# Re: Fuzzy relations vs. Mamdani model

Subject: Re: Fuzzy relations vs. Mamdani model
From: Jon Williams (jon@williams-home.demon.co.uk)
Date: Wed Nov 15 2000 - 12:32:43 MET

----- Original Message -----
From: <WSiler@aol.com>
To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>
Sent: Tuesday, November 14, 2000 4:42 PM
Subject: Re: Fuzzy relations vs. Mamdani model

> In a message dated 11/14/00 1:37:53 AM Central Standard Time,
>
> << 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.
>
A discussion which covers this point quite well can be found in

@ARTICLE{Koczy,
author = "L.T. K\'{o}czy and Hirota, K.",
title = "Ordering and closeness of fuzzy sets",
journal = " Fuzzy Sets and Systems",
year = "1993",
volume = "59",
pages = "282 - 293"

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Jon Williams,
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e-mail   jon@williams-home.demon.co.uk
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Coventry University
e-mail: J.Williams@coventry.ac.uk
Address: Control Theory and Applications Centre, School of Mathematical and
Information Sciences, Coventry University, Priory Street, Coventry CV1 5FB
United Kingdom.
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