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


Subject: fuzzy -- in disquise? *A Step-Down Analogy*
From: pramit sarma (pramits@vsnl.com)
Date: Sat Jun 17 2000 - 04:41:56 MET DST


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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