**Subject: **fuzzy mapping rules and fuzzy implication rules

*albert@massivbau.tu-darmstadt.de*

**Date: **Fri Nov 24 2000 - 03:38:33 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Scott Ferson: "Re: does fuzzy bound probability?"**Previous message:**Breezy: "Re: Imprecise Probability"

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

TU Darmstadt

Alexanderstraße 35

Tel.: 06151/16-7036

Fax : 06151/16-7034

http://www.massivbau.tu-darmstadt.de/user/albert/web/default.htm

*********************************

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**Next message:**Scott Ferson: "Re: does fuzzy bound probability?"**Previous message:**Breezy: "Re: Imprecise Probability"

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