**Subject: **Re: Fuzzy relations vs. Mamdani model

**From: **Pim van den Broek (*pimvdb@cs.utwente.nl*)

**Date: **Fri Nov 24 2000 - 03:16:25 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Breezy: "Re: Imprecise Probability"**Previous message:**Vladik Kreinovich: "Re: does fuzzy bound probability?"**Maybe in reply to:**albert@massivbau.tu-darmstadt.de: "Fuzzy relations vs. Mamdani model"**Maybe reply:**Pim van den Broek: "Re: Fuzzy relations vs. Mamdani model"

WSiler@aol.com wrote:

*>
*

*>
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*> Actually, the fuzzy relation involved a fuzzy implication operator, and is
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*> completely unworkable for any true fuzzy implication operator (one which
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*> collapses to the classical implication for crisp values).
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Why unworkable?

The result B' from the rule IF A THEN B and the fact A' is given by

B'(y) = sup_x min (A'(x), J(A(x),B(y)))

where J is an implication operator or, for the Mamdani model, J = min.

When we consider this equation as an algorithm, we see that in order to compute B

we have, for all y, to iterate over all x.

In case J = min the equation can be rewritten as

B'(y) = sup_x min ( min (A(x),A'(x)) , B(y))

This algorithm needs only a single iteration over all x.

Such a rewriting cannot be done for arbitrary J, and I suppose that this is the

reason

you consider an implication operator to be unworkable.

However, for several implication operators (Kleene-Dienes, Early-Zadeh, Willmott

for example)

a rewriting as above can be done, leading to algorithms which need a single

iteration over all x.

Therefore I would certainly consider these implication operators to be

"workable".

Details can be found in my paper "Efficient Algorithms for Approximate Reasoning"

which

appeared in the Proceedings of the ICONIP'99 conference, Volume 1, page 292.

Yours,

Pim van den Broek

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