**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.

---------

*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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