BISC: Zadeh/ Crossing into Uncharted Territory — The

From: masoud nikravesh (
Date: Fri Jan 11 2002 - 08:07:59 MET

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    Berkeley Initiative in Soft Computing (BISC)

    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 <> with cc to


    With my warm regards
    Lotfi Zadeh


    Crossing into Uncharted Territory — The Concept of Approximate X

    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
    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

    * 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: 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 Administrator

    BTExact 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: URL:

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

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