Re: Kosko's Information Wave Mechanics


Subject: Re: Kosko's Information Wave Mechanics
From: Stephen Paul King (stephenk1@home.com)
Date: Fri Apr 07 2000 - 19:57:56 MET DST


Hi Zunt and Friends,

        Unfortunately I am unable to reproduce the full content of
Kosko's ideas here. :-( But l will try to give an abbreviated quote
that generates sufficient background information. All typos are mine!
;-)

Quoting from Fuzzy Engineering, Prentice Hall, 1996 (ISBN
0-13-124991-6) Chapter 12, pg. 403-

"A fuzzy cube contains all fuzzy subsets of a set X of n objects. The
2^n bivalent subsets of X lie at the 2^n corners of the n-cube [0,
1]^n. The continuum of fuzzy sets fill the cube. ... The fuzzy mutual
(Kullback) entropy of a fuzzy set F acts as a type of distance measure
between F and its set complement F^c. It stems from the logarithm of a
unique measure of the fuzziness of the set F. ... A deeper result
shows that fuzzy mutual entropy gives back the standard Shannon
entropy H(P) of a probability vector P if we integrate the fuzzy
mutual entropy. The set of all probability vectors of length n defines
the simplex in the fuzzy n-cube. We allow the Shannon entropy to
extend beyond the simplex and range over the entire fuzzy cube. We
then compute the Shannon entropy H(F) of any fuzzy set F of length n.
This shows in turn that fuzzy mutual entropy has a fluid-mechanical
structure and leads to the concept of an information field in a fuzzy
cube. Fuzzy mutual entropy equals the negative of the divergence of
Shannon entropy. Uncertainty descriptions define points in the
fuzzy-cube parameter space. Versions of both extended Shannon entropy
and fuzzy mutual entropy define vector fields on the fuzzy cube. The
field equations show that Shannon entropy acts as the potential of the
conservative mutual entropy vector field. This implies a dynamical
form of the "second law of thermodynamics" for flows on the fuzzy
cube: Shannon entropy can only grow in time in the fuzzy mutual
entropy field. It also suggests that a simple reaction-diffusion
equation may hold in fuzzy cubes."

On 2 Apr 2000 13:38:40 GMT, Zunt@aol.com wrote:

>Perhaps one might express the idea for the benefit of those of us who have
>not yet seen that chapter--
>
>In a message dated 00-04-02 08:58:07 EDT, you write:
>
>> In chapter 12 of Fuzzy Engineering, Bart Kosko discusses the
>> notion of Information wave mechanics. Does anyone have a comment of
>> follow up on Kosko's idea?
>>
>> TIA!

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