# 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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