Re: methods for approximate reasoning


Subject: Re: methods for approximate reasoning
From: Pim van den Broek (pimvdb@cs.utwente.nl)
Date: Sun Nov 19 2000 - 22:25:57 MET


albert@massivbau.tu-darmstadt.de wrote:

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

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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