Wolfgang Bibel Automated Theorem Proving
Wolfgang Bibel Automated Theorem Proving Friedr. Vieweg & Sohn Braunschweig IWiesbaden
CIP-Kurztitelaufnahme der Deutschen Bibliothek Bibel, Wolfgang: Automated theorem proving I Wolfgang Bibel. - Braunschweig; Wiesbaden: Vieweg, 1982. 1982 All rights reserved Friedr. Vieweg & Sohn Verlagsgesellschaft mbh, Braunschweig 1982 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright holder. ISBN-13: 978-3-528-08520-9 e-isbn-13: 978-3-322-90100-2 DOl: 10.1007/978-3-322-90100-2
Preface Among the dreams of mankind is the one dealing with the mechanization of human thought. As the world today has become so complex that humans apparently fail to manage it properly with their intellectual gifts, the realization of this dream might be regarded even as something like a necessity. On the other hand, the incredible advances in computer technology let it appear as a real possibility. Of course, it is not easy to say what sort of thing human thinking actually is, a theme which over the centuries occupied many thinkers, mainly philosophers. From a critical point of view most of their theories were of a speculative nature since their only way of testing was by Gedanken-experiments. It is the computer which has opened here a wide range of new possibilities since with this tool we now can model real experiments and thus test such theories like physicists do in their field. About a quarter of a century ago, scientific activities of that sort were started under the label of artificial intelligence Today these activities establish a wide and prosperous field which the author, in lack of any better name, prefers to call intellectics. Without any doubt, the computer programs developed in this field have tought us much about the nature of human thinking. One of its prominent features is the ability for logical reasoning which had been studied extensively by the logicians of many centuries. In particular, their contributions within the last hundred years have prepared the grounds for the mechanization of this special feature. Although reasoning certainly is part of most intellectual activities, it naturally plays a particularly important role in mathematics. Not surprisingly then, the first attempts towards automatic reasoning were made in mathematical applications focusing on generating
VI proofs of mathematical theorems. For this historical reason, this subarea within intellectics is still identified as automated theorem proving although proving mathematical theorems is just one in a wide variety of applications. The purpose of this book is to prov ide a comprehens i ve development of the most advanced basic deductive tools presently available in this area and to give an idea of their usefulness for many important applications. Because of the rapid expansion of this field, which in a wider sense also is termed automated deduction, it is certainly not possible any more to cover all its aspects in a single book. Hence our attention will focus on the classical tool of proof procedures for firstorder logic which in our opinion are to be regarded as basic for the whole field, at least for the time being. In the 1970' s much research in this area has concentrated on how to el iminate the enormous redundancy experienced in running computer systems which realized such proof procedures. Much of it was based on resolution, but some was carried out also with a rather different background. With our uniform treatment based on what we call the connection method we hope to have re-combined these various attempts into one single stream of research, which culminates in the description of what, accord ing to current technology, appear to be the features of a most advanced proof procedure for first-order logic. Unfortunately, these features have become so complex that any author dealing with this topic faces a real problem of presentation. On the one hand, because of this complexity a rigorous treatment is of essential importance in order to avoid serious errors or misjudgements. On the other hand, many readers will be frightened by the resultant formalism, thus creating the need for plenty of illustrations and informal descriptions. We have made an attempt to serve both these needs by pairing the rigorous definitions, thorems and proofs with informal descriptions and discussions, illustrated with many examples. If this attempt has been successful then the book might actually serve for a wide spectrum of readers. On one extreme, there would be those who just want to understand the ideas behind all the formalism and thus study the examples the
VII guided by the informal discussions without going much into the details of formal definitions, theorems and proofs. On the other extreme, well-trained logicians might easily skip much of the informal text. And in the middle there are those readers who are grateful for informal explanations but also acknowledge the necessity of preciseness for such a complex topic, and thus read both these approaches in parallel. The ability to read mathematical definitions, theorems and proofs together with some basic knowledge about elementary set theory and about algorithms are actually all the prerequisites needed for a full understanding of most parts of the book. However, some familiarity with mathematical logic and/or some previous training in abstract mathematical thinking will certainly be helpful for coping with the intrinsic complexity of some of the results. Although this book has not been explicitly designed as a textbook it may well be used in instructor-student settings. For such purposes a number of exercises of varied difficulties may be found at the end of each chapter 1 isted in the sequence of the presented topics. The selection of material for such a course should be easy with the following hints. Chapter I provides a short introduction into logic as the formal structure of natural reasoning. The basic connection method is then developed, first, in chapter II, on the level of propositional logic and second, in a strictly parallel treatment in chapter III, on the level of first-order logic. This, together with the first two sections in chapter IV, which introduce resolution and embed it into the connection method, is regarded as the basis for the field of automated theorem proving. The rest of chapter IV contains more specialized material on the connection method towards a most advanced proof system for first-order logic, which will be of particular interest for researchers specializing in this field. Readers with a more general interest might rather consider the material in chapter V, perhaps even at an earlier stage of their reading. It briefly introduces some of the possible applications and extensions of first-order theorem proving.
VIII Each chapter is preceded by a more detailed overview of its contents for further orientation. Moreover, the many references to previous or later parts of the book within the text should ease to begin reading at any of its parts. For this purpose we use a familiar numbering scheme. For instance, (III.3.5) refers to the item labeled 3.5 in chapter III. By convention, the number of the chapter is deleted for any reference within the actual chapter, that is, within chapter til the previous reference is simply (3.5) rather than (III.3.5). The same applies to figures and tables which, however, are numbered independently. The abbreviations used are generally familiar and are listed in table below. Also with our denotations we have tried to follow common practice as listed in table 2 and 3. Both, the historical remarks at the end of each chapter and the bibliography as a whole are by no means comprehensive. Rather, they reflect both, the author's limited knowledge of an exploding literature and their direct relevance to the topics we consider in this book. Finally, we hope that the reader acknowledges the author's difficulty in expressing the material in a non-native language. Munchen, Dezember 1981 w. Bibel ACKNOWLEDGEMENTS Man ist geneigt, die Vollendung eines solchen Buches als ein personlich wichtiges Teilziel zu interpretieren, das stellvertretend fur vieles andere im eigenen Leben steht. Deshalb sieht man sich bei solcher Gelegenheit auch zum Ruckblick auf die Einflusse veranlaj3t, (He den Weg bis hierher mitbestimmt haben. lch mul3 gestehen, dal3 mir jede Auswahl unter solchen Einflussen und die damit verbundene Gewichtung zumindest anfechtbar, wenn nicht sogar willkurlich erscheint. Deshalb
IX mochte ich nur feststellen, da~ ich dankbar an viele Menschen denke, die mich in Liebe, Freundschaft, manche auch in Ha~ oder Gegnerschaft auf meinem Weg gefordert haben. Die vorbildliche Gestaltung des Textes selbst verdanken wir aile dem auf3erordentlichen Geschick von Frl. H. Hohn, die mit unermudlichem Einsatz aile Schwierigkeiten zu meistern verstand. Bei den zeichnungen und Sonderzeichen war zudem Frau A. Bussmann behilflich. Dr. K.-M. Hornig sowie Herrn A. Muller bin ich fur viele Korrekturen und Verbesserungsvorschlage dankbar. Ihnen verdanke ich auch manche Anregung aus der gemeinsamen projektarbeit. Oem Fachbereich Informatik der Hochschule der Bundeswehr Munchen, insbesondere Herrn Prof. W. Hahn, bin ich fur die Erlaubnis zur Benutzung eines Textautomaten verpflichtet. Herrn Prof. K. Mehlhorn sei fur die an den Verlag gegebene Anregung eines solchen Buches gedankt. Meine Musikfreunde, jedoch besonders meine Frau und meine Kinder haben mir die mit der Niederschrift verbundenen Muhen ertraglicher gemacht, wodurch sie einen nicht unbetrachtlichen Anteil an der Fertigstellung haben. li.bbreviation Intended meaning ATP fol w.r.t. iff A iff B iff C D. T. L. C. F. q.e.d. o Automated Theorem Proving first-order logic with respect to if and only if A iff Band B iff C Definition Theorem Lemma Corollary Formula quod erat demonstrandum (what had to be proved) end of proof or definition Table 1. List of abbreviations
x Kind of objects propositional variables constant symbols function symbols terms predicate symbols signum or arity literals object variables formulas, matrices clauses paths connections sets of connections connection graphs natural numbers indices sets of indices occurrences, positions Standard Symbols P, Q, R a, b, c f, g, h s, t P, Q, R n K, L, M x, y, z 0, E, F c, d, e p, q u, v, w U, V, W G, H m, n, I i, j, k I, J r substitutions p, (J truth values T Comment. All symbols may be decorated with indices etc. Table 2. Standardized denotations
XI N'otation Meaning n \;' m, i~1 1 1N o v,n X\.:JY i~ Xi, b, xi X x Y Xn, X*, X+ 2X n mod m sum, product set of natural numbers with 0 EN empty set union, intersection set difference union in the special case XnY o union, intersection with ~ Xl' = 0, A Xl' 0 i= 1 i= 1 number of elements in set X, cardinality of X cartesian product of X and Y n-fold product, ~ set of subsets in X n modulo m CD Xi, M Xi i.e. Table 3. Standard notations
Contents Preface... V Acknowledgements List of abbreviations Standardized denotations Standard notations Contents...... VIII IX X XI XII CHAPTER I. Natural and formal logic 1 Logic abstracted from natural reasoning 2. J~gical rules.... 1 6 CHAPTER II. The connection method in propositional logic 1. The language of propositional logic 2. The semantics of propositional logic 3. 4. A basic syntactic characterization of validity The connection calculus 5. Consistency, completeness, and confluence 6. 7. 8. Algorithmic aspects...... Exercises.... Bibliographical and historical remarks 11 11 21 25 32 40 45 53 55 CHAPTER II I. The connection method in first-order logic 1. 2. 3. 4. '5. 6. The language of first-order logic The semantics of first-order logic..... A basic syntactic characterization of validity Transformation to normal form Unificatiofl.... The connection calculus..... '57 58 67 70 84 88 97
XIII 7. Algorithmic aspects 109 8. Exercises........................................... 116 9. Bibliographical and historical remarks 118 CHAPTER IV. Variants and improvements 1 2. 3. 4. 5. 6. 7. 8. 9. 10. n. 12. 13. Resolution Linear resolution and the connection method On performance evaluation Connection graph resolution and the connection method A connection procedure for arbitrary matrices Reduction, factorization, and tautological circuits Logical calculi of natural deduction An alternative for skolemization Linear unification Splitting by need Summary and prospectus Exerc ises.... Bibliographical and historical remarks 119 120 134 138 144 155 162 170 178 186 195 207 214 216 CHAPTER V. Applications and extensions 1. Structuring and processing knowledge 2. programming and problem solving 3. The connection method with equality 4. Rewrite rules and generalized unification 5. The connection method with induction 6. The connection method in higher-order logic 7. Aspects of actual implementations 8. 9. 10. Omissions Exercises.... Bibliographical and historical remarks 218 219 225 234 242 247 254 262 271 273 275 REFERENCES INDEX LIST OF SYMBOLS 277 288 293