Table Of ContentWolfgang 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, with
out 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 mecha
nization 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-experi
ments. 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 activi
ties 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 com
puter programs developed in this field have tought us much
about the nature of human thinking.
One of its prominent features is the ability for logi
cal reasoning which had been studied extensively by the logi
cians 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 sur
prisingly 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 present
ly available in this area and to give an idea of their useful
ness 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 first
order 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 concen
trated on how to el iminate the enormous redundancy experienced
in running computer systems which realized such proof proce
dures. Much of it was based on resolution, but some was car
ried out also with a rather different background. With our uni
form 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 fea
tures 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 the
one extreme, there would be those who just want to understand
the ideas behind all the formalism and thus study the examples
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 pre
requisites 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 think
ing 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 connec
tion method is then developed, first, in chapter II, on the
level of propositional logic and second, in a strictly par
allel 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 connec
tion method, is regarded as the basis for the field of auto
mated 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 partic
ular 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, how
ever, 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 chap
ter and the bibliography as a whole are by no means comprehen
sive. Rather, they reflect both, the author's limited knowl
edge 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 stellver
tretend 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 an
fechtbar, 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 verdan
ken 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 ge
meinsamen projektarbeit. Oem Fachbereich Informatik der Hoch
schule der Bundeswehr Munchen, insbesondere Herrn Prof. W.
Hahn, bin ich fur die Erlaubnis zur Benutzung eines Textauto
maten 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 unbe
trachtlichen Anteil an der Fertigstellung haben •
.l I.bbreviation Intended meaning
ATP Automated Theorem Proving
fol first-order logic
w.r.t. with respect to
iff if and only if
A iff B iff C A iff Band B iff C
D. Definition
T. Theorem
L. Lemma
C. Corollary
F. Formula
q.e.d. quod erat demonstrandum
(what had to be proved)
o end of proof or definition
Table 1. List of abbreviations
x
Kind of objects Standard Symbols
propositional variables P, Q, R
constant symbols a, b, c
function symbols f, g, h
terms s, t
predicate symbols P, Q, R
signum or arity n
literals K, L, M
object variables x, y, z
formulas, matrices 0, E, F
clauses c, d, e
paths p, q
connections u, v, w
sets of connections U, V, W
connection graphs G, H
natural numbers m, n, I
indices i, j, k
sets of indices I, J
occurrences, positions 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, sum, product
i~1 1
set of natural numbers with 0 EN
1N
o empty set
v,n union, intersection
set difference
X\.:JY union in the special case XnY o
b,
i~ union, intersection with
Xi , xi
A
~ Xl' = 0, Xl' 0
i= 1 i= 1
number of elements in set X, i.e.
cardinality of X
X x Y cartesian product of X and Y
Xn, X*, X+ n-fold product, ~ Xi , MCD Xi
2X set of subsets in X
n mod m n modulo m
Table 3. Standard notations
