**Subject: **fuzzy -- in disquise? *A Step-Down Analogy*

**From: **pramit sarma (*pramits@vsnl.com*)

**Date: **Sat Jun 17 2000 - 04:41:56 MET DST

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**anilo1@my-deja.com: "Fuzzy Logic = approximation right?"**Previous message:**Y'uni'ar: "fuzzy learning"

Having read some of Dr. Li's interesting papers, it appears even more

interesting that such

an approach should exist in his mind. In fact, one gets the impression of a

deliberate statement to evoke and invoke some really good

counterargumentation.

If anything is clear to many of those involved in studying/using/hybridising

things

involving FLS (fuzzy logic sets & systems), it is the fact that FLS is by

nature and

definition, a formal extension of crisp sets, hence all crisp structures

that evolve from

that - sets generate number theories generate algebras generate continuous

frameworks

generate calculus and so on ad infinitum. It is clear, but not obvious, that

the fundament involves simple, or plain numbers. The thread that extends the

crisp

structures and maps, is the well-known Zadeh *Extension* Principle (ZEP).

Much of the murk and countermurk is clarified via the utilisation of ZEP on

the crisp

number - the building block of (crisp) mathematics. It's well known that FLS

produces

a clearcut theoretical (yet practical) *generalisation* of one of the key

set theoretic

bricks - Modus Ponens (and it's various cousins) to Generalised MP (GMP).

Ergo

nonsemantically we can suggest that Extension \equiv Generalisation. Many

might

protest at this elementary exposition, but the point is eventually semantic:

the crisp number theory (and adjuncts and consequents) is then bootstrapped

up

in terms of generalisation - so that the Fuzzy Number (FN) directly subsumes

and

generalises the crisp number (CN). At the possible risk of being too

obvious, the

CN is simply a (very) special case of FN (as is also any finite real

interval) . In fact

there exist undoubtedly a vast number of earlier investigated number systems

that

do subsume crisp sets, and some later (like rough sets). Few, if any, of

those have

had practical uses, outside pure research.

There is scarce new theory in the above. However it germinates a point: All

of

CN theory adjuncts (calculus, mathematical programming, clustering, ...)

depend on the

fundament of the number per se. So much so, that it is almost disregarded as

being

too obvious. Doing this amounts to, in a gentler way perhaps, to missing the

people in

the crowd, or the trees for the forest, to invert a bromide. Without the

tree, there is no

forest at all; without the CN no adjuncts. The true point here is that the

FN is identically

the same fundament in FLS - for all the adjuncts.

Recast, every FLS application rests clearly on a basis of FN's. In FLC's the

numbers are still fuzzy, of course though they have names which are so well

known that they almost become ... crisp? {ZO, PS,...} are finally just FN's.

Or so on

for other FLC formats - {M, TS}. Now it is clear that the FN is not a "mere

interpolation"

of the CN, any more than GMP is so, with respect to MP.

A trivial, and in fact crisp, example. Consider the set of all polynomials,

say of finite

degree, say P. Consider, as a microformal system, the set of all straight

lines, say L.

It can always be shown that L has a certain reasonable set of mathematical

and

geometric properties. For the sake of local argument, let us say that this

set L is

a well-defined one, and can be used exclusively for a wide variety of

tasks:

linear interpolation, numerical integration, spline-fits, regression inter

alia. Stand-alone,

it may appear to be "sufficient" ... but this is mainly an intuitive

reasoning. Then if

the set P is discovered later on, it is eventually clear that P generalises

L, in the sense

of extension, subsuming but certainly not by being an interpolation! Every L

can be

found in P, but there are *infinite* members of P that fail to satisfy the

criteria for L.

The construction of P is not arbitrary, it is thoroughly formal, as much as

so as L.

It is more than a tautology to affirm that P (superset) L, P extends L, P

generalises L.

This is entirely a different matter from mere interpolation. Clearly, the

set P opens up

new vistas of functional analysis which were simply unreachable, due to

constructional

limitations. A beautiful example of a P property being the Weierstrass

Approximation

Theorem (WAT).

These arguments can be intuitively bootstrapped into the realm of FN and CN.

Ergo,

it is more than a tautology to affirm that FN (superset) CN, FN extends CN,

FN

generalises CN. This is entirely a different matter from mere

interpolation. Clearly,

the set FN opens up new vistas of functional analysis which were simply

unreachable,

due to constructional limitations. The analogous extension of WAT in the

domain of FLS

could perhaps be a future product of Fuzzy Mathematicians.

Pramit

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

From: HX Li

To: Multiple recipients of list

Sent: Thursday, June 08, 2000 2:20 PM

Subject: fuzzy -- in disquise?

Dear Scientists and Engineers

Unfortunately, fuzzy set theory is nothing

else but an interpolation algorithm in disguise.

Prof. Dr. Hongxing Li of Beijing Normal

University has proven it in his paper:

"Relationship Between Fuzzy Controllers and PID Controllers"

SCIENCE IN CHINA SERIES E-TECHNOLOGICAL SCIENCES

1999, Vol 42, Iss 2, pp 215-224

In fact, he is a fuzzy expert, see

http://www.crcpress.com/index.htm?catalog/8931

we must believe his analysis.

You can also write to

Prof. Hongxing Li

Dept. of Mathematics

Beijing Normal University

Beijing 100875, China

for future discussing

Sincerely yours

Explore world

________________________________________________________________________

Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com

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