**Subject: **Re: Kosko's Information Wave Mechanics

**From: **Stephen Paul King (*stephenk1@home.com*)

**Date: **Fri Apr 07 2000 - 19:57:56 MET DST

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Steven Lim: "Fuzzy Operators"**Previous message:**Matthias Klusch: "CIA-2000 Call for Participation"**Maybe in reply to:**Stephen Paul King: "Kosko's Information Wave Mechanics"**Maybe reply:**Stephen Paul King: "Re: Kosko's Information Wave Mechanics"

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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**Next message:**Steven Lim: "Fuzzy Operators"**Previous message:**Matthias Klusch: "CIA-2000 Call for Participation"**Maybe in reply to:**Stephen Paul King: "Kosko's Information Wave Mechanics"**Maybe reply:**Stephen Paul King: "Re: Kosko's Information Wave Mechanics"

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