# Re: Fuzzy relations vs. Mamdani model

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

WSiler@aol.com wrote:

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

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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