Conditions for equivalence between the compositional rule of inference and the compatibility modification inference

Abstract

This paper shows that the compositional rule of inference reduces to the compatibility modification inference, when the antecedent of a fuzzy rule and its input fulfil some properties. Two main types of implications are investigated, those generalising the classical material conditional (residuated and strong implications), called in this paper m-implications, and those generalising the classical Cartesian product (t-norms and pseudo-conjunctions), called t-implications. For m-implications, when their maximum modus ponens generating function is used, the compatibility measure depends obviously on the relationship between input and antecedent, but also on the t-norm the implication comes from. Similarly, for t-implications, also used as their own modus ponens generating function, the compatibility measure depends again on the input and antecedent, but also on the t-norm they come from. As it could be expected, in the first case the compatibility measure is a degree of the inclusion of the input into the antecedent, while in the second one it is a degree of the intersection of both.