Table Of ContentTHE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
LOGIC PROGRAM SYNTHESIS
FROM INCOMPLETE INFORMATION
LOGIC PROGRAM SYNTHESIS
FROM INCOMPLETE INFORMATION
by
Pierre Flener
Bilkent University
"~.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
ISBN 978-1-4613-5925-8 ISBN 978-1-4615-2205-8 (eBook)
DOI 10.1007/978-1-4615-2205-8
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1995 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint ofthe hardcover lst edition 1995
AU rights reserved. No part of this publication may be reproduced, stored in
a retrieval system or transmitted in any form or by any means, mechanical,
photo-copying, recording, or otherwise, without the prior written permission of
the publisher, Springer Science+Business Media, B. V.
Printed on acid1ree pap er.
Table of Contents
Lists of Logic Algorithms, Figures, and Tables ....... .. . · .. ix
Foreword. · .. xi
Preface .. · . xiii
I STATE OF THE ART
1 Automatic Programming. . . . . . . . . . . . . . ... 3
1.1 The Grand Aim of Automatic Programming . . . . 3
1.2 Specification, Algorithm, and Programming Languages 6
1.3 A Classification of Synthesis Mechanisms. . . . . . . . .10
1.4 Requirements and Promises of Automatic Programming .11
1.5 Sample Problems . . . . . . . . . . . . . . . . . . .13
2 Deductive Inference in Automatic Programming ... . .15
2.1 Specifications by Axioms ..... .15
2.2 Deductive Inference . . . . . . . . . . .18
2.2.1 Transfonnational Synthesis. . . 18
2.2.2 Proofs-as-Programs Synthesis .19
2.2.3 Schema-Guided Synthesis . . .21
2.3 Functional Program Synthesis from Axioms .21
2.3.1 Transfonnational Synthesis of LISP Programs . . .21
2.3.2 Proofs-as-Programs Synthesis of LISP Programs .22
2.3.3 Schema-Guided Synthesis of LISP Programs .. .23
2.4 Logic Program Synthesis from Axioms . . . . . . . . . .23
2.4.1 Transfonnational Synthesis of Prolog Programs . .24
2.4.2 Proofs-as-Programs Synthesis of Prolog Programs .26
2.4.3 Schema-Guided Synthesis of Prolog Programs .. .27
2.5 Conclusions on Program Synthesis from Axiomatizations .28
vi Table of Contents
3 Inductive Inference in Automatic Programming . ........... · .29
3.1 Specifications by Examples ........ .29
3.2 Inductive Inference. . . . . . . . . . . . . .32
3.2.1 Components of a Learning System . .33
3.2.2 Rules of Inductive Inference .... .33
3.2.3 Empirical Learning from Examples .34
3.2.4 Algorithm Synthesis from Examples as a Niche of Learning .39
3.2.5 Pointers to the Literature . . . . . . . . .41
3.3 Functional Program Synthesis from Examples .42
3.3.1 Synthesis from Traces ......... .42
3.3.2 Trace-Based Synthesis from Examples. .42
3.3.3 Heuristic Synthesis from Examples .45
3.4 Logic Program Synthesis from Examples ... .46
3.4.1 Shapiro's Model Inference System ... .46
3.4.2 Other Model-Based Synthesis Systems .51
3.5 Conclusions on Program Synthesis from Examples · .52
4 A Logic Program Development Methodology · .55
....
4.1 Elaboration of a Specification .56
4.2 Design of a Logic Algorithm . . . . . .56
4.3 Transformation of a Logic Algorithm .61
4.4 Derivation of a Logic Program . . . .61
4.5 Transformation of a Logic Program .62
. . . . . . . . . . . . . . . .
5 Objectives . . . . . . . . . . . . . . ... .. . 63
5.1 Precise Statement of the Objective. .63
5.2 Motivating Examples. . . . . . . . .64
5.2.1 Sample Logic Algorithms . .64
5.2.2 Some Comments on Logic Algorithms. .71
5.3 Challenges .................. . · .75
IT BUILDING BLOCKS
6 A Specification Approach ............. ...... . .. . 79
6.1 Specifications by Examples and Properties ... . .79
6.2 Sample Specifications by Examples and Properties .80
6.3 Future Work . .83
6.4 Related Work . .83
6.5 Conclusion.. .84
Table of Contents vii
7 A Framework for Stepwise Logic Algorithm Synthesis · .. 85
7.1 Correctness of Logic Algorithms ..... .85
7.1.1 Logic Algorithm and Intentions .. .87
7.l.2 Logic Algorithm and Specification . .89
7.1.3 Specification and Intentions .90
7.2 Comparison of Logic Algorithms .90
7.2.1 Semantic Generalization . . .91
7.2.2 Syntactic Generalization . . .91
7.3 Stepwise Logic Algorithm Synthesis Strategies .93
7.3.1 An Incremental Synthesis Strategy ... .93
7.3.2 A Non-Incremental Synthesis Strategy. .94
7.4 Future Work .99
7.5 Related Work . .99
7.6 Conclusion .. .100
8 Algorithm Analysis and Algorithm Schemata · . 101
8.1 Introduction to Algorithm Schemata . . . · 101
8.2 A Divide-and-Conquer Logic Algorithm Schema . · 103
8.2.1 Divide-and-Conquer Logic Algorithm Analysis .104
8.2.2 Integrity Constraints on Instances · 107
8.2.3 Justifications ................. . .108
8.2.4 Discussion .................. . .109
8.3 Stepwise, Schema-Guided Logic Algorithm Synthesis · 110
8.4 Future Work · 111
8.5 Related Work . · 112
8.6 Conclusion .. · 114
9 The Proofs-as-Programs Method • • 115
9.1 The Problem · 115
9.2 A Method. . · 116
9.3 Correctness. · 122
9.4 Illustration · 123
9.5 Future Work · 130
9.6 Related Work. .130
9.7 Conclusion .. · 132
10 The Most-Specific-Generalization Method · .133
10.1 Most-Specific-Generalization of Terms · 133
10.2 Objective and Terminology .134
10.3 The Ground Case ... .136
lOA The Non-Ground Case · 139
10.5 Future Work . .142
10.6 Related Work . · 143
10.7 Conclusion.. · 144
viii Table o/Contents
ITI A LOGIC ALGORITHM SYNTHESIS MECHANISM
11 Overview of the Synthesis Mechanism ..... . . . . . . . . . . . . . . 147
11.1 Desired Features .... · . 147
11.2 Preliminary Restrictions · 151
11.3 A Sample Synthesis .. · . 152
12 The Expansion Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
12.1 Step I: Syntactic Creation of a First Approximation .. · . 160
12.2 Step 2: Synthesis of Minimal and NonMinimal .... . · . 161
12.3 Step 3: Synthesis of Decompose ........... . · 166
12.4 Step 4: Syntactic Introduction of the Recursive Atoms . · . 171
13 The Reduction Phase ............................. 175
13.1 Step 5: Synthesis of Solve and the SolveNonMink . · 175
13.2 Step 6: Synthesis of the Processk and Composek · 181
13.3 Step 7: Synthesis of the Discriminatek ...... . · 186
14 Conclusion .................................. 191
14.1 Summary.... . 191
14.2 Extensions . . . . . . . . . . 196
14.2.1 Negation ...... . 196
14.2.2 Auxiliary Parameters . 197
14.2.3 Supporting Other Schemata . 198
14.2.4 The Synthesis Method . . . . 198
14.3 A Methodology for Choosing "Good" Examples and Properties . 204
14.4 The SYNAPSE System ........ . 206
14.4.1 The Architecture of SYNAPSE .. . 206
14.4.2 Target Scenarios for SYNAPSE .. . 207
14.5 Evaluation . . . . . . . . . . . . . . . . . . 211
14.5.1 Insights . . . . . . . . . . . . . . . . 211
14.5.2 Comparison with Related Systems. . 213
Appendix: Conventions, Abbreviations, and Symbols ............. 215
References ..................................... 219
SUbject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
List of Logic Algorithms
Logic Algorithm 4-1: LA(compress) .................................. 57
Logic Algorithm 4-2: lA(firstPlateau) ............................... 57
Logic Algorithm 4-3: LA(compress-int-L) ............................. 61
Logic Algorithm 5-1: LA(compress-ext-L) ............................. 64
Logic Algorithm 5-2: LA(compress-int-C) ............................. 65
Logic Algorithm 5-3: LA(compress-ext-C) ............................. 65
Logic Algorithm 5-4: LA(delOddElems-L) ............................. 66
Logic Algorithm 5-5: LA(efface-L) ................................... 66
Logic Algorithm 5-6: LA(firstN-L) ................................... 66
Logic Algorithm 5-7: LA(firstN-N) .................................. 67
Logic Algorithm 5-8: LA(insert-L) ................................... 67
Logic Algorithm 5-9: LA(insert-R) ................................... 67
Logic Algorithm 5-10: LA(mernber-L) ................................. 68
Logic Algorithm 5-11: LA(merge-<A,B» .............................. 68
Logic Algorithm 5-12: LA(parity-L) ................................... 69
Logic Algorithm 5-13: LA(partition-L) ................................ 69
Logic Algorithm 5-14: LA(permutation-L) .............................. 69
Logic Algorithm 5-15: LA(plateau-N) ................................. 70
Logic Algorithm 5-16: LA(sort-int-L) {Insertion-Sort} .................... 70
Logic Algorithm 5-17: LA(sort-ext-L) {Quick-Sort} ...................... 70
Logic Algorithm 5-18: LA(sort-ext-L) {Merge-Sort} ...................... 70
Logic Algorithm 5-19: LA(split) ...................................... 71
Logic Algorithm 12-1: LA} (firstPlateau) .............................. 161
Logic Algorithm 12-2: exp(LA} (firstPlateau» .......................... 161
Logic Algorithm 12-3: LA2(firstPlateau) .............................. 166
Logic Algorithm 12-4: exp(LA2(firstPlateau» .......................... 166
Logic Algorithm 12-5: LA3(firstPlateau) .............................. 170
Logic Algorithm 12-6: exp(LA3(firstPlateau» .......................... 170
Logic Algorithm 12-7: LA4(firstPlateau) .............................. 174
Logic Algorithm 12-8: exp(LA4(firstPlateau» ............. " ........... 174
Logic Algorithm 13-1: LAs(firstPlateau) .............................. 180
Logic Algorithm 13-2: exp(LAs(firstPlateau» .......................... 180
Logic Algorithm 13-3: LA6(firstPlateau) .............................. 185
Logic Algorithm 13-4: exp(LA6(firstPlateau» .......................... 185
Logic Algorithm 13-5: LA7(firstPlateau) .............................. 190
Logic Algorithm 13-6: exp(LA7(firstPlateau» .......................... 190
x Lists of Logic Algorithms, Figures, and Tables
List of Figures
Figure 1-1: The traditional software life-cycle ........................... 4
Figure 1-2: A new software life-cycle .................................. 5
Figure 1-3: A practical view of program development ..................... 6
Figure 3-2: Part of the specialization graph for the member/2 predicate ....... 48
Figure 4-1: Spec(compress) ......................................... 56
Figure 4-2: Spec(firstPlateau) ....................................... 57
Figure 4-3: LP(compress-int-L) ...................................... 62
Figure 4-4: A transformed version of LP(compress-int-L) ................. 62
Figure 7-1: Upward and downward progression ......................... 94
Figure 10-1: MSG Method (ground case): The first two steps .............. 137
Figure 10-2: MSG Method (ground case): Example ...................... 139
List of Tables
Table 3-1: Algorithm synthesis from examples as a niche of learning ....... 40
Table 5-1: Summary of the features of some sample relations ............. 73
Table 5-2: Summary of the features of some sample logic algorithms ....... 74
Table 12-1: Database of instances of Minimal/l and NonMinimal/1 ......... 163
Table 12-2: Database of intrinsic instances of Decompose/l +h+t .......... 168
Table 12-3: Database of extrinsic instances of Decompose/l +h+t .......... 168
Table 12-4: Database of logarithmic instances of Decompose/l+h+t ........ 168
Table 14-1: Summary of the synthesis steps and tasks ................... 192
Table 14-2: Summary of the features of the synthesized logic algorithms .... 193
