On the representation of fuzzy rules in terms of crisp rules

There are two classical ways of viewing a fuzzy set in terms of a combination of crisp sets; either as a weighted max-aggregation of its level-cuts, or as a convex combination of the latter. These views are applied to the particular case of fuzzy relations which are induced by fuzzy rules, where the rules are modeled by implication connectives (defined on a finite subset of the unit interval). The paper studies more closely the representation of such fuzzy rules in terms of a convex combination of gradual rules, a special type of implication-based fuzzy rule inducing a crisp relation. This representation, which might be interpreted in a probabilistic way, is shown to be unique on the assumption that the implication operator used for modeling the fuzzy rule does not satisfy a special kind of strict monotonicity condition. In this case, the crisp relations induced by the involved gradual rules correspond to level-cuts of the fuzzy relation associated with the fuzzy rule. However, other representations might exist if the aforementioned property is not satisfied. Under a slightly stronger (strict) monotonicity condition, the existence of further (non-consonant) representations is even guaranteed. Then, the crisp relations induced by gradual rules do not necessarily correspond to level-cuts of the underlying fuzzy relation. In this sense, the decomposition of a fuzzy rule into a class of crisp implication-based rules is more general than the classical decomposition of fuzzy sets. (C) 2002 Elsevier Science Inc. All rights reserved.