Topics in Artificial Intelligence

292 Pages·1976·20.54 MB·English

by  A. Marzollo (eds.)

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES No. 2'6
TOPICS IN
ARTIFICIAL INTELLIGENCE
EDITED BY
A. MARZOLLO
UNIVERSITY OF TRIESTE
SPRINGER-VERLAG WIEN GMBH
This work is subject to copyright.
AII rights are reselVed,
whether the whole or part of the material is concemed
specificaIly those of translation, ,reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data hanks.
©1976 by Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1976
ISBN 978-3-211-81466-6 ISBN 978-3-7091-4358-2 (eBook)
DOI 10.1007/978-3-7091-4358-2
PREFACE
The mechanization of some tasks which are considered as typical of the
human intelligence has been a challenge for a long time, at least since the times of
Leibniz and Pascal. At present, within the vast developments of computer science,
the discipline called Artificial Intellingence (A.I.) has inherited this challenge and is
now confronting itself with matters like automatic problem solving, "under
standing" and translating natural languages, recognizing visual images and spoken
sounds, etc ... It does so by borrowing its theoretical tools from other sciences and,
also, by trying to build its own theoretical bases. The present volume reflects this
situation and aims at offering an appreciation of the wide scope of A.I. by collecting
contributions of various researchers active in this field.
The two first contributions, by Meltzer and by Marzollo and Ukovich are
both concerned with the general problem of giving appropriate definitions of
mathematical objects and its importance in connection with the task of automatic
theorem proving. B. Meltzer tends to broaden our view of the reasoning activity
involved in mathematics and gives a closer look at what a proof is, trying to discover
general principles governing its design. The contribution of Marzollo and Ukovich
deals with computable functions and proposes a new look at their properties. These
are considered as independent of any description of the functions themselves and
simply follow from the existence of a finite sentence unambiguously indicating their
behaviour in correspondence to each natural number.
The three next contributions (by Mandrioli, Sangiovanni Vincentelli and
Somalvico, by Kulikowski and by Levelt) consider more specific aspects 0rA.I.:
problem solving, visual images recognition and natural languages understanding.
The first one extensively illustrates an algebraic approach to problem
solving, based on the state-space, naive, syntactic and semantic descriptions.
}. Kulikowski shows an interesting approach to the problem of pattern
recognition which is based on a linguistic description and which treats a picture as a
set of local and global features. He suggests a general scheme ofp attern recognition
based on semi-ordering relations between pictures. The last contribution by W.
Levelt deals with the theory of formal grammars and the linguistic aspects of
mathematical psychology. It clarifies the empirical domain of linguistic theory and
the empiric interpretation of the elements and relations which appear in it.
A. Marzollo
March 1978
CONTENTS
Page
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Proof, abstraction and semantics in mathematics and artificial intellingence
B. Meltzer. . . . . . . . . .. .............................. 1
An introduction to some 1?asic concepts and techniques in the theory of
computable functions
A. Marzollo, W. Ukovich. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
Toward a Theory of problem solving
D. Mandrioli, A. Sangiovanni Vincentelli, M. Somalvico. . . . . . . . 47
Recognition of composite patterns
J.L. Kulikowski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Formal grammars and the natural language user: a review
W.J.M. Levelt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 291
BERNARD MELT ZER(*)
PROOF, ABSTRACTION AND SEMANTICS
IN
MATHEMATICS AND ARTIFICIAL INTELLIGENCE
(*) School of Artificial Intelligence, Department of Computational Logic. Fniversity of Edinburgh.
2 B. Meltzer
Abstract
The paradigm of a chain of inference from axioms is inadequate as an
account of proof. A re-analysis of what proof is leads to a proposal for a semantic
framework, which in a natural way accommodates abstraction, generalization and
specialization involved in proof. It can be applied to other languages than that of
mathematics, and in fact appears likely to be useful in considering the problems of
free-ranging reasoning in programming languages of artificial intelligence.
Introduction
Researchers in artificial intelligence, in designing their programs and
examining their results, are constantly made aware of how rich and how poorly
understood are the cognitive activities. involved in doing mathematics and other
kinds of reasoning. Demonstrating a truth in mathematics involves far more than
drawing a chain of inferences from a given collection of axioms. For example,
introspection of one's mental activities in solving a school geometry problem might
reveal that all of, and maybe more than, the following are involved: a perception, a
thought experiment, a guess, rationalization of the guess by means of a vague
concept, failure to intuit a proof, induction and generalization, drawing from
memory a related complex concept, back-tracking because a line of attack looks like
being clumsy and tedious, feeling of discomfort that a proof arrived at is
peripherally not water-tight, examining special cases and carrying out a proof only in
outline. In fact, it is often the case that one finally has a satisfying demonstration
without even having carried through the small nitty-gritty chains of inference
which constitute the major part of many modern automatic theorem- proving
programs.
of course it is the aim of A.!, research in the first place to develop
computational models of such and other activities and try to discover general
principles governing their design. The considerations below constitute an attempt to
broaden our view of t~e reasoning activity involved in mathematics and similar kinds
of reasoning beyond the framework of the proof and model theory of formal logic.
No pretence is made that the kind of conceptual framework this leads to is adequate
for all the rich structure of cognitive activity we are interested in, but it is a tentative
in this direction. The point of departure is a closer look at what proof is,
epistemologically speaking.
Mathematics and Artificial Intelligence 3
-----------------------------------------
What are Proofs About?
Suppose one proves a theorem in plane geometry about all squares. If
the proof made use of the fact that a square is a four-sided rectilinear figure and its
sides are equal, but not that its angles are right-angles, then -in any strict sense - the
theorem proved is not about squares but about rhombuses. And this is a general
feature of mathematical proofs: they do not use ali the properties of the object the
theorem is supposed to be about, but only a selection of them. In fact they could
not use them all, since mathematical objects have in general an infinite number of
properties and a finite proof can use only a finite number of them. So, inherent in a
proof of a theorem there is involved some abstraction, namely a selection of features
of the object the theorem is supposed to be about. The rhombus is obtained by
abstraction from the square when the feature of right- angledness is left out of the
selection; and our theorem is about this 'more general' object, its application to the
square involving a 'special instance' of this object.
The implication of this insight is more far-reaching than migh t at first
appear. To illustrate: suppose in arithmetic we prove something about the number 2.
for instance that its square root is irrational. Now 2 certainly has an infinite number
of properties (including, for example, that it is less than 3, that it is less than 4, that
it is less than 5, etc.) and only a finite number of them are used in the proof. The
theorem proved is therefore essentially not about 2 as we intuitively understand it
but about some other 'more general' object 21 , say, for which unfortunately we may
not have a generally accepted name like 'rhombus' in the previous example. In this
case actually a proof might use of 2 only the facts that it is a natural number and
that it is a prime; so that it would be more correct to say that the theorem proved is
not about 2 but about any prime number. Even this would not be entirely correct since
only some of the properties of prime numbers are used. as indeed for that matter in
the earlier example only some of the properties of a rhombus are used; but, provided
we are clear about the epistemological situation, we have to reconcile ourselves to
the inade(juacy and ambiguity of natural language in such cases.
Thus. proof implicitly involves not only deduction but abstraction and
generalization, traditionally thought of as the province of induction: and the proof
of any theorem implicitly creates a new mathematical object. namely that one
defined by the (explicit and implicit) premises actually used in the deduction.
Recognizing this feature of mathematical reasoning. or for that m;ltter of the kind of
reasoning implemented in many A.!. programs, has interesting COllSCljUCnces.
4 B. Meltzer
Semantics
For a fuller grasp of the epistemological situation we need to have a
general notion of what we mean by 'mathematical object'. The above discussion and
a classical definition of Leibniz's suggest what this might be. Leibniz defined
identity as follows: two things are identical if everything that can be truthfully
asserted about either can be similarly asserted about the other. It seems natural then
to take the meaning of a mathematical object, like a number or a triangle or a vector
or whatnot, to be the collection of true assertions about it. Then the operation of
abstraction, which occurs in proofs as illustrated" above, consists in making a
selection from this collection. The meaning of 'a square' is the collection of true
statements about squares, the meaning of a 'rhombus' is a sub-collection of the
latter. The square, being a special case of the rhombus, has to have ~ said about
it than the rhombus, to help identify it from among all possible types of rhombus.
One has to say, though it sounds at first a little odd, that the 'meaning' of 'rhombus'
is contained in that of 'square', when the latter is a special case of the former. (But it
seems sound epistemology to do so, since selection is of the essence of the
acquisition of knowledge at all levels, starting at the most elementary levels of
perception ).
A few comments: firstly, this notion of meaning is on the face of it
language-dependent; for example, the meaning of 'triangle' would be a different
collection of assertions if one's language was English than if it was French. This is
not a serious defect. The important consequences and issues depend only on the fact
that one language is posited, not on which it is; and in any case one might have some
standard language into which other languages can be translated.
Secondly, it has been left open whether the collection is of all true
assertions about the object, or only of all known true ones, or even of all known
true but putatively not irrelevant ones. This is deliberate, since the choice should be
determined by the purpose, mathematical, philosophical or practical of the user. For
instance, in some of the most successful modern A.1. programs the implicit
semantics used is of the third kind.
The Role ofAxiomatization
From the present point of view, the long rich history ofaxiomatization
in mathematics from Greek times to the present, might be looked upon as an effort
to put in a more manageable form the general notion of a mathematical object. That
Mathematics and Artificial Intelligence 5
is to say, since the object's meaning is in general infinite, deal instead with its
definition - the latter being a hopefully finite subset of the assertions constituting
the meaning, from which by 'logical inference' all the other true assertions could, in
principle, be derived. If A be the collection of axioms constituting the defmition of
the object, and A * its 'logical closure', i.e. the collecting of assertions consisting of
those of A and all others which can be truly inferred from A, then A* constitutes
the meaning of the object. Proofs in an axiomatized theory can then be looked upon
as efforts to explore and display some of the meaning of the objects the theorems
are about.
The analysis of the nature of logical closure for formal languages is one
of the important achievements of 20th century logic, so that we have a good
understanding of the relationship between definition and meaning in the above
sense 1 .. But, in spite of the triumphs ofaxiomatization in algebra and other fields, it
is clear from other logical studies that it is not possible or even desirable to replace
the notion of meaning by that of defmition, since - for example - for integer
arithmetic one would need a non-denumberably infinite number of axioms in the
definition!
The formal representation of niathematical objects requires, in our
semantics, a systematic method of naming them, since in fact - as we have
seen - only some of them, like rhombus and 2, have conventional names. This
requirement is similar to that met by Church's lambda-calculus in the naming of
functions, and can be dealt with in a similar way. We need an operator, say 11 ('eta')
which binds a variable in an assertion or conjunction of assertions. Thus: let S(x) be
an assertion or conjunction of assertions about the unspecified object represented by
the variable x. Then by
nx . S(x)
we mean that object which has the properties asserted by S. Note that this is the
1 It is perhaps a misnomer to call this traditional (Tarskian) model theory 'semantics', since it does not really
deal with thl' mea-iogs of terms or sentences, but only with conditions those meanings satisfy if they are to
agree without intuitive notion of logical consequences: in fact it deals essentially only with the meanings of
logical constant and quantifiers. But it is interesting that some of the contributors to this theory seem to have
used Tarskian models which are collections of assertions; cf. Henkin's proofs of completeness of lst-order and
type logic, and Hintikka's modt'l sets_