BISC: Prof. Zadeh's Abstract for: "A Critical View of the Foundations of

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Date: Fri May 11 2001 - 14:46:02 MET DST

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    A Critical View of the Foundations of Control and Decision Analysis

    Lotfi A. Zadeh *

    Abstract

            Practitioners of classical control and decision analysis can point with pride
    to a multitude of brilliant successes, among them the conquest of space and
    optimal allocation of resources involving linear programs with tens of thousands
    of variables and constraints.
            But alongside the brilliant successes stand much less grandiose problems which
    so far have eluded solution. We cannot automate driving in city traffic, build
    robots which can play basketball or construct programs which can do economic
    forecasting without human intervention.
            The debut of fuzzy logic was motivated in large measure by a realization that
    classical-logic-based theories do not have the capability to solve problems of
    this type. What is becoming clearer now is that to deal with such problems it
    is necessary to develop theories which are capable of processing
    perception-based information. At the center of these theories is the recently
    developed methodology of computing with words (CW).
            As a methodology, computing with words brings to light some basic flaws in the
    foundations of control and decision analysis. One such flaw relates to what may
    be called the illusion of crisp definability.

            More specifically, almost all concepts in classical control, probability theory
    and decision analysis are crisp, that is, are based on Aristotelian logic. For
    example, under Lyapounov's definition of stability, a system is either stable or
    unstable, with no shades of gray allowed; a process is either random or
    unrandom; random variables are either dependent or independent; and causal
    relations are categorical rather than a matter of degree.
            What can be shown is that crisp definitions lead to counterintuitive
    conclusions in much the same way as the ancient Greek sorites paradox - a
    paradox which involves successive removal of grains of sand from a heap. As an
    illustration, consider Lyapounov's definition of stability in application to a
    ball of diameter D which is placed on the mouth of an open bottle of diameter
    d. When D is slightly larger than d, the system is clearly stable. As D
    increases and eventually becomes much larger that d, the system becomes less and
    less stable and eventually becomes unstable. This contradicts Lyapounov's
    definition of stability, according to which the system is stable no matter how
    large D is. In this example, gradual increase in D is analogous to gradual
    decrease in the size of the heap.
            A related problem arises in the case of a concept which plays a basic role in
    decision analysis - the concept of the expected value of a random variable. It
    is well known that the widely used principle of minimization of expected utility
    leads to paradoxes such as the Allais paradox. A basic reason is that the
    crisply-defined expected value of a random variable is its average value, which
    may or may not coincide with our intuitive perception of the value which the
    variable is most likely to take. To capture this concept, what is needed is the
    fuzzy-logic-based concept of the usual value. Manipulation of usual values
    falls outside the scope of classical probability theory; it falls however,
    within the methodology of computing with words under the rubric of dispositional
    logic.
            The illusion of crisp definability is not the only basic flaw in classical
    control and decision analysis. There are others. In particular, both classical
    control and decision analysis founder on the rocks of what may be called the
    dilemma of "it is possible but not impossible."
            Von Neumann, Morgenstern, Wald and other founders of decision analysis were
    driven by a quest for a mathematical theory which is rigorous, precise and
    prescriptive. The dilemma of "possible but not probable" is a major obstacle to
    the development of theories in this spirit. More specifically, decision
    analysis and control rest on the tacit assumption that the worst-case scenario,
    though possible, is not probable. The problem is that the probability of a
    worst-case scenario does not lend itself to precise assessment. As a
    consequence, validity of many basic concepts centering on optimality, stability
    and causality is called into question. What is needed to deal with this basic
    problem is the methodology of computing with words.
            Computing with words is not a panacea. In essence, it opens the door to a
    potentially radical enlargement of the role of natural languages in science and,
    in particular, in information processing, decision and control.

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