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It is interesting to note that the continuous model of the binary set, or

the Heaviside step function S(x) \in [0,1], satisfies the basic criteria for

inclusion in the general collection of "sigmoidal" functions, \sigma(x),

that are used in Cybenko's classic universal function approximator (UFA)

paper. By definition, S(x) is Lebesgue-integrable; and clearly S(x) ---> 1

as x ---> \infty, S(x) ---> 0 as x ---> - \infty.

Therefore, according to Cybenko's theorems, S(x) also forms the basis for a

UFA. Since Cybenko's paper generates necessary conditions only, there is no

specification on the upper bound of number of nodal \sigma(x) units required

to actually approximate some smooth function F(x) within a prespecified

tolerance, \epsilon, except that it is finite, for all such \sigma(x). Thus

a finite, though possibly large, collection of binary number-step functions

may be used to model any smooth function, including, eg. some fuzzy

membership function. This generates one interrelationship between binary

and fuzzy variables.

Corrections, insight, comments are very welcome.

Pramit

----- Original Message -----

From: "Greg Chien" <gchien@protodesign-inc.com>

To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>

Sent: Wednesday, November 21, 2001 5:42 AM

Subject: Re: Stupid question

*> "Ricky" <01900990R@polyu.edu.hk> wrote in message
*

*> news:9tcggh$1bd3@hkpa05.polyu.edu.hk...
*

*> > It may be a stupid question, but it really a trouble in my mind
*

*> these few
*

*> > days. We use the mathematics expression to formulate the algorithm,
*

*> and we
*

*> > use the programme to apply the algorithm, then what is the
*

*> relationship
*

*> > between mathematics and computer programme?
*

*>
*

*> How about "a computer program is a machine readable/executable
*

*> implementation of certain logic and mathematics."
*

*>
*

*> > In the world of computer programme, binary nos. (1 and 0) are used
*

*> but as we
*

*> > all know that there is a lot of things between 1 and 0 in the world
*

*> of fuzzy
*

*> > algorithm. Then, contradition is here.
*

*>
*

*> A single binary bit may not be fuzzy, but a series of 0's and 1's that
*

*> forms different patterns may be used to model fuzzy ranges, and,
*

*> perhaps even, to simulate analog devices.
*

*>
*

*> Regards,
*

*> Greg Chien
*

*> http://protodesign-inc.com
*

*>
*

*>
*

*>
*

*>
*

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