BISC: abstract - L. A. Zadeh


Subject: BISC: abstract - L. A. Zadeh
From: Michelle T. Lin (michlin@eecs.berkeley.edu)
Date: Wed Aug 23 2000 - 19:05:56 MET DST


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Berkeley Initiative in Soft Computing (BISC)
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To: BISC Group
From: L. A. Zadeh <zadeh@cs.berkeley.edu>

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Toward the Concept of Generalized Definability

Lotfi A. Zadeh*

(Abstract of a lecture presented at the Rolf Nevanlinna Colloquium,
University of Helsinki, Helsinki, Finland, August 8, 2000)

        Attempts to formulate mathematically precise definitions of
basic concepts such as causality, randomness, probability and
intelligence have a long history. The concept of generalized
definability which is outlined in this lecture suggests that such
definitions exist. Furthermore, it suggests that existing definitions
of many basic concepts, among them those of stability, statistical
independence and Pareto-optimality, may be in need of redefinition.

        In essence, definability is concerned with whether and how a
concept, X, can be defined in a way that lends itself to mathematical
analysis and computation. In mathematics, definability of mathematical
concepts is taken for granted. But as we move farther into the age of
machine intelligence and automated reasoning, the issue of
definability is certain to grow in importance and visibility, raising
basic questions which are not easy to resolve.

        To be more specific, let X be the concept of, say, a summary,
and assume that I am instructing a machine to generate a summary of a
given article or a book. To execute my instruction, the machine must
be provided with a definition of what is meant by a summary. It is
somewhat paradoxical that we have summarization programs which can
summarize, albeit in a narrowly prescribed sense, without being able
to formulate a general definition of summarization. The same applies
to the concepts of causality, randomness and probability. Indeed, it
may be argued that these and many other basic concepts cannot be
defined within the conceptual framework of classical logic and set
theory.

        The point of departure in our approach to definability is the
assumption that definability has a hierarchical structure.
Furthermore, it is understood that a definition must be unambiguous,
precise, operational, general and co-extensive with the concept which
it defines.

        The hierarchy involves five different types of definability.
The lowest level is that of c-definability, with c standing for crisp.
Thus, informally, a concept, X, is c-definable if it is a crisp
concept, e.g., a prime number, a linear system or a Gaussian
distribution. The domain of X is the space of instances to which X
applies.

        The next level is that of f-definability, with f standing for
fuzzy. Thus, X is a fuzzy concept if its denotation, F, is a fuzzy set
in its universe of discourse. A fuzzy concepts is associated with a
membership function which assigns to each point, u, in the universe of
discourse of X, the degree to which u is a member of F. A fuzzy
concept is defined by its membership function or in terms of other
fuzzy concepts. Examples of fuzzy concepts are small number, strong
evidence and similarity. It should be noted that many concepts
associated with fuzzy sets are crisp concepts. An example is the
concept of a convex fuzzy set. Most fuzzy concepts are
context-dependent.

        The next level is that of f.g-definability, with g standing
for granular, and f.g denoting the conjunction of fuzzy and granular.
Informally, in the case of a concept which is f.g-granular, the values
of attributes are granulated, with a granule being a clump of values
which are drawn together by indistinguishability, similarity,
proximity or functionality. f.g-granularity reflects the bounded
ability of the human mind to resolve detail and store information. An
example of an f.g-granular concept which is traditionally defined as a
crisp concept, is that of statistical independence. This is a case of
misdefinition -- a definition which is applied to instances for which
the concept is not defined, e.g., fuzzy events. In particular, a
common misdefinition is to treat a concept as if it were c-definable
whereas in fact it is not.

        The next level is that of PNL-definability, where PNL stands
for Precisiated Natural Language. Basically, PNL consists of
propositions drawn from a natural language which can be precisiated
through translation into what is called precisiation language. An
example of a proposition in PNL is: It is very unlikely that there
will be a significant increase in the price of oil in the near future.

        In the case of PNL, the precisiation language is the
Generalized Constraint Language (GCL). A generic generalized
constraint is represented by Z isr R, where Z is the constrained
variable, R is the constraining relation and r is a discrete-valued
indexing variable whose values define the ways in which R constrains
Z. The principal types of constraints are: possibilistic (r=blank);
veristic (r=v); probabilistic (r=p); random set (r=rs); usuality
(r=u); fuzzy graph (r=fg); and Pawlak set (r=ps). The rationale for
constructing a large variety of constraints is that of conventional
crisp constraints are in capable of representing the meaning of
propositions expressed in a natural language -- most of which are
intrinsically imprecise -- in a form that lends itself to computation.

        The elements of GCL are composite generalized constraints
which are formed from generic generalized constraints by combination,
modification and qualification. An example of a generalized constraint
in GCL is ((Z isp R) and ((Z,Y) is S) is unlikely.

        By construction, the Generalized Constraint Language is
maximally expressive. What this implies is that PNL is the largest
subset of a natural language which admits precisiation. Informally,
this implication serves as a basis for the conclusion that if a
concept, X, cannot be defined in terms of PNL, then, in effect, it is
undefinable or, synonymously, amorphic.

        In this perspective, the highest level of definability
hierarchy, which is the level above PNL-definability, is that of
undefinability or amorphicity. A canonical example of an amorphic
concept is that of causality. More specifically, is it not possible to
construct a general definition of causality such that given any two
events A and B and the question, "Did A cause B?" the question could
be answered based on the definition. Equivalently, given any
definition of causality, it will always be possible to construct
examples to which the definition would not apply or yield
counterintuitive results.

        In dealing with an amorphic concept, X, what is possible --
and what we generally do -- is to restrict the domain of applicability
of X to instances for which X is definable. For example, in the case
of the concept of a summary, which is an amorphic concept, we could
restrict the length, type and other attributes of what we want to
summarize. In this sense, an amorphic concept may be partially
definable or, p-definable, for short. The concept of p-definability
applies to all levels of the definability hierarchy.

        The theory of generalized definability is not a theory in the
traditional spirit. The definitions are informal and conclusions are
not theorems. Nonetheless, it serves a significant purpose by raising
significant questions about a basic issue -- the issue of definability
of concepts which lie at the center of scientific theories.

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*Professor in the Graduate School and Director, Berkeley Initiative in
 Soft Computing (BISC). Computer Science Division and the Electronics
 Research Laboratory, Department of EECS, University of California,
 Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712;
 E-mail: zadeh@cs.berkeley.edu
 Research supported in part by ONR Contract N00014-99-C-0298, ONR
 Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the BISC Program
 of UC Berkeley.

------
To post your comments to the BISC Group, please send them to
me(zadeh@cs.berkeley.edu) with cc to Michael Berthold
(berthold@cs.berkeley.edu)
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