fuzzy mapping rules and fuzzy implication rules

Subject: fuzzy mapping rules and fuzzy implication rules
Date: Fri Nov 24 2000 - 03:38:33 MET

Hi!

I got a lot of good answers to my question regarding Fuzzy
Relations and the Mamdani Model. However a few questions still
remain. I want to apologize if these questions are answered in one
of the many papers which have been recommended to me but I
could not read all of them yet.
I have learned that there are two types of fuzzy rules which have to
be distinguished, i.e. fuzzy mapping rules and fuzzy implication
rules. Fuzzy mapping rules (e.g. Mamdani) use the minimum or
product as the implication operator (which are no real implication
operators since they do not verify their properties). In the case of
fuzzy mapping rules the rule base can be seen as a disjunction of
conjunctions or as a fuzzy graph but not as implications. As far as
I understand compared to the compositional rule of inference there
are two major simplifications possible using the operators of the
Mamdani model. First of all it is possible to calculate the output of
every rule (every relation) individually and then aggregate the
results to the final result instead of having to build the overall
relation for the rule set and then apply the input to this relation.
And second using the Mamdani operators the following equations
are valid:
B'(v) = sup_{u in U} min(A'(u),R(u,v))
= sup_{u in U} min(A'(u),A(u),B(v))
= min ( sup_{u in U} min(A'(u),A(u)), B(v))
= min(alpha, B(v)).
Therefore for an input A' its degree of compatibility with the fuzzy
set A can be calculated independently of the fuzzy set B of the
conclusion instead of applying A' to the relation R(u,v). This is even
further simplified for crisp inputs.
I will now come to my questions:
As far as I understand both simplifications (calculating the rules
individually and calculating the input independently from the output)
are only valid for the operators of the Mamdani model, i.e. for fuzzy
mapping rules. If I want to apply fuzzy implication rules with real
implication operators (Goedel, Lukasiewicz etc.) do these
simplifations still hold or am I forced to calculate the relation? For
which operators and/or for which type of input values (fuzzy/crisp)
are the simplifications still valid?
As far as I understand fuzzy mapping rules are only meaningful as
sets of rules. Is the same true for fuzzy implication rules or are
they also meaningful as single rules? Is defuzzification meaningful
for fuzzy implication rules or is it rather true that I have to apply
some method of linguistic approximation for the result?

These are a lot of questions but I still hope that some of you can
help me to clarify these points. Thank you very much in advance.

Best Regards
Andrej
**********************************
Dipl.-Ing. Andrej Albert
Institut für Massivbau
Alexanderstraße 35
Tel.: 06151/16-7036
Fax : 06151/16-7034
*********************************

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