Stephen P. King (email@example.com)
Fri, 19 Mar 1999 23:42:16 -0500
A very nice paper!
> Stephen P. King wrote:
> > This, to me, indicates how hierarchical ordering can come into play,
> This article called
> "The ordering of spatial states and temporal events":
> is how I see spatial order arising from the degree of dependence
> between temporal events. I suppose the reverse is true as well,
> that temporal ordering arises from spatial state dependences ?
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
fuzzy subsethood <=> mutual entropy of basis
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.
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,
"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
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."
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
This is very sketchy and I am working with several people on a
mathematical description, so I beg your indulgence. :)
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