Re: Fuzzy relations vs. Mamdani model


Subject: Re: Fuzzy relations vs. Mamdani model
WSiler@aol.com
Date: Tue Nov 14 2000 - 17:47:30 MET


In a message dated 11/14/00 1:37:53 AM Central Standard Time,
albert@massivbau.tu-darmstadt.de writes:

<< Originally a fuzzy rule "if A, then B" was defined to be seen as a
 fuzzy relation between the input A and the output B. The
 compositional rule of inference was used to calculate the output for
 an input value A'. This meant to calculate the intersection between
 the cylidrical extension of A and the fuzzy set of the relation and
 then to project the resulting fuzzy set onto the domain of B.
 The Mamdani/Assilian model for fuzzy systems, however,
 calculates the degree of compatibility of the input A' with A then
 uses the minimum for implication which cuts the fuzzy set B and
 so on...
 My question now is: are these two methods equivalent or is the
 second one a simplified model of the first one? >>

I think this question was answered a couple of days ago, qbut since it is an
important question I answer it again.

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). However, an
unspoken non-aggression pact among fuzzy mathematicians meant that nobody
called attention to this embarassing fact. The Mamdani "implication" operator
works, but since it does not collapase to the classical for crisp values it
is not an implication operator at all. Unfortunately I do not know of a
thorough discussion of this point in the literature, although Klir and Yuan
give a correct although brief discussion of the Mamdani method.

Almost all the theoretical literature assumes that a rule is a fuzzy logical
proposition. In fact, a rule in any working system is not a fuzzy logical
proposition; instead, the antecedent is a fuzzy logical proposition, and the
consequent is a set of instructions to be executed with the combined
antecedent and rule confidence if the antecedent is sufficiently true. In
executing the consequent we must have a convention for handling any prior
truth values of data to be modified by the rule. Those who actually construct
fuzzy expert system shells and systems face several theoretical problems
here, which (again unfortunately) almost no fuzzy theoreticians have
addressed.

William Siler

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