Re: Fuzzy relations vs. Mamdani model


Subject: Re: Fuzzy relations vs. Mamdani model
From: Ulrich Bodenhofer (ulrich.bodenhofer@scch.at)
Date: Mon Nov 13 2000 - 13:40:35 MET


Dear Andrej,

> 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.

Ok, I agree.

> 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...

Before I come to my answer, please note that it is not justified to
say that the minimum is used as implication. This is a common
misconception which has even led to the rejection of Mamdani's
model by logicians. You should rather consider the Mamdani model
as conditional assignments with fuzzy transitions than as a
model where logical implications in the strict sense are involved.
Some crazy guys (even prominents ones among them) have denoted
the minimum Mamdani or Zadeh implication which is nothing else
than nonsense. I would recommend you to read some of the following
publications:

@incollection{GodoHajek:99,
  author = {L. Godo and P. H\'ajek},
  title = {A Note on Fuzzy Inference as Deduction},
  booktitle = {Fuzzy Sets, Logics and Reasoning about Knowledge},
  editor = {D. Dubois and E. P. Klement and H. Prade},
  publisher = {Kluwer Academic Publishers},
  address = {Dordrecht},
  volume = {15},
  series = {Applied Logic Series},
  year = {1999},
  pages = {237--241}
}

@book{Hajek:98,
  author = {P. H\'ajek},
  title = {Metamathematics of Fuzzy Logic},
  publisher = {Kluwer Academic Publishers},
  volume = {4},
  series = {Trends in Logic},
  address = {Dordrecht},
  year = {1998}
}

@article{HajekGodo:97,
  author = {P. H\'ajek and L. Godo},
  title = {Deductive Systems of Fuzzy Logic},
  journal = {Tatra Mt. Math. Publ.},
  volume = {13},
  pages = {35--66},
  year = {1997}
}

> My question now is: are these two methods equivalent or is the
> second one a simplified model of the first one? Is a solution of the
> Mamdani model for multiple rules one but not the smallest solution
> of the system of relational equations? Or is the Mamdani model
> only valid for crisp input values?

The latter is the case: the two models coincide if you model the fuzzy
relation as the union of Cartesian products, i.e. if you model the rule
base by a disjunction of conjunctions (and not as implications, see above).
However, this is only one possibility. The other is to consider the
rule base as a list of logical implications (deduction rules) which have
to hold simultaneously. Then the fuzzy relation has to be constructed
as a conjunction of implications (see the references above or
some of Vilem Novak's papers). In both cases, it is not guaranteed that
the resulting relation is a solution of the corresponding system of
relational equations. I would recommend the following references:

@incollection{BauerKlementMoserLeikermoser:95,
  author = {P. Bauer and E. P. Klement and B. Moser and
                  A. Leikermoser},
  title = {Modeling of Control Functions by Fuzzy Controllers},
  booktitle = {Theoretical Aspects of Fuzzy Control},
  publisher = {John Wiley \& Sons},
  address = {New York},
  chapter = {5},
  pages = {91--116},
  year = {1995},
  editor = {H. T. Nguyen and M. Sugeno and R. M. Tong and
                  R. R. Yager}
}

@techreport{DeBaets:96,
  author = {B. {De Baets}},
  title = {A Note on {M}amdani Controllers},
  institution = {University of Gent},
  year = {1996},
  number = {FUM.BDB.96.12}
}

@book{KruseGebhardtKlawonn:94,
  author = {R. Kruse and J. Gebhardt and F. Klawonn},
  title = {Foundations of Fuzzy Systems},
  publisher = {John Wiley \& Sons},
  address = {New York},
  year = {1994}
}

I hope I could help to clarify these points a little bit.

Regards,
Ulrich

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