Table Of ContentLecture Not ens i
Computer Science
Edited by .G Goos and .J Hartmanis
90
David .M Sandford
Using Sophisticated Models ri
noituloseR Theorem Proving
galreV-regnirpS
Berlin Heidelberg New kroY 1980
Editorial Board
W. Brauer .P Brinch Hansen .D Gries .C Moler G. re110mgeeS
1 Stoer .N Wirth
Author
David .M Sandford
Visiting Assistant Professor
Laboratory for Computer Science ,hcraeseR
Department of Computer Science
Rutgers University
New Brunswick, .J.N 08903/USA
AMS Subject Classifications :)9791( 68A40, 68A45
CR Subject Classifications :)4791( ,12.5 3.60
ISBN 3-540-10231-0 Berlin Heidelberg Springer-Verlag NewYork
ISBN 0-387-10231-0 NewYork Springer-Verlag Heidelberg Berlin
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ABSTRACT
Two related topics are treated dealing with methods for
mechanical theorem proving in the first order predicate
calculus using the resolution system of J. A. Robinson.
The first topic deals with a new strategy, called
Hereditary Lock Resolution (HLR) which is a refinement of the
original resolution inference rule. HLR is composed of two
interacting refinements. One is a modification of the locking
refinement (a syntactic refinement) of Boyer, and the other is
a strengthening of the model strategy (a semantic refinement)
of Luckham. Previously known strategies combining syntactic
and semantic components either used a weaker syntactic
strategy than lock resolution, or used weaker semantic
notions, or were incomplete (i.e. unable to prove some
theorems). HLR is complete and sound (i.e~ never constructs
fallacious proofs). HLR generates a search space involving
clauses, as does ordinary resolution, but each clause has
attached to it an additional data structure which contains
information about the deduction leading to that clause. This
data structure is called an FSL (False Substitution List) and
consists of a set of literals all of which must be falsifiable
according to some model (which initially can be chosen
arbitrarily). The FSL mechanism is applicable to other
semantic refinements of resolution besides HLR, and this is
illustrated specifically for the case of the model strategy of
Luckham.
The second topic concerns the specification and use of
models in resolution inference systems. The usual requirement
in semantic refinements of resolution has been that the model
used must be a Herbrand interpretation, which is an abstractly
defined way of considering models. However, in pragmatic
situations where implemented procedures must utilize models,
Herbrand interpretations which capture the relevant structure
of the domain to be modeled usually are both difficult to find
and computationally costly to use. We take the position that
the essence of the difficulty is that Herbrand interpretations
require the specification of details which are mostly
irrelevant to the theorem proving task, and that the way out
of this difficulty is to develop a theory of models which are
based on incomplete specification. The key to doing this is
to focus on the interface between a semantic refinement of
resolution (e.g. HLR) and the model. This interface is simple
and is adequately summarized by the notion of a semantic
function, which is a function mapping logical sentences into
the values "true" or "false". Once this is done a simple
theory of incompletely specified models can be developed which
defines semantic functions with the appropriate properties.
The completeness of HLR and other semantic refinements can be
demonstrated using these semantic functions instead of the
usual Herbrand interpretations.
PREFACE
This work is a slightly edited version of the authors
Ph.D. thesis, which was originally entitled
"HL-Resolution: A Semantic Refinement of Resolution and a
Theory of Model Specification". Thanks are due to several
people who had a constructive influence on the completion of
this work.
Foremost I wish to thank my adviser Professor
Chitoor V. Srinivasan for patience, faith, and encouragement
in a (seemingly) endless writing process.
My sincerest appreciation to Professor Ann Yasuhara for
careful readings of this material, both in its present and
in previous forms.
Special thanks are due to Professor Donald W. Loveland
of Duke University for literally (physically) going out of
his way to serve on the thesis committee. I also thank
Professors Martin Davis of the University of California and
Wolfgang Bibel of the Technische Universitat Munchen for
reading and offering comments on an earlier draft. I thank
Professor Juris Hartmanis in his capacity as editor of
Lecture Notes in Computer Science, and Professor David
Luckham of Stanford University for a favorable review.
This work was supported in part by the Advanced
Research Projects Agency of the Office of the Secretary of
Defense under grant DAHCIS-73-G6.
CONTENT~
Abstract ii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface .................................................. iii
List of Figures .......................................... vi
Chapter I
1.0 Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Conventions and Definitions ........................ 8
Chapter 2
2.0 Hereditary Lock Resolution ......................... 12
2.1 Examples of HL-Resolution .......................... 13
2.2 Combining LR and TMS: The Combination Issue ........ 34
2.3 TMS and History Tracing ............................ 43
2.4 Formal Definition of HL-Resolution ................. 53
2.5 Soundness of HL-Resolution ......................... 77
Chapter 3
3.0 Completeness of HL-Resolution ...................... 81
3.1 Completeness Theorems .............................. 81
3.2 Herbrand Model Completeness Proof .................. 84
3.3 Completeness of the Inference Rule HLR - t.v.d ..... 103
3.4 HLR False Permissive Completeness Proof ............ II0
lllV
Chapter 4
4.0 Models ............................................. 114
4.] Notation and Preliminary Definitions ............... 118
4.2 Theoretical Construction
of Sound Semantic Functions .................... 127
4.3 Example III -- A Simple Model Structure ............ 143
4.4 Pragmatic Construction
of Sound Semantic Functions .................... 149
4.5 Example IV --- A Sophisticated Model ............... 164
4.6 General Discussion of Models ....................... 193
Chapter 5
5.0 Discussion of HLR .................................. 207
5.1 Refinement Strength of HLR ......................... 207
5.2 Overhead of HLR .................................... 215
5.3 Experimental Results -- Domain of Applicability .... 219
5.4 Compatibility with other Strategies ................ 222
5.5 Areas for Future Research .......................... 227
Bibliography ............................................. 229
Table of Symbol Conventions .............................. 234
Index of Definitions ..................................... 237
JX
LIST OF FIGURES
Figure I--I A TMS Search ............................... 5
Figure 2--I Example I .................................. 19
Figure 2--2 Example I£ ................................. 25
Figure 2--3 A TMS + FSL Search ......................... 45
Figure 2--4 An Infeasible Deduction .................... 45
Figure 4--I Relationship of Herbrand
Interpretations and Model
Schemes in Different Languages ........ 126
l×
"There are no solved problems; there are only problems
that are more or less solved."
H. Poincare
Chapter 1
INTRODUCTION
Despite limited success to date there remains strong
interest in mechanizing theorem proving for the first order
predicate calculus. The areas of application of a
pragmatically effective automatic theorem prover include the
purely logical (Morgan, 1976), as well as areas of
considerable economic importance, such as program
verification, and areas of application in artificial
intelligence such as reasoning about actions. While in
certain proposed application areas there remain unresolved
issues concerning the appropriateness of a first order
theorem proving approach (and sometimes of a theorem proving
approach altogether) the primary difficulties are related to
the efficiency of current techniques, even in those areas
where the first order logic seems quite appropriate. The
efficiency problem is currently a totally debilitating
problem with respect to achieving most pragmatic
applications.
As humans would intuitively Judge problem complexity,
the theorems arising in many application areas seem only
several times harder than what can currently be handled by
automatic methods. Thus one would expect that relatively
modest efficiency increases (or even increases in machine
hardware capability) would bring application areas into the
realm of both economic feasibility and real time
feasibility. This is not the case however. Human intuitive
estimates of how difficult it is to prove a particular
theorem are based on features quite distinct from the
features of theorems which determine the difficulty of proof
finding for current automatic techniques. The level of
theorem difficulty that can be handled by current techniques
is that level that can be just proven before driving the
search space into a comblnatoric explosion of search
possibilities. In such a situation, even small increases in
theorem difficulty by human standards involves massive
increases in effort to prove the theorem by automatic
procedures. For this reason the only way that automatic
theorem proving can become a pragmatic tool is through
techniques which reduce the search space size, and not by
increasing the rate at which a given search space is
explored.
For several reasons the resolution inference rule of
J. A. Robinson (Robinson, 1965) seems to be the most
promising current approach to dealing with the efficiency
issue. These reasons are not concerned with the intrinsic
efficiency of the resolution inference system, but rather
with its relative tractability in being analyzed and
modified, and include:
i. the simplicity of the inference rule
(consisting of only I or 2 primitive types of
operations);
