**Stephen P. King** (*stephenk1@home.com*)

*Fri, 19 Mar 1999 23:42:16 -0500*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 38] Re: [time 37] Re:The ordering of spatial states and temporal events"**Previous message:**Hitoshi Kitada: "[time 36] RE: [time 27] Turing machines under difeomorphisms?"**Maybe in reply to:**Stephen P. King: "[time 27] Turing machines under difeomorphisms?"**Next in thread:**Matti Pitkanen: "[time 38] Re: [time 37] Re:The ordering of spatial states and temporal events"

Dear Robert,

A very nice paper!

ca314159 wrote:

*>
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*> Stephen P. King wrote:
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*> > This, to me, indicates how hierarchical ordering can come into play,
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*>
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*> This article called
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*> "The ordering of spatial states and temporal events":
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*> http://www.bestweb.net/~ca314159/CEREAL.HTM
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*> is how I see spatial order arising from the degree of dependence
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*> between temporal events. I suppose the reverse is true as well,
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*> that temporal ordering arises from spatial state dependences ?
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I have long suspected that "spaces" are constructed from relations

between local systems and temporal orderings, e.g. histories (!) are

constructed from relations between spaces. This "degree of dependence",

I believe, is what Kosko's subsethood is all about! Let me see if I can

argue this:

fuzzy subsethood <=> mutual entropy of basis

states

First a quote:

"This is Kosko's view that fuzzy sets represent points in an

n-dimensional hypercube. Each edge of the hypercube represents one

element of the fuzzy set. The membership value of that element

determines a point on the edge. When all the elements of the set are

considered, these points on the edges define a point in n-space within

the hypercube. If all the membership values are 0 or 1 then the points

define the corners of the cube. This is consistent with Boolean logic.

If all the memberships values are 1/2 then the point is the midpoint of

the cube.This violates Aristotle's law of the excluded middle, A and

not-A. The sets as points view an important concept because Kosko uses

it to derive equations for fuzzy entropy and fuzzy subsethood."

Note that this hypercube is isomorphic with R^n, thus it is Euclidian!

"Fuzzy entropy measures the fuzziness of a set or how fuzzy a fuzzy set

is. Given a fuzzy set, or a point in the hypercube, Kosko shows that the

fuzzy entropy is the distance between the point and the nearest corner

divided by the distance between the point and the farthest corner. If

the set is a crisp set then this is 0/1 = 0, the set is not fuzzy at

all. If the set is fuzzy with all membership values = 1/2 then, in the

2D case, the entropy is (1/2)/(1/2) = 1, or maximum entropy. Therefore,

this set is as fuzzy as a set can be. Kosko notes that this agrees with

the probabilistic entropy measure."

Note that ordinary set membership values are the basis vectors of the

space... We need to look more into measure theory, M.C. Mackey's work

becomes very relevant!:)

M. C. Mackey. Time's Arrow: The Origins of Thermodynamic Behaviour.

Springer-Verlag, 1992.

A. Lasota & M.C. Mackey. Probabilistic Properties of Deterministic

Systems, Cambridge University Press,

New York-Cambridge, 1985.

A. Lasota & M.C. Mackey. Chaos, Fractals and Noise: Stochastic Aspects

of Dynamics. Springer-Verlag,

1994.

etc.

http://www.cnd.mcgill.ca/bios/mackey/mackey_publ.html

"Subsethood is the measure of how much one set is a subset of another.

If two sets contain the same elements and one has membership values (or

fit-values) that are element for element lower than the other then that

set is a subset of the first. However, what if one set is almost a

subset but has 1 fit-value greater, or in violation, then the other set.

This cries out for a fuzzy subset measure, how much is one set a subset

of another. Kosko provides 2 derivations, one standard and another using

geometric orthogonality (Pythagorean style) to show that fuzzy

subsethood can be measure as the sum of subset violations over the

number

of elements in the set (a normalizing factor). He further shows that

this is consistent with the probability measure of subsethood."

Here we have a method to think about how sets of LS interactions can be

modeled, but this formalism is, I believe, incomplete; we also need the

formalisms of hypersets and ultrametrics.

"Kosko then develops the Entropy-Subsethood theorem than reveals a

relationship between entropy and subsethood. He proves that the

entropy-subsethood theorem describes how much the whole is contained

within the part, turning Venn diagrams inside-out. This relationship is

unique to fuzzy logic in that probability and other mathematical systems

contain no analog to this theorem."

http://www.afit.af.mil/ENGgraphics/annobib/kosk92a.mme.html

Ok, back to my conjecture. We can see easily that a set of relations

between LS, acting as clocks, can be used to map out a space. Please

forgive my lack of math to back this up... :( The usual ideas used in

special relativity, where distances are defined using pulses of light.

By comparing many such spaces to each other, we can see that histories

of events will be defined by ordering according to minimum difference.

This is analogous to building a feature lenght movie out of many

individual photos. Each "movie" is a history; each "photo" is defined by

spatial distances defined by the differences in the clocks. There are

more details to work through, such as the statistics and partitioning

methods...

This is very sketchy and I am working with several people on a

mathematical description, so I beg your indulgence. :)

Later,

Stephen

Later,

Stephen

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