**Subject: **Re: methods for approximate reasoning

**From: **Pim van den Broek (*pimvdb@cs.utwente.nl*)

**Date: **Sun Nov 19 2000 - 22:25:57 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Masoud Nikravesh: "BISC: Call FOR PAPER"**Previous message:**Iris Gilsdorf: "Re: Fuzzy relations vs. Mamdani model"**Maybe in reply to:**albert@massivbau.tu-darmstadt.de: "methods for approximate reasoning"**Maybe reply:**Pim van den Broek: "Re: methods for approximate reasoning"

albert@massivbau.tu-darmstadt.de wrote:

*> Hi everybody!
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*>
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*> I have a (maybe very simple) question for which I cannot find an
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*> answer in any reference.
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*> Originally fuzzy rules have been defined to be fuzzy relations
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*> between the input A and the output B. The result for a (fuzzy or
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*> crisp) input is calculated by applying the compositional rule of
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*> inference, i.e. by intersecting the cylindrical extension of an input
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*> value A' with the fuzzy set of the relation and then projecting the
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*> resulting fuzzy set onto the domain of B.
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*> Most fuzzy systems use models which appear to be different. They
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*> first calculate the degree of compatibility of the input values with
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*> the fuzzy sets on the left hand side of the fuzzy rule. Then they
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*> aggregate the degrees of compatibility (e.g. using the min-
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*> operator). Using an implication operator (e.g. again min) they
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*> calculate the result of the rule and then they accumulate the
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*> results to get one fuzzy result of the rule set. Finally a crisp value
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*> is obtained using a defuzzification method.
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*> My question is: are these two methods equal for both crisp and
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*> fuzzy input parameters? Or is the second one a simplification of
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*> the original one?
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*>
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Hello Andrej,

The two methods are similar mathematically, but actually quite different.

They are similar in the sense that they both use the equation

B'(y) = sup_x min (A'(x), J (A(x),B(y)))

to compute the result B' from the rule IF A THEN B and the fact A'.

In the first method J is equal to min, and in the second J is some implication

operator.

That the methods are actually quite different can be seen for instance in the

case

where A and A' have no overlap, i.e. sup_x min (A(x),A'(x)) = 0.

The first method gives B'(y) = 0 for all y.

The second method usually gives B'(y) = 1 for all y. This is true for instance

when

A' is normal (there exists a x with A'(x) = 1) and J(0,B(y)) = 1 for all y (true

for

almost all of the proposed implication operators).

Yours,

Pim van den Broek.

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**Next message:**Masoud Nikravesh: "BISC: Call FOR PAPER"**Previous message:**Iris Gilsdorf: "Re: Fuzzy relations vs. Mamdani model"**Maybe in reply to:**albert@massivbau.tu-darmstadt.de: "methods for approximate reasoning"**Maybe reply:**Pim van den Broek: "Re: methods for approximate reasoning"

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