Re: Stupid question

From: P. Sarma (psarma@seas.upenn.edu)
Date: Wed Dec 05 2001 - 22:44:12 MET

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    That is correct. However, the point was that this same {0,1} constructional
    codification is regularly used to represent continuous numbers in R, the
    real domain, within "machine precision", eg. a double-precision real
    variable representing a real number such as \pi. Here, this may be extended
    to cover smooth real functions, including fuzzy MF's.

    It is a simple extension of this, via Cybenko's theorem, where we need only
    map a binary number in B={0,1} to the shifted Heaviside step function with
    the code

    IF (x < s) THEN f = -1 ELSE
    IF (x > s) THEN f = +1

    which extends the concept of binary numbers to the real numbers. This
    produces a real function f(x;s) with s \in R as parameter, so that f : R ->
    {0,1}. Then, this function f appears to satisfy all the Cybenko-necessary
    conditions to be a "sigmoidal", and accordingly, a finite (though possibly
    large) collection of these "step-sigmoidals" should permit the approximation
    of any real smooth function, say F(x), to a prespecified precision. If this
    holds, this extends B to the field of smooth functions in R, of which the
    class of fuzzy membership functions with \Phi : R -> [0,1] is a subfield.
    It is only a conjecture, and certainly the point is well taken that this is
    constructional, not conceptual. It was simply an alternate way of looking at
    these things.

    The direct conceptual equivalence, of course, is seen immediately from the
    original Zadeh (1965) paper. The fuzzy MF's are intentionally taken, by
    Zadeh, from crisp, binary logic {0,1} and simply extended or smoothed out
    into fuzzy continuous truth variables. Smooth, "sigmoidal" monotonic
    functions are regularly used in FLS, \Phi : R -> [0,1], which directly
    extend the step function f, which in turn is a continuous model of the
    binaries B.

    It seemed interesting that this "adds to the loop" of these equivalences, in
    a way.

    Pramit

    ----- Original Message -----
    From: "Francisco Bernal Rosso" <pacob@mixmail.com>
    To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>
    Sent: Wednesday, December 05, 2001 6:35 AM
    Subject: Re: Stupid question

    > Binariy in computers affects only to the codification of information, not
    at
    > yhe mean of it.
    >
    >
    >
    >
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