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Berkeley Initiative in Soft Computing (BISC)

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The attached abstract "Crossing into Uncharted Territory — The Concept

of Approximate X" is for your information and comments, if any.

Should you like your comment to be ported to the BISC mailing list,

please e-mail it to Dr. Nikravesh <nikravesh@cs.berkeley.edu> with cc to

me.

With my warm regards

Lotfi Zadeh

==========================

1-03-02

Crossing into Uncharted Territory — The Concept of Approximate X

Abstract

Lotfi A. Zadeh*

In science — and especially in mathematics — it is a universal practice

to express definitions in a language based on bivalent logic. Thus, if C

is a concept, then under its definition every object, u, is either an

instance of C or it is not, with no shades of gray allowed. This

deep-seated tradition — which is rooted in the principle of the excluded

middle—is in conflict with reality. Furthermore, it rules out the

possibility of graceful degradation, leading to counterintuitive

conclusions in the spirit of the ancient Greek sorites paradox.

In fuzzy logic — in contrast to bivalent logic — everything is, or is

allowed to be, a matter of degree. This is well known, but what is new

is the possibility of employing the recently developed fuzzy-logic-based

language PNL (Precisiated Natural Language) as a concept-definition

language to formulate definitions of concepts of the form “approximate

X,” where X is a crisply defined bivalent-logic-based concept. For

example, if X is the concept of a linear system, then “approximate X”

would be a system that is approximately linear.

The machinery of PNL provides a basis for a far-reaching project aimed

at associating with every — or almost every — crisply defined concept X

a PNL-based definition of “approximate X,” with the understanding that

“approximate X” is a fuzzy concept in the sense that every object x is

associated with the degree to which x fits X, with the degree taking

values in the unit interval or a partially ordered set. A crisp

definition of “approximate X” is not acceptable because it would have

the same problems as the crisp definition of X.

As a simple example, consider the concept of a linear system. Under the

usual definition of linearity, no physical system is linear. On the

other hand, every physical system may be viewed as being approximately

linear to a degree. The question is: How can the degree be defined?

More concretely, assume that I want to get a linear amplifier, A, and

that the deviation from linearity of A is described by the total

harmonic distortion, h, as a function of power output, P. For a given

h(P), then, the degree of linearity may be defined in the language of

fuzzy if-then rules – a language which is a sublanguage of PNL. In

effect, such a definition would associate with h(P) its grade of

membership in the fuzzy set of distortion/power functions which are

acceptable for my purposes. What is important to note is that the

definition would be local, or, equivalently, context-dependent, in the

sense of being tied to a particular application. What we see is that the

standard, crisp, definition of linearity is global (universal,

context-independent, objective), whereas the definition of approximate

linearity is local (context-dependent, subjective). This is a basic

difference between a crisp definition of X and PNL-based definition of

“approximate X.” In effect, the loss of universality is the price which

has to be paid to define a concept, C, in a way that enhances its

rapport with reality.

In principle, with every crisply defined X we can associate a PNL-based

definition of “approximate X.” Among the basic concepts for which this

can be done are the concepts of stability, optimality, stationarity and

statistical independence. But a really intriguing possibility is to

formulate a PNL-based definition of “approximate theorem.” It is

conceivable that in many realistic settings informative assertions about

“approximate X” would of necessity have the form of “approximate

theorems,” rather than theorems in the usual sense. This is one of the

many basic issues which arise when we cross into the uncharted territory

of approximate concepts defined via PNL.

A simple example of “approximate theorem” is an approximate version of

Fermat’s theorem. More specifically, assume that the equality

x**n+y**n=z**n is replaced with approximate equality. Furthermore,

assume that x, y, z are restricted to lie in the interval [I,N]. For a

given n, the error, e(n), is defined as the minimum of a normalized

value of |x**n+y**n-z**n|over all allowable values of x, y, z.

Observing the sequence {e(n)}, n = 3,4,…, we may form perceptions

described as, say, “for almost all n the error is small;” or “the

average error is small;” or whatever appears to have a high degree of

truth. Such perceptions, which in effect are summaries of the behavior

of e(n) as a function of n, may qualify to be called “approximate

Fermat’s theorems.” It should be noted that in number theory there is a

sizeable literature on approximate Diophantine equations. There are many

deep theorems in this literature, all of which are theorems in the usual

sense.

In a sense, an approximate theorem may be viewed as a description of a

perception. The concept of a fuzzy theorem was mentioned in my 1975

paper “The Concept of a Linguistic Variable and its Application to

Approximate Reasoning.” What was missing at the time was the concept of

PNL.

* Lotfi A. Zadeh is Professor in the Graduate School and director,

Berkeley initiative in Soft Computing (BISC), Computer Science Division

and the Electronics Research Laboratory, Department of EECS, Univeristy

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 Grant N00014-00-1-0621, ONR Contract N00014-99-C-0298,

NASAContract NCC2-1006, NASA Grant NAC2-117, ONR Grant N00014-96-1-0556,

ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the BISC Program

of UC Berkeley

-- Dr. Masoud Nikravesh BISC Associate Director and Program AdministratorBTExact Technologies (British Telecom-BT) Senior Research Fellow Chairs: BISC-SIG-FLINT,ES, RT Berkeley Initiative in Soft Computing (BISC) Computer Science Division- Department of EECS University of California, Berkeley, CA 94720 Phone: (510) 643-4522; Fax: (510) 642-5775 Email: Nikravesh@cs.berkeley.edu URL: http://www-bisc.cs.berkeley.edu/

Staff Scientist Lawrence Berkeley National Lab, Imaging and Collaborative Computing Group Email: Masoud@media.lbl URL: http://vision.lbl.gov/

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