On measuring the specificity of IF-THEN rules

We consider the problem of how to measure the specificity of knowledge represented as a collection of if-then (production) rules. The following two most general types of rules are considered: (1) if A ϵ a then B ϵ b, and (2) if A ϵ a then {(B ϵ b1, ν1), …, (B ϵ bk, νk)}, to be interpreted as: (1) if a primary variable A takes on its value in a set a, then a secondary variable B may take on its value in a set b, and (2) if A takes on its value in a set a, then B may take on its values in diverse sets, b1, …, bk, each with its associated degree of belief ν1, …, νk ϵ (0, 1], respectively. Simpler cases of these two rules are also considered in which the sets a and / or b (bi) collapse to single elements. First, these if-then rules are represented by the so-called compatibility relations as proposed by Kacprzyk [1–4]. Then Yager's idea of specificity, introduced initially in the context of fuzzy sets and possibility distributions, is applied to define some new measures of specificity of if-then rules (their corresponding compatibility relations). In the derivation of these measures of specificity we also use Yager's concept of a real number subsuming a fuzzy number.