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Lectu re Notes in
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28
Theoretical Approaches to
Non-Numerical Problem Solving
Proceedings of the IV Systems Symposium
at Case Western Reserve University
Edited by R. B. Banerji and M. D. Mesarovic
Systems Research Center
Case Western Reserve University, Cleveland, Ohio
•
Springer-Verlag
Berlin· Heidelberg · New York 1970
Advisory Board
H. Albach A. V. Balakrishnan F. Ferschl
W. Krelle . N. Wirth
ISBN-13: 978-3-540-04900-5 e-ISBN-13: 978-3-642-99976-5
DOl: 10.1007/978-3-642-99976-5
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher,
the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin· Heidelberg 1970. Library of Congress Catalog Card Number 79-121996.
Title No. 3777
INTRODUCTION
Advances in computer technology have pointed out the next important area of
computer applications: solution of non-numerical problems. It is hardly necessary
to emphasize the importance of these kind of problems. First of all most of the
decisions one has to make in real-life situations are non-numerical in the first
instance and can be represented as numerical problems only as approximations which
are often only partially valid. Second, to use the computer to its full potential
it should be employed as a logical machine, capable of deduction, and not just as
a numerical calculating machine. Thus the computer would extend man's capability
for logical reasoning and not just for his capability to do fast and accurate
calculation.
It is not a new area; indeed non-numerical problems are central in fields
such as artificial intelligence, heuristic programming, pattern recognition,
classification and information-processing (and retrival) etc. However, it is fair
to assess that progress in the area has not been quite as expected. One of the
reasons was a lack of conceptual and theoretical framework in which to investigate
different classes of non-numerical problems to improve understanding of various
types of problems and methods for their solutions and furthermore to enable the
methods which have been proven as effective in one situation to be used in another
situation with appropriately similar structure.
To give an impetus to the process of further theoretical developments in this
direction, the Fourth Systems Symposium of the Systems Research Center at Case
Western Reserve University was organized on the topic of "Theoretical Approaches
to Non-Numerical Problem Solving". The meeting started on November 19, 1968 and
lasted two days. It essentially had five parts. First there were critical over
views of the major past developments; Second, there were presentations of some
basic conceptual approaches which appear to have potential for the developments of
some theories of non-numerical problem solving which are both broader in appli
cation and deeper than currently available; Third, there was a selection of current
research projects to indicate what is going on in the field; Last, but not the
iv
least, a potpounni of new areas of applications, ranging from psychology to
chemistry was presented. The symposium was concluded with a round table discus-
sion devoted to the prospect for future developments.
The Symposium was organized by a committee which consisted of ourselves,
Dr. Saul Amarel of The Rutgers University, Dr. Allen Newell of Carnegie-Mellon
University and Prof. Edward Glaser of Case Western Reserve University.
The present volume contains edited versions of the talks at the symposium.
This way, the insights of some works in the field and notably the contributors to
the Symposium are made available to an audience broader than the participants of
the Symposium. Some of the participants were unfortunately not able to submit
the written version of their talks before the publication deadline; it is with
great regret that we had to reconcile ourselves to do without these. In order to
make the proceedings more complete some of the researchers who were not able to
participate actively in the conference have been invited to contribute to the
printed volume. Because of the nature of the papers included in the volume (e.g.
containing the overviews of the various approach and the proposals for new
developments) it is expected that the book will be used as supplementary reading
in various courses in the graduate computer science programs - in particular those
dealing with artificial intelligence and non-numerical information processing.
Thanks are due to many people in addition to the organizing committee, the
Session Chairmen and the Speakers. Among these we record our gratitude to
Dr. George Ernst for his editorial help and to Mrs. Mary Lou Cantini for the
diligence and patience in preparing the master copy.
The symposium was sponsored by The Systems Research Center of the Case
Western Reserve University with financial assistance from The Thompson Ramo
Wooldridge Foundation.
R.B. Banerji and
M. D. Mesarovi c
TABLE OF CONTENTS
R.B. BANERJI, M.D. MESAROVIC, Introduction
iii
PART I: OVERVIEWS
J.A. ROBINSON, An Overview of Mechanical Theorem Proving 2
R.B. BANERJI, Game Playing Programs: An Approach and An Overview 21
G.W. ERNST, GPS and Decision Making: An Overview 59
R.F. SIMMONS, Natural Language Question Answering Systems: 1969 108
PART I I : PROBLEMS IN FOUNDATIONS 140
C.W. CHURCHMAN, The Role of Weltanschauung in Problem Solving
and Inqui ry 141
H. WANG, Remarks on Mathematics and Computers 152
1<l.D. MESAROVIC, Systems Theoretic Approach to Formal Theory
of Problem Solving 161
S. AMAREL, On the Representation of Problems and Goal-Directed
Procedures for Computers 179
PART III: CURRENT RESEARCH 245
J.R. SLAGLE, Heuristic Search Programs 246
R. BELLMAN, Dynamic Programming and Problem-Solving 274
R.L. LONDON, Computer Programs Can Be Proved Correct 281
E. THORP, A Computer-Assisted Study of GO on M x N Boards 303
R.E. FIKES, Stating Problems as Procedures to a General Problem
Solving Program 344
PART IV: NEW APPLICATIONS 362
A. NEWELL, Remarks on the Relationship Between Artificial
Intelligence and Cognitive Psychology 363
J. LEDERBERG, G.L. SUTHERLAND, B.G. BUCHANAN, E.A. FEIGENBAUM,
A Heuristic Program for Solving Scientific Inference Problem:
Summary of Motivation and Implementation 401
vi
S. LIN, Heuristic Techniques for Solving Large Combinatorial
Problems on a Computer 410
w.
JACOBS, Help Stamp Out Programming 419
B. RAPHAEL, The Relevance of Robot Research to Artificial
Intelligence 455
PART 1: OVERVIEWS
AN OVERVIEW OF MECHANICAL THEOREt·' PROVING
J.A. Robinson
Syracuse University
Syracuse, New York
The majority of the theorem proving algorithms which have been studied over
the past decade are for the first order predicate calculus. One or two very
tentative papers (Gould (1966), Robinson (1968 IFIPS), Robinson (1969)) discuss
theorem proving algorithms for the higher order predicate calculus. We shall deal
in this paper only with the first category. It is too early yet to form any use
ful appraisal of the work now being done on algorithms in the second category.
It is customary to distinguish between the first order predicate calculus
with, and the first order predicate calculus without, equality. In the spirit of
an overview we attempt in this paper to treat both languages at once without
unduly emphasizing the distinction between the two.
Since it was introduced by Davis and Putnam (1960) a certain special formu
lation of the first order predicate calculus has become accepted as the standard
framework for the development of theorem proving algorithms for the computer.
Its extremely simple syntactical and semantical structure are deliberately con
trived to facilitate this prupose. We have relegated to the Appendix a full
description of this language, together with definitions of associated nomenclature.
of this material is in fairly wide currency by now; some of it is rather new.
~1uch
Experienced readers can probably skip the Appendix, consulting it only when in the
body of the paper they encounter terminology whose meaning is not standard.
We propose here the following further terminology. We shall say that a set
C of clauses is contradictory, or that it is a contradiction, iff every
valuation of C falsifies at least one clause in C. Furthermore we say that C
is a minimal contradiction iff C is a contradiction but no proper subset of C
is a contradiction.
It is clear that if C is finite then we can effectively decide whether or
not C is a contradiction, and if so whether or not it is a minimal contradiction.
3
We say that a set C of clauses is inconsistent iff the Herbrand expansion
C* is contradictory.
Since C is always a subset of C*, contradictoriness of C implies incon
sistency of C. The converse, however, is not always true. For example, the set
of cl auses
(Px Qxy) (PA) (QAB)
is not contradictory because the following valuation, among others, satisfies it:
Px -+ true, Qxy -+ true, PA -+ true, QAB -+ true
However, it is inconsistent, since its Herbrand expansion contains the clauses:
(PA QAB) (rA) ((JAB)
The two notions contradictory, inconsistent correspond to the two ways of
interpreting clauses in which there occur one or more individual symbols. The
first way is to interpret each individual symbol as denoting some particular
individual. The second way is to interpret each individual symbol as being
universally quantified by a quantifier whose scope is the entire clause. These
two interpretations are known as the particularity interpretation and the
generality interpretation of individual symbols. So a set of clauses is contra
dictory iff it is unsatisfiable under the particularity interpretation, and incon
sistent iff it is unsatisfiable under the generality interpretatiOn.
The general theory proving problem has the form: given a finite set C of
clauses, show that C is inconsistent, i.e., show that C* is contradictory.
An algorithm is demanded which will solve sll such problems.
It is clear that in general we cannot effectively decide whether or not a set
of clauses is inconsistent. However, inconsistency (as opposed to consistency)
can be detected by exploiting the fundamental fact of first order logic: every
infinite contradiction includes a finite contradiction. This means that if we
systematically enumerate C* and test the successive initial segments for being
contradictory, we must eventually discover the inconsistency of C (if C is in
fact inconsistent) since all sufficiently long initial segments will be
contradictions.
