Table Of ContentElmarEder
Relative Complexities of
First Order Calculi
Artificial Intelligence
Kiinstliche Intelligenz
edited by Wolfgang Bibel and Walther von Hahn
Artificial Intelligence aims for an understanding and the technical realization of intelligent
behaviour.
The books of this series are meant to cover topics from the areas of knowledge processing, know
ledge representation, expert systems, communication of knowledge (language, images, speach, etc.),
AI machinery as well as languages, models of biological systems, and cognitive modelling.
In Eng/ish:
Automated Theorem Proving
by Wolfgang Bibel
Parallelism in Logic
by Franz KurfeS
Relative Complexities of First Order Calculi
by Elmar Eder
In German:
Die Wissensrepriisentationssprache OPS 5
by Reinhard Krickhahn and Bernd Radig
Prolog
by Ralf Cordes, Rudolf Kurse, Horst Langendorfer and Heinrich Rust
LISP
by RUdiger Esser and Elisabeth Feldmar
Logische Gmndlagen der Ktinstlichen Intelligenz
by Michael R. Genesereth and Nils J. Nilsson
WlSsensbasierte Echtzeitpianung
by Jiirgen Dorn
Modulare Regelprogrammierung
by Siegfried Bocionek
Automatisierung von Terminierungsbeweisen
by Christoph Walther
Logische und Fnnktionale Programmierung
by Ulrich Furbach
Schlie8en bei unsicherem Wissen in der Ktinstlichen Intelligenz
byUaSomb6
ElmarEder
Relative Complexities of
First Order Calculi
I I
Vleweg
AMS Subject Classification: 68T27, 68T15, 03B35, 03 BI0, 03 F20
Verlag Vieweg· P. O. Box 5829· D-6200 Wiesbaden· FR Germany
Vieweg is a subsidiary company of the Bertelsmann Publishing International.
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1992
Softcover reprint of the hardcover 15t edition 1992
No part of this publication may be reproduced, stored in a retrieval system
or transmitted, mechanical, photocopying or otherwise, without prior per
mission of the copyright holder.
Cover design: L. Markgraf, Wiesbaden
ISBN-13: 978-3-528-05122-8 e-ISBN-13: 978-3-322-84222-0
DOl: 10.1007/978-3-322-84222-0
Acknowledgements
The first ideas for this book developed during research at the Technische Univer
sitat Munchen with Wolfgang Bibel. Major advances were made during a short
stay granted to me at Duke University in Durham, North Carolina, and during
stays as a visiting professor at various German Universities. The ideas and the
main results of Chapter 4 developed during a stay granted to me at the University
of British Columbia (UBC) in Vancouver, Canada. Chapter 3 is an outcome of my
stay as a visiting professor at the Technische Hochschule Darmstadt. Espescially
fruitful were discussions with my colleagues at the ICOT in Tokyo where I was
granted a stay as a visiting researcher.
I want to thank Armin B. Cremers for his support. I thank Kurt Schutte
for teaching me the foundations of mathematical logic. lowe special thanks to
Wolfgang Bibel from whome I learned a lot about automated theorem proving,
and through whose initiative and support my visits to many renowned universities
and institutes were possible. I want to thank my colleagues here and abroad vor
valuable discussions.
Contents
Introduction
1 Calculi for First Order Logic 7
1.1 Basic Concepts and General Remarks. 7
1.1.1 First Order Predicate Logic . 7
1.1.2 Substitutions and Unification 9
1.2 Resolution............... 10
1.3 The Connection Method . . . . . . . 12
1.3.1 The Connection Method in Propositional Logic 14
1.3.2 The Connection Method in First Order Predicate Logic . 19
1.3.3 Splitting.............. 21
1.4 Consolution . . . . . . . . . . . . . . . . 25
1.4.1 Consolution in propositional logic 25
1.4.2 Consolution in first oder logic . . 27
1.4.3 Simulation of resolution and connection calculus by conso-
lution ....... 29
1.5 The Tableau Calculus TC 31
1.6 The Sequent Calculus. . 36
1. 7 Natural Deduction . . . 38
1.8 A Frege-Hilbert Calculus 42
2 Comparison of Calculi for First Order Logic 44
2.1 Known Results on the Complexity of Calculi. 44
2.2 Transformation to Clausal Form . . . . . . . . 48
2.3 Complexity Measures for Resolution Refutations . 52
2.4 Simulation of the Connection Calculus by Resolution 60
2.5 Non-Simulatability of Resolution in the Connection Calculus 64
2.6 Variants of the Tableau Calculus TC . . . . . . . . 67
2.6.1 The Tableau Calculi TCae, TCiae and TCm • 68
2.6.2 Comparison of Tableau Calculi ...... . 70
2.6.3 The Tableau Calculus with Unification, TC 74
u
2.7 The Method of Tableaux and the Connection Method . 77
3 The Extension Rule in First Order Logic 79
3.1 The Extension Rule. . . . . . . . . . . . . 80
3.2 Complexities of Formulas and Derivations 83
3.3 Occurrences in the Sequent Calculus .. 86
3.4 Application of Substitutions to Formulas . 92
3.5 Transformation of Sequents to Clauses . . 96
3.5.1 The Definition Set for a Derivation in the Sequent Calculus. 96
3.5.2 Transforms ........ 97
3.5.3 Equivalence of Transforms . . . . . . . . . . . . . . . 101
3.5.4 Inheritance of Transforms ............... 103
3.6 Simulation of the Sequent Calculus in Extended Resolution . 109
3.6.1 Tree Derivations 109
3.6.2 Arbitrary Derivations . . . . . . . . . . . . . . . . . . 113
3.7 Gentzen's Transformations. . . . . . . . . . . . . . . . . . . 116
3.7.1 Simulation of the Frege-Hilbert Calculus in Natural Deduction1l8
3.7.2 Simulation of Natural Deduction in the Sequent Calculus .. 120
3.7.3 Simulationofthe Sequent Calculus in the Frege-Hilbert Cal-
culus 125
3.8 Definitions...... 130
4 Connection Structures 136
4.1 Unifier Sets ..... 136
4.2 From Resolution to Connection Proofs 141
4.3 Connection Structures . . . . . . . . 147
4.4 The Connection Structure Calculus . 151
4.5 Splitting with Connection Structures 156
4.6 Extended Definitional Calculi 157
Conclusion 159
Bibliography 162
Introduction
In this paper, a comparison is made of several proof calculi in terms of the lengths
of shortest proofs for some given formula of first order predicate logic with function
symbols. In particular, we address the question whether, given two calculi, any
derivation in one of them can be simulated in the other in polynomial time. The
analogous question for propositional logic has been intensively studied by various
authors because of its implications for complexity theory. And it seems there has
not been as much endeavour in this field in first order logic as there has been in
propositional logic. On the other hand, fOr most of the practical applications of
logic, a powerful tool such as the language of first order logic is needed.
The main interest of this investigation lies in the calculi most frequently used
in automated theorem proving, the resolution calculus, and analytic calculi such as
the tableau calculus and the connection method. In automated theorem proving
there are two important aspects of complexity. In order to have a good theorem
proving system, we must first have some calculus in which we can express our
derivations in concise form. And second, there must be an efficient search strategy.
This book deals mainly with the first aspect which is a necessary condition for
the second since the length of a shortest proof always also gives a lower bound to
the complexity of any strategy. Moreover, the search space increases exponentially
with the length of a proof. Especially in interactive theorem proving systems where
the search is partly left to the user's intuition, the naturalness and conciseness of
the representation of proofs is among the main issues. This motivates our interest
in lengths of shortest proofs in a given first order calculus, and in its ability to
simulate proof techniques which are expressible in other proof calculi.
In Chapter 1, several calculi for first order predicate logic are described.
The calculi considered here are Robinson's resolution calculus, Bibel's connection
method, Beth's and Smullyan's method of tableaux, Gentzen's sequent calculus,
Gentzen's natural deduction calculus, and a Frege-Hilbert calculus, and the con
solution calculus. Since the connection method is not a fixed calculus, but rather
a design philosophy for calculi, I had to make a choice for one particular calculus.
The calculus chosen contains most of the features relevant to the lengths of short
est proofs that are included in the various connection calculi described so far. On
the other hand, it is simple enough to be easily seen to be sound. One feature
that is not contained in this calculus is splitting. Since it is not obvious how split
ting can be done when it is nested, there was no connection calculus available in
which splitting is formally defined in a completely satisfactory way. The connec-
tion structure calculus introduced in Chapter 4 can be viewed as such a calculus,
however, since it allows to simulate splitting in any form in which it is known to
be correct; and it is, in fact, a generalization of splitting. It is shown that the
consolution calculus can simulate resolution as well as the connection calculus in
a very simple and streight-forward way.
In Chapter 2, most of these calculi are compared with each other. After a
brief review of some of the major results known so far in this field, a polynomial
time transformation of a formula to clausal form, called its definitional form, is
described which is needed for comparing calculi for full first order logic with calculi
for clausal form logic. Measures are given and investigated for measuring the
complexities of resolution refutations. In the course of a resolution refutation, the
cardinalities of the clauses may grow exponentially; so the number of resolution
steps is not always an adequate measure of complexity. The first simulation result
presented here is a step by step simulation of the chosen connection calculus by
resolution. The converse is not possible. There is a class of formulas for which the
length of the shortest proof in the connection calculus grows superpolynomially
with respect to the length of their shortest resolution proof. The two main reasons
for this inefficiency are the unability of the connection calculus considered here
to forget variables that are not needed any more, and the fact that the use of
lemmata is possible only in a very restricted way in this calculus. Then variant!>
of the tableau calculus are considered, and it is shown that the version of the
connection method considered here, and a suitable variant of the tableau calculus,
are essentially identical.
In Chapter 3, Tseitin's extension rule is generalized to first order logic, and
it is proved that Tseitin's and Reckhow's result, stating that resolution can sim
ulate the cut-free sequent calculus at polynomial cost, and that resolution with
extension can simulate the sequent calculus with cut at polynomial cost, carries
over from propositional logic to first order logic. More precisely, the calculus of
extended definitional resolution is introduced. A derivation in this calculus con
sists of a transformation of a given arbitrary formula of first order predicate logic
to its definitional form, and a subsequent derivation of this definitional form by
resolution and extension. It is then proved that extended definitional resolution
can p-simulate the sequent calculus, and that definitional resolution (without ex
tension) can p-simulate the sequent calculus without the cut rule. Moreover, the
simulation maps tree derivations in the sequent calculus to tree derivations in ex
tended definitional resolution. From Gentzen's simulation results, it then follows
that extended definitional resolution can also p-simulate natural deduction and
the Frege-Hilbert calculus. Moreover, it is proved that, even when these calculi
are augmented by the feature of definition, they can be p-simulated by extended
definitional resolution.
In Chapter 4, the connection structure calculus is introduced. The connection
structure calculus is a calculus which is derived from the connection method. In
stead of the usual concept of unification it uses the concept of a unifier set which
presents a way to incorporate the process of forgetting of variables in the concept of
unifiers and of unification. It is shown how each resolution proof can be expanded
(at exponential cost) to a connection proof and how such connection proofs can be
represented more efficiently as dags, called connection structures. In particular,
multiple nested use of lemmata does not cause the exponential explosion in this
representation that it does in the connection calculus described in Section 1.3.
We give a formal definition of connection structures, and we give some theorems
holding for connection structures. A proof calculus, the connection structure cal
culus, is presented that is based on connection structures. To this end, annotations
are introduced for each node of the dag. These annotations are needed to keep
track of the unification process during derivation and of the set of paths still to
be checked for complementarity at each instant of time (in the language of the
connection method). In resolution, the object corresponding to this set of paths
is the resolvent of the considered resolution step. It is shown how splitting can be
done in the connection structure calculus.
