Mathematical Analysis of Fuzzy Classifiers

. We examine the principle capabilities and limits of fuzzy classifiers that are based on a finite set of fuzzy if--then rules like they are used for fuzzy controllers, except that the conclusion of a rule specifies a discrete class instead of a (fuzzy) real output value. Our results show that in the two--dimensional case, for classification problems whose solutions can only be solved approximately by crisp classification rules, very simple fuzzy rules provide an exact solution. However, in the multi--dimensional case, even for linear separable problems, max--min rules are not sufficient. 1 Introduction Fuzzy controllers are well examined as function approximators. Piecewise monotone functions of one variable can be exactly reproduced by a fuzzy controller [1, 9] and for the multi--dimensional case fuzzy controllers are known to be universal approximators [2, 6, 12]. Although a lot of approaches for automatically learning fuzzy classifiers are proposed in the literature (see for insta..

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