**Previous message:**Greg Heath: "Re: coupling neural networks and fuzzy logic for forecasting purposes"**Maybe in reply to:**Ricky: "Stupid question"**Next in thread:**Tadeusz Dobrowiecki: "Re: Stupid question"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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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*>
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**Next message:**Kyriakos Deliparaschos: "Ooops!!! Forgot the address!"**Previous message:**Greg Heath: "Re: coupling neural networks and fuzzy logic for forecasting purposes"**Maybe in reply to:**Ricky: "Stupid question"**Next in thread:**Tadeusz Dobrowiecki: "Re: Stupid question"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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