:
tidy_and_count(label(N) ,N,N,Code\Code).
tidy_and_count(no_op,N,N,Code\Code).
Declaratively, the clauses are identical. Procedurally, the unification of the label number with the current address causes a major effect in the program. Every reference to the label address is filled in. This program is another illustration of the power of the logical variable.
The predicate allocate(Dictionary,M,N) has primarily a procedu- ral interpretation. During the code generation as the dictionary is con- structed, storage locations are associated with each of the PL variables
477 A Compiler
Program testi: instr(load,11);instr(divc,2);instr(store,12);instr(load,1O);
instr(sub,12);instr(add,9);instr(write,O);instr(halt,O);block(4)
Program test2: instr(load,1O);instr(sub,11);instr(jumple,7);instr(load,1O);
instr(store, 12) ; iristr(jurnp,9) ïnstr(load, 11) instr(store, 12)
instr(halt 0) ;block(3)
Program factorial: instr(read,21);instr(loadc,1);instr(store,19);instx(loadc,1);
instr(store,20);instr(load,19);instr(sub,21);instr(jumpge,16);
instr(load,19);instr(addc,1);instr(store,19);instr(load,20);
instr(mul,19);instr(store,20);instr(junip,6);instr(load,20);
instr(write,0) instr(halt »0) ;block(3)
Figure 24.7 The compiled object code
in the program, plus any temporary variables needed for computing ex- pressions. The effect of allocate is to assign actual memory locations for the variables and to fill in the references to them in the program.
The variables are found by traversing the Dictionary. M is the address of the memory location for the first variable, and N is i more than the address of the last. The order of variables is alphabetic corresponding to their order in the dictionary. The code also completes the dictionary as a data structure.
allocate(void,N,N).
allocate(dict(Name,N1,Before,After) ,NO,N)
allocate (Bef ore,NO,N1),
N2 is N1+1,
allocate(After,N2,N).
Because the dictionary is an incomplete data structure, the predicate allocate can succeed many times. The variables at the end of the tree match both the fact and the recursive clause. For the compiler, the easi- est way to stop multiple solutions is to add a cut to the clause for assem- ble/3, which conmits to the first (and minimal) assignment of memory locations for variables.
The compiled versions of the test programs given in Program 24.2 appear in Figure 24.7.
478 Chapter 24
Exercises for Chapter 24
(i) Extend the compiler so that it handles repeat loops. The syntax is repeat (statement) until (test). Extensions to both the parser and the compiler need to be made. Test the program on the following:
program repeat;
begin
i : 1;
repeat
i begin
ii write(i);
ii i i+1
i end
until i = 11
end.
(ii) Extend the definition of arithmetic expressions to allow arbitrary ones. In the encoder, you will have to cater for the possibility of needing several temporary variables.
24.5 Background
The compiler described is based on a delightful paper by Warren (1980).
A Operators
An operator is defined by its name, specifier, and priority. The name is usually an atom. The priority is an integer between i and 1200 inclusive. The specifier is a mnemonic that defines two things, class and asso- ciativity. There are three classes of operators: prefix, infix, and postfix. Associativity, which determines how to associate terms containing mul- tiple operators, can be one of three possibilities: left-associative, right- associative, and non-associative.
There are seven possible operator types, which are given in Table AJ. A left-associative prefix operator is not possible, nor is a right-associative postfix operator. An operator specifier yfy does not make sense as it would lead to ambiguity. Consequently Standard Prolog does not allow such a specifier.
To explain the associativity, consider a term a :: b :: c. If the infix operator :: was left-associative, the term would be read as (a :: b) :: c. If the operator :: was right-associative, the term would be read as a :: (b :: c). If the operator :: was non-associative, the term would be illegal.
If uncertain about priorities when using operators, terms can always be bracketed. If you prefer not to bracket terms, you must take into account the associativity of the operator(s) involved and the priorities of terms. For example, the following three rules apply.
An operand with the same priority as a non-associative operator must be bracketed to avoid a syntax error by the Prolog reader. An operator with the same (or smaller) priority as a right-associative operator that follows that operator need not be bracketed. An operator with smaller priority than a left-associative operator that precedes that operator need not be bracketed.
480 Appendix A
Table A. i Types of Operators in Standard Prolog
Specifier Class Associativity
fx prefix non-associative fy prefix right-associative xfx infix non-associative xfy infix right-associative yfx infix left-associative xf postfix non-associative yf postfix left-associative
Table A.2 Predefined Operators in Standard Prolog
Priority Specifier Operator(s)
1200 xfx (:- -->) 1200 fx 1100 xfy 1000 xfy
700 xfx = \= 700 xfx == \== 700 xfx 700 xfx 500 yfx +- 400 yfx 200 xfy 200 fy
Standard Prolog specifies some predefined operators. The priorities and specifiers of the operators which have been used in the text are given in Table A.2.
New operators are added with the directive
op (X ,Y, Z)
where X is the priority, Y is the operator specifier, and Z is the operator name. These were used in Chapter 17 when defining a new rule language.
481 Operators
The system of operator declarations in Prolog is straightforward and can be used effectively for applications. The reader should be aware, however, that there are some subtle semantic anomalies in how opera- tors are defined and handled. The anomalies, best discovered by trial and error, should not cause problems and can be "programmed around"
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Index
Abstract interpretation, 147 Abstract interpreter, 22-24, 91-96,
119-123 Accumulator, 146-147, 155-157,
166, 240, 254, 266, 287-288, 309
bottom-up construction of lists, 14 5-146
definition, 155 generalization to difference-list,
28 7-288 Abramson, Harvey, 388 Ackermann's function, 53-55, 85 Ait-Kaci, Hassan, 127, 213 All-solutions predicates, 302-304 Alpha-beta pruning, 405-407 Alphabetic variant, 88, 200 Alternating Turing machine, 115 ANALOGY, 270, 272, 281 Ancestor cut, 354 Anonymous variable, 234 APES expert system shell, 355 append, 60-62, 93-94, 104-105, 109,
122, 131, 133, 140, 196-197, 288, 311
Appending. See Concatenation Apt, Krzystof, 99, 1056 arg, 167-168, 173, 174 Argument, 13, 27, 171 Arithmetic
logic programs for, 45-55
Prolog programs for, 150-161 Arithmetic evaluation, 150-151 Arithnietic expression, 151, 467 Arity, 13, 27, 32 Askable goals for expert systems,
344-346 Assembler, 475-477 assert, 220-221, 231-232, 304 Assignment, 125, 170 Atom, 11, 27, 164 atom, 163-164 atomic, 164 Automata, 3 19-323
Backtracking, 120-122, 125, 190, 192, 218, 231, 249-250, 280, 324
deep, 124 intelligent, 280 shallow, 124
bagof, 303, 317 Bansal, Arvind, 280 Barklund, Jonas, 187 Beer, Randall, 373 Benchmark, 209 Ben-David, Arie, 438 Berge, Claude, 423 Bergman, Marc, 457 Best-first search, 396-400 Binary predicate, 38, 44 Binary tree, 72-77, 85, 295
498 Index
Binding, 91 Bips, 150 Bird, R., 245 Bloch, Charlene, 162 Blocks world problem, 269-271 Body of a clause, 18 Boolean formula, 82-83 Bottom-up construction, 145-146,
287 Bottom-up evaluation, 44 Bottom-up implementation, 237 Bottom-up parser, 388 Bowen, Dave, 127 Bowen, Ken, 85 Boyer, Robert, 85 Breadth-first search, 266, 305-307,
396, 398 Broderick, David, 317 Bruffaerts, A., 355 Bruynooghe, Maurice, 212, 282, 300 Builtm predicates, 150, 326-32 7 Bundy, Alan, 85, 99, 281, 457 Burstall, Rod, 373 Byrd, Lawrence, 457
call, 186 car, 56 Cartesian product, in relational
algebra, 43 cdr, 56 Certainty factor, 330 Certainty threshold, 330-331 Chain rule, for derivatives, 80-81 Character I/O, 216-218 Character strings, 216, 218 Chen, Weidong, 318 Chikayama, Takashi, 212 Choice point, 208 Chow, D., 99 Circuit, logic program, 32-34 Circular definition, 133 Clark, Keith, 115, 127, 147, 212, 282,
300, 355 Clause, 18, 24, 27-28, 95, 120, 219
body, 18 definition, 18, 27 head, 18 Horn, 18 indexing, 196-197, 209-2 10 iterative, 18, 28 nondeterministic choice, 24, 120 order. See Rule order reduction, 95, 120, 326 unit, 18, 28
clause, 219, 324-325 Clause reduction level, 326 Clocksin, William, 85, 128 Closed world assumption, 115 Code generation (for PL), 470-475 Codish, Michael, 457 Coelho, Helder, 85, 282, 354 Cohen, Jacques, 127, 128 Colmerauer, Alain, 99, 127, 187, 212,
354, 388 Coloring planar maps, 255-257 Comments, 33, 235 Common instance, 16, 88 Commutative relation, 132 Comparison of Prolog with conven-
tional languages, 124-126 Comparison operator, 153 Compiler for PL, 459-475 Compiler for Prolog, 213 Complete list, 107, 133 Complete natural number, 107 Complete structure, 133 Complete type, 107 Completeness
of program, 26, 28, 46, 48, 106 of recursive data structure, 107
Complexity, 108-109 depth, 109 goal-size, 109 length, 108 time, 108, 209
COMPOSERS, 244 Composition, 239, 366-368, 374 Composition specification, 367
499 Index
compound, 163-164
Compound term, 13-14, 27, 35, 37, 108, 167-173
size of, 108 Computation
deterministic, 95, 112 goal, 92, 96, 120 nondetermthistic, 112 nonterminating, 92, 106-107, 111,
132-133, 332, 386.
of logic program, 22, 28, 92 output, 91 parallel, 99 of Prolog program, 120-12 3 redundant, 138
Concatenation of lists, 60-61 of difference-lists, 284
Conceptual dependency, 276, 281 Concurrent logic language, 99, 231 Concurrent Prolog, 120, 179, 186,
457 Conjunctive goal, 92, 219 Conjunctive query, 16-18 Cons pair, 58 Consing, 211, 284 Constant, 27, 108, 164 constant, 164
consult, 230-23 1
Context-dependent information, 385 Context, finding keywords in, 311-
313 Context-free grammar, 369-371,
3 75-3 77
translation to Prolog program, 3 72-3 73
Continuation-style meta-mterpreter, 326
Control flow, 124, 133, 238-240 Conventional language, comparison
with Prolog, 124-126 copy_term, 224
Co-routining, 147, 187, 280
Correctness of program, 26, 28, 46-49, 106, 153
Cotta, J., 85, 282 Counterexample, 334 Credit evaluation system, 439-457 Cut, 189-2 13, 327, 449
definition, 190 effect of, 190-192, 202-205 effect on storage space, 192, 208-209
in expressing mutually exclusive test, 189
-fail combination. See Cut-fail combination
green. See Green cut in if-then-else statement, 205 incompleteness in implementing
negation as failure, 199 loss of flexibility, 194, 204, 205 red, 195, 202-205 restriction on rule order, 198 rule order in negation as failure,
198
scope, 195, 327 search tree pruning, 190-191 simulation in meta-interpreter, 327
Cut-fail combination, 201-202 implementing meta-logical
predicate, 201-202 safe use, 201
DAG, 264, 305-306 Dahl, Veronica, 394 Darlington, John, 373 Data abstraction, 35-38, 258 Data manipulation, in Prolog, 125 Data structure, 125, 143-147, 171,
211, 274, 283-300
creation in Prolog, 125 cyclic, 298
incomplete, 146, 274, 283-300 in Prolog, 143-147
500 Index
Database, deductive, 44 logical, 29 relational, 42-44
Davis, Ernie, 281 DCG. See Definite clause grammar Debugging, enhanced meta-
interpreter for, 33 1-340 Declarative reading, 19, 57, 65-66,
324 Declarative semantics, 104 Deductive database, 44 Deep backtracking, 124 Default rule, 206-208 Definite clause grammar, 375-388,
466 generative feature, 386 relationship with context-free
grammar, 376, 377 Denotational semantics, 105 Depth-bounded meta-interpreter,
3 3 2-3 3 3 Depth-complexity, 109 Depth-first graph traversal, 266 Depth-first search, 112, 120, 130,
264-266, 389, 396 Depth-first traversal of search tree,
112 Derivative, 79-81, 172 Dershowitz, Nachum, 354 Destructive assignment, 125, 170 Deterministic computation, 95,
111-112 Deville, Yves, 115, 147, 244, 245 Dictionary, 293-296, 300, 470, 477 Difference-list, 283-292, 299-300,
304, 305, 371, 373 compatibility of, 284 concatenation, 284-285 head, 283 tail, 284
Difference-structure, 291-293, 457, 476
Difference-sum, 291
Differentiation, 79-81, 172 Dincbas, Mehmet, 354 Directed acydic graph, 264, 305-306 Directed graph, 40-4 1, 264 Disjunction, 21, 186 Disjunctive goal, 186 Disjunctive relationship, 42 Disz, Terry, 127 Divide-and-conquer, 69 Divide-and-query, 338 Domain, 106, 133, 332
intended, 332 termination, 106
Don't-care nondeterminism, 263, 280
Don't-know nondeterminism, 263- 264, 280
Double recursion, 73-74, 165, 287 Drabent, w., 354 Dudeney, H., 401 Dutch flag problem, 289-290 Dwork, Cynthia, 99 Dynamic predicate, 232
Editor, 223-226 Edinburgh Prolog, 120, 127, 208 Eggert, Paul, 99 Elcock, E., 280 Eliza, 273-275, 281 Empty difference-list, 284 Empty list, 56 Empty queue, 298 Empty tree, 72 Enhancement, 239-241, 326 Equation solving, 439-457
definition, 440 factorization, 440, 448-449 homogenization, 441, 454-456 isolation, 440-441, 449-452 overview, 439-441 polynomial, 452-454 quadratic, 441 simplification, 451-452
Ernst, George, 281
501 Index
Error condition, 126, 151, 164 Error handling, 1 26, 151 Error, runtime, 151, 153 Euclidean algorithm, 54, 152 Evaluable predicate. See Builtin
predicates Evaluation of arithmetic expression,
150-151 Evaluation function, 395, 403 Evaluation, partial, 360-365 Exception handling, 126, 151 Execution mechanism, 120 Execution model of Prolog, 119-122 Existential quantification, 20 Existential query, 14-15 Expert system
for credit evaluation, 429-438 enhanced interpreters for, 341-
354 Evans, Thomas, 281 Explanation shell for expert systems,
341-35 3 Extra-logical predicate, 215-231
for i/o, 215-219 for program access and manipula-
tion, 219-221 types of, 215
Fact, 11-12, 15-16,27 Factorial, 5 1-52, 155-156 Factorization, 440, 448-449 Failure-driven loops, 229-231, 423 Failure node, 110 False solution, 334-339 FCP, 457 findall, 302, 317 Finite failure, 113-114 Fixpoint of a logic program, 105 Fixpoint semantics, 105 Flattening a list, 164-166, 285-288,
298-299 Fold/unfold, 357-360, 373 Freezing terms, 183-185 Früwïrth, Thom, 280
Function algebraic, 440 relationship to relation, 49
Functional programming, 3 Functor, 4, 13-14, 167 functor, 167-174 Furukawa, Koichi, 373 Futamura, Y., 373
Gallagher, John, 300, 373 Gallaire, Herve, 44, 354 Game playing framework, 402 Game tree, 402-403 Garbage collection, 197, 231, 423 Generalization, 14 Generate-and-test, 69, 249-262, 280
in game playing, 412 optimization, 252
Generator, 250, 252, 254 get_char, 2 17-218 GI-IC, 120, 127 Goal
conjunctive, 22, 92 computation of, 92, 120 definition, 12 derived, 23, 92 disjunctive, 186 dynamic construction of, 315 ground, 25-26 invocation, 92, 128 parent, 92 reduction, 22, 92, 95-96 selection, 24 sibling, 92 size, 108
Goat order, 95, 129, 133, 136, 178, 209, 324
and left recursion, 136 and nonterminating computation,
13 5-136 comparison with clause order,
135 effect on termination, 135 effect on trace, 95
502 Index
Grammar See Context-free grammar; Definite clause grammar
Grammar rule, 369-3 72 Granularity, of a meta-interpreter,
326 Graph, 40-4 1, 2 64-266, 305, 306
connectivity, 40-4 1, 266 cyclic, 305-306 directed, 40-41, 264-266 directed acyclic, 264-266, 305-306
Greatest common divisor, 54, 152 Green cut, 184-195, 212, 449
effect on declarative meaning, 194 Gregory, Steve, 127, 198 Ground
definition, 14 goal, 2 5-26, 93 instance, 24, 26 object, 183-183 reduction, 23 representation, 187 term, 27 query, 14
Hammond, Peter, 355 Haridi, Seif, 127 Head of a clause, 18 Head of a list, 58 Heap, 208, 211 Heapify a binary tree, 7 5-77 Heap property, 75 Heintze, Nevin, 187 Hemn, Eric, 355 Herbrand base, 102-105 Herbrand universe, 102 Heuristic search, 395 Hill-climbing, 396-398 Hill, Pat, 187 Hill, R., 115 HiLog, 324 Hirschmann, Lynette, 388 Homogenization, 441, 454-456 Horn clause, 18
¡C-Prolog, 120, 127, 147, 280 Identity, 13 If-then-else statement, 205, 212 Incomplete data structure, 146, 274,
283-300 Incomplete list, 107, 133, 136, 141,
144, 283, 293 Incomplete structure, 146, 287 Incomplete type, 107 Indexing, 196-197, 210 Infinite graph, 266, 305 Infinite search tree, 96, 111, 131 morder traversal of binary tree, 75 Input/output at the character level,
2 16-2 18 Input/output for reading in a list of
words, 217-2 18 Insertion sort, 69-70, 333 Instance,
common, 16, 88 definition, 14, 27, 88 ground, 23, 26
Instantiation, 16 Intelligent backtracking, 280 Intended domain, 332 Interactive
loop, 223 program, 223-230 prompting, 346
integer, 163-164 Interchange sort, 194-195 Interpretation of a logic program,
103-104 Interpreter
abstract. See Abstract mterpreter of automata, 319-322 meta-, 227, 323-341
Intersection in relational algebra, 43 Isolation, 440-441, 449-452 Isomorphism of binary trees, 74,
264 Iteration, 154-159 Iterative clause, 18, 28, 154
503 Index
Jaffar, Joxan, 99, 115 Jam, Ashish, 245 Janson, Sverker, 127 Join in relational algebra, 43
Kahn, Ken, 317 Kahn, Gilles, 245 Kalah, 420-427 Karninski, Steven, 280 Kanoui, H., 457 Key-value pairs, 294 Kirschenbaum, Marc, 244, 245, 374 Knuth, Donald, 85 Komorowski, Jan, 373 Kowaiski, Robert, 44, 115, 127, 147,
280 Kunen, Ken, 115 KWIC, 311-313, 318
Lakhotia, Arun, 245, 374 Lambda calculus, 119 Lambda expression, 316, 318 Lambda Prolog, 324 Lasserre, Claudine, 354 Lassez, Jean-Louis, 99 Last call optimization, 196-197, 209 Lee algorithm, 306-311, 317 Lee,Y., 354 Left recursive rule, 132 Lemma, 221 Length complexity, 109 Lengthof list, 64, 160-161, 177 Le Pape, J., 354 Lexical analysis, 461 Li, Deyi, 44 Um, Pierre, 187 Undholm, Tim, 232 Linear recursive, 40 LIPS, 209 Lisp, 119, 317, 373 List, 56-64, 125-126, 133, 135,
137-146, 158 complete, 107, 133 defimtion, 57
empty, 56 flattening, 164-166, 285-288,
298-299 head, 56 incomplete, 107, 133, 136, 141,
144, 283, 293 length, 64, 160-161, 177 merging, 137-138, 189-192 splitting, 61 tail, 56 type definition, 57
Lloyd, John, 99, 115, 187, 300 Logging facility, 227-228 Logic program,
definition, 20, 27 interpretation, 103-104 meaning, 25-26, 28
Logic puzzles, 2 58-261, 280 Logical consequence, 17, 20 Logical deduction, 16, 20, 27 Logical database, 29 Logical disjunction, 186 Logical implication, 13, 16 Logical variable, 13, 91, 126, 156,
287, 383, 476 LOGLISP, 147
Maier, David, 44, 213 Maher, Michael, 99 Maintenance, 242 Mapping of list, 143, 314-315 Marriott, Kim, 99, 300 Marseilles Prolog, 127-128, 212 Martelli, Alberto, 99 Mastermind, 411-414, 423 Matsumoto, Y., 388 McCabe, Frank, 127, 147, 282, 355 McSAM, 274, 276-278, 281 Mellish, Chris, 85, 128, 147 Meaning, 2 5-26, 28
declarative, 104 definition, 25 intended, 25, 28, 105-106, 332,
340
504 Index
Meaning (cont.) of logic program, 2 5-26
Melting frozen terms, 184, 296 niember, 57-58, 59, 61, 130-131, 133,
138, 140, 205, 250, 251 Memo-function, 221-222, 232, 396 Merge sort, 69 Merging sorted lists, 137-138, 189-
192 Meta-arguments, 373 Meta-interpreter, 227, 242, 323-341
for debugging, 33 1-340, 354 definition, 323 depth-bounded, 332-333 enhanced, 324, 328-331, granularity, 326 proof tree, 329-330, 337-338 run-time overhead, 373
Meta-logical predicate, 175-186, 201, 317
Meta-logical test, 152, 176-178 Meta-prograrnrning, 186, 319, 354,
366 Meta-variable facility, 155-156, 215,
304 Mgu, 88 Michie, Donald, 232 Miller, Dale, 318 Minimal model, 104 Minimal recursive program, 46, 132,
140 Minimax algorithm, 404-407 Minker, Jack, 44 Minsky, Marvin, 281 Missing solution, 3 39-340 Model of a logic program, 104 Mode of use, 243 Module, 244 Monotomc mapping, 105 Montanari, Ugo, 99 Moore, J., 85 Moss, Chris, 212, 232 Most general unifier, 88 MU-Prolog, 120, 127, 147
Multiple solutions, 49-50, 243, 302 Multiple success node, 11
N queens problem, 252-25 5, 280 Nadathur, Gopalan, 318 Naish, Lee, 127, 147, 213, 317 Nakashima,, H., 186-187 Natural number, 46-5 1, 102, 104 NDFA, 319-321 Negation as failure, 113-115, 198-
200 Negation in logic programs, 113-115,
199 Nilsson, Martin, 162 Nim, 415-420, 423 Nonground representation, 187 Nonterminal symbols in grammar,
369-3 70 Nontermmating computation, 92,
111, 131-133, 153, 332-334, 386 Nondeterminism, 24, 95-96, 112,
249-280 combining with negation as failure,
303 combining with second-order, 306 definition, 24, 95-96 don't-care, 263, 280 don't-know, 263-264, 280 in game playing, 412
Nondeterministic choice, 24, 95, 119 Nondeterministic computation, 111-
112 Nondeterministic finite automata,
3 19-32 1 nonvar, 176, 178 not, 198-200 NPDA, 322 Number, 46-51, 152
parsing, 386-387 recursive structure, 152
Occurs check, 89, 90-91, 99, 109, 179, 298
Offenders set, 441, 455
505 Index
O'Keefe, Richard, 174, 186, 212, 213, 232, 244, 281, 300, 317, 354
Operational semantics, 102 Operator, 31, 150, 479-481 Oracle, 335
Palindrome, 322 Parent goal, 92 PARLOG, 120, 127 Parser for PL, 466-469 Parse tree, 383 Parsing with DCGs, 375-388 Partial evaluation, 360, 373-3 74 Partial reduction, 360-365 Pascal, 3 78-379, 459 Pattern matching, 273-275, 278 Pereira, Fernando, 128, 388 Pereira, Luis, 280, 354 Permutation sort, 68-69, 252 Perpetual process, 231, 300 PL, 459-460 Plaisted, David, 99 Plümer, Lutz, 147 Poker, 74, 87 Polynomial, 78-79, 452-454
coefficient representation, 452 using cut, 193
Postorder traversal, 75, 335 Power, John, 244 Powers, David, 423 Predicate
definition, 11 dynamic, 232 evaluable, 149 extra-logical, 215-231 names, 29, 233-234 static, 232 structure inspection, 163-174 system, 149
Preorder traversal, 75 PRESS, 439, 441, 457 Priority of operators, 479-480 Problem solving
depth-first framework, 390
best-first framework, 399-400 bill-climbing framework, 397 searching state space, 389-398
Procedural semantics, 102 Procedure
definition, 21 invocation, 124
Program access and manipulation, 219-221 completeness, 26, 28, 46, 48, 105 complexity, 108-109 correctness, 26, 28, 46-49, 105-
106, 153 definition, 12, 20, 27 development, 235-238 functional, 3, 49 maintenance, 242 relational, 3, 49 termination, 106, 131-133, 147
Programming with side-effects, 220, 231, 237 bottom-up, 237 interactive, 223-230 style and layout, 233-23 5
Projection in relational algebra, 43 Prolog
comparison with conventional languages, 124-126
computation, 120-122 execution mechanism, 120 higher order extension, 314-3 18 program efficiency, 208-211 pure, 119, 235, 326, 332
Prolog!!, 120, 127 Proof tree, 25, 47, 58, 60-61, 63, 112,
329-330, 337-338 ProSE group, 244 Prototyping, 237 Pushdown automata, 32 1-322 Puzzle solver, 258-261
Quantification, 15, 18-20 Query, 12-18, 28
conjunctive, 16-18
506 Index
Query (cont.) definition, 12 existential, 14-15 simple, 12, 17
Queue, 297-300 negative elements, 299
Quicksort, 69-71, 122-123, 135, 288-289
read, 215-216 real, 164 Record structure, 126 Recursion, 154-162, 195-198
tail, 154-162, 192, 209, 213 Recursive computation, 154 Recursive data structure, 107, 291
heuristics for goal order, 136 Recursive rule, 39-42, 132-13 3, 386
left, 132 linear, 40
Red cut, 195, 202-205 Reduced term, 441, 455 Reduction, 22-23, 92-95, 111 Redundant computation, 138 Redundant solution, 136-138, 192 Redundancy, 264 Reiter, Raymond, 115 Relation
definition, 11, 30 relationship to function, 49
Relational algebra, 42-44 Relational database, 42-44 Relational Language, 127 Relation scheme, 29, 242 Relocatable code, 461 Renaming, 92 repeat, 230 Repeat loop, 230-231 Resolvent, 22, 92, 96 retract, 220-221, 23 1-232, 304 Reusable code, 242, Reversing a list, 62-64, 136 Robinson, Alan, 98, 115, 147 Ross, Peter, 232, 244, 317
Rotating of list, 312 Roussel, Phillipe, 127 rplacd, 284-285 Rule, 18-21, 39-41, 130
body, 18, 27 default, 206-208 definition, 18, 27 head, 18, 27 recursive, 39-4 1
Rule order, 129-131, 198 effect on solution, 129-130
Runtime error, 151, 153
Safra, Shmuel, 232 Sahlin, Dan, 374 Sato, T., 373 Scheduling policy, 93, 119 Schemas, 244 Scope of cut in meta-interpreter, 327 Scope of variable, 17 Script, 274, 276-2 78 Search
best-first, 396-400 breadth-first, 266, 305-307, 396,
398 depth-first, 112, 120, 130, 264-
266, 389, 396 state space, 389-398
Searching game tree, 401-407 Searching state space, 284-294 Search tree, 96, 110-112, 130-131,
191 pruning using cut, 190-192
Second-order predicate, 314-318 Second-order programming, 301-
318 combining with nondeterministic
programming, 306 Selection in relational algebra, 43 Semantics, 101-105
declarative, 104 denotational, 105 fixpoint, 105 operational, 102
507 Index
procedural, 102 Sergot, Marek, 355 Set difference, in relational algebra,
42 setof, 303, 317 Set predicate, 303, 315-314
implementation, 304 S-expression, 285 Shared variable, 15, 17, 43, 126, 199,
256 in conjunctive query, 17, 43 in negation as failure, 199 instantiation, 17
Shapiro, Ehud, 115, 127, 186, 231, 281, 354, 423
Shell, 226-228, 341-353, 355 Sibert, Ernie, 147 Sibling goal, 92 Sicstus Prolog, 187 Side effects, 215, 220, 227,
231 Silver, Bernard, 457 Simple query, 12-18 Simplification of expression, 374 Skeleton, 238, 240-242, 244-245,
366-368 SLD tree. See Search tree SLD resolution, 115 Smith, D., 374 Snips, 212 Software engineering, 242 Sondergaard, Harald, 300 Sort
insertion, 69-70, 333 interchange, 194-195 merge, 69 permutation, 68-69, 252 quick, 69-71, 122-123, 135, 288-
289 Specification, 242-243 Specification formalism, 236 Stack, 89, 166, 208-209, 321-322
overflow, 333 scheduling policy, 93, 119
Standard Prolog, 128, 150, 232, 244, 317
State space graph, 389, 402 State space search, 380-398 Static predicate, 232 Steel, Sam, 213 Stepwise refinement, 67 Stepwise enhancement, 240-242,
244, 366 Sterling, Leon, 85, 244, 245, 280,
281, 355, 373, 374, 438, 457 Stimulus/response pair, 273-274 Stream, 231, 300 String manipulation, 216 Structure
incomplete, 146, 287 incremental buildup, 292 recursive, 291
Structured data, 3 5-38 Structure inspection, 163-174, 178,
315 Stuckey, Peter, 187 Subject/object number agreement in
grammar, 384-385 Substitution, 14, 27, 88
in a list, 71 inaterm, 169-171, 182-183 ¡n a tree, 75
Successor function, 46 Success node, 110 Symbolic expression manipulation,
78-81, 439, 457 System predicates, 149-152
Tail of a list, 58 Tall recursion optimization, 192,
196-197, 209, 213, 231, 423 Tait recursive loop, 231, 423 Term, 11, 13-14, 27, 35, 88, 167-169,
180-18 5 accessing, 167-168 building given list, 173 compound, 13-14, 27, 35, 37, 108,
16 7-173
508 Index
Term (cont.) copying, 183, 184 definition, 13 finding subterm, 168-169 identity, 180 reading, 215 size, 108 substitution, 169-171 unification. See Unification writing, 215
Takeuchi, Akikazu, 373 Tamaki, H., 373 Tarniund, Sten-Ake, 300 Techniques, 239-242, 244-245 Tester, 250, 252, 254 Thawing terms, 184, Thompson, Henry, 281 Tick, Evan, 280 Time complexity, 108, 209 Top-down construction of struc-
tures, 144-145, 237, 286-288, 293, 383
Top-down development, 65-67 Top-down evaluation, 44 Towers of Hanoi, 81-82, 97-98,
22 1-222 Trace, 22-23, 92, 96-98, 120-123,
328, 287 as meta-program, 328 of meta-interpreter, 325 of Prolog computation, 120-12 3
Transformation, of recursion to iteration, 154-159
Transitive closure of relation, 40-41 Tree
binary, 72-77, 85, 295 empty, 72 game, 404-403 isomorphism of, 74, 264 parse, 383 search, 96, 110-112, 130-131, 191 traversal, 75, 335
Turing machine, 115, 323 Type, 45, 242
complete, 107 condition, 50-51, 58 definition, 45 incomplete, 107 me ta-logical predicate, 176 predicate, 163-164, 176 recursive, 45
Ueda, Kazunori, 127 Unary relation, type of a term,
163-164 Uncertainty reasoning, 330-331,
3 54-3 5 5 Unfolding, 286, 288, 3 57-360, 373 Unification, 87-91, 98-99, 109, 125-
126, 127, 143, 179-180, 251, 286
algorithm, 88-90, 179-180 including occurs check, 179-180
Unifier, 88 Union in relational algebra, 42 Unit clause, 18, 28 univ, 171-173 Universal modus ponens, 19, 101 Universal quantification, 15, 18-19 Universally quantified fact, 15-16
Van Emden, Maarten, 99, 115, 212 Vanilla meta-interpreter, 324 var, 176, 317 Variable
anonymous, 234 binding, 91 definition, 13 difference in Prolog, 13 identity testing, 180-182 logical, 13, 91, 126, 156, 287, 383,
376 mnemonics, 29 as object, 182-185 predicate names, 315 renaming, 88, 92 scope, 17 shared, 15, 17, 43, 126, 256
509 Index
size, 108 type, 126 type checking, 256
Van Caneghem, Michel, 127, 213 Venken, Raf, 373 Verification, 236
WAM, 213 Warren Abstract Machine, 213 Warren, David H. D., 127, 128, 213,
232, 282, 300, 317, 318, 388, 478
Warren, David Scott, 127, 213 Weak cut, 212 Weizenbaum, Joseph, 281 Weiham, Bob, 457 Why explanation, 346 Wilson, Walter, 407 Winograd, Terry, 388 write, 215
Yalçinalp, Umit, 354, 355
Zebra puzzle, 262-263, 280 Zero-sum game, 404
Logic Programming Ehud Shapiro, editor Koichi Furukawa, Jean-Louis Lassez, Fernando Pereira, and David H. D. Warren, associate editors
The Art of Prolog: Advanced Programming Techniques, Leon Sterling and Ehud Shapiro, 1986
Logic Programming: Proceedings of the Fourth International Conference (vol- umes i and 2), edited by Jean-Louis Lassez, 1987
Concurrent Prolog: Collected Papers (volumes 1 and 2), edited by Ehud Shapiro, 1987
Logic Programming: Proceedings of the Fifth International Conference and Symposium (volumes i and 2), edited by Robert A. Kowaiski and Kenneth A. Bowen, 1988
Constraint Satisfaction in Logic Programming, Pascal Van Hentenryck, 1989
Logic-Based Knowledge Representation, edited by Peter Jackson, Han Re- ichgelt, and Frank van Harmelen, 1989
Logic Programming: Proceedings of the SiNth International Conference, edited by Giorgio Levi and Maurizio Martelli, 1989
Meta-Programming in Logic Programming, edited by Harvey Abramson and M. H. Rogers, 1989
Logic Programming: Proceedings of the North American Conference 1989 (volumes 1 and 2), edited by Ewing L. Lusk and Ross A. Overbeek, 1989
Logic Programming: Proceedings of the 1990 North American Conference, edited by Saumya Debray and Manuel Hermenegildo, 1990
Logic Programming: Proceedings of the Seventh International Conference, edited by David H. D. Warren and Peter Szeredi, 1990
The Craft of Prolog, Richard A. O'Keefe, 1990
The Practice of Prolog, edited by Leon S. Sterling, 1990
Eco-Logic: Logic-Based Approaches to Ecological Modelling, David Robertson, Alan Bundy, Robert Muetzelfeldt, Mandy Haggith, and Michael Uschold, 1991
Warren's Abstract Machine: A Tutorial Reconstruction, Hassan Aït-Kaci, 1991
Parallel Logic Programming, Evan Tick, 1991
Logic Programming: Proceedings of the Eighth International Conference, edited by Koichi Furukawa, 1991
Logic Programming: Proceedings of the 1991 International Symposium, edited by Vijay Saraswat and Kazunori Ueda, 1991
Foundations of Disjunctive Logic Programming, Jorge Lobo, Jack Minker, and Arcot Rajasekar, 1992
Types in Logic Programming, edited by Frank Pfenriing, 1992
Logic Programming: Proceedings of the Joint International Conference and Symposium on Logic Programming, edited by Krzysztof Apt, 1992
Concurrent Constraint Programming, Vijay A. Saraswat, 1993
Logic Programming Languages: Constraints, Functions, and Objects, edited by K. R. Apt, J. W. de Bakker, and J. J. M. M. Rutten, 1993
Logic Programming: Proceedings of the Tenth International Conference on Logic Programming, edited by David S. Warren, 1993
Constraint Logic Programming: Selected Research, edited by Frédéric Ben- hamou and Alain Colmerauer, 1993
A Grammatical View of Logic Programming, Pierre Deransart and Jan Maluszyuiski, 1993
Logic Programming: Proceedings of the 1993 International Symposium, edited by Dale Miller, 1993
The Gödel Programming Language, Patricia Hill and John Lloyd, 1994
The Art of Prolog: Advanced Programming Techniques, second edition, Leon Sterling and Ehud Shapiro, 1994
- Cover
- Contents
- Figures
- Programs
- Series Foreword
- Foreword
- Preface
- Preface to First Edition
- Introduction
- I Logic Programs
- 1 Basic Constructs
- 2 Database Programming
- 3 Recursive Programming
- 4 The Computation Model of Logic Programs
- 5 Theory of Logic Programs
- II The Prolog Language
- 6 Pure Prolog
- 7 Programming in Pure Prolog
- 8 Arithmetic
- 9 Structure Inspection
- 10 Meta-Logical Predicates
- 11 Cuts and Negation
- 12 Extra-Logical Predicates
- 13 Program Development
- III Advanced Prolog Programming Techniques
- 14 Nondeterministic Programming
- 15 Incomplete Data Structures
- 16 Second-Order Programming
- 17 Interpreters
- 18 Program Transformation
- 19 Logic Grammars
- 20 Search Techniques
- IV Applications
- 21 Game-Playing Programs
- 22 A Credit Evaluation Expert System
- 23 An Equation Solver
- 24 A Compiler
- A Operators
- References
- Index