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 follow ing two most general types of rules are considered: (1) IF A is-an-ele ment-of a THEN B is-an-element-of b, and (2) IF A is-an-element-of a T HEN {(B is-an-element-of b1, v1),..., (B is-an-element-of b(k), V(k))} , 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 se t b, and (2) if A takes on its value in a set a, then B may take on it s values in diverse sets, b1, . . . , b(k), each with its associated d egree of belief v1,..., v(k) is-an-element-of (0, 1], respectively. Si mpler cases of these two rules are also considered in which the sets a and / or b (b(i)) collapse to single elements. First, these IF-THEN r ules are represented by the so-called compatibility relations as propo sed by Kacprzyk [1-4]. Then Yager's idea of specificity, introduced in itially 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 the se measures of specificity we also use Yager's concept of a real numbe r subsuming a fuzzy number.