Toward the representation of implication-based fuzzy rules in terms of crisp rules

Studies the representation of implication-based fuzzy rules in terms of a convex combination of gradual rules, which are a special type of (implication-based) fuzzy rule inducing a crisp relation. This representation, which can 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 have some strict monotonicity property. 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 satisfied. Under a slightly stronger (strict) monotonicity condition, the existence of further (non-consonant) representations is even guaranteed. The usefulness of the proposed interpretation of fuzzy rules is exemplified in connection with the evaluation of rules in data mining