A Decision Procedure for Rough Set Equalities

We use the notion of the rough set diagram introduced by Wasilewska and Vigneron 8] to present a general decision procedure for validity of equations in rough boolean algebra. First, we establish equivalence of validity in rough boolean algebra to validity in so called simple rough boolean algebra. Second, we propose a decision method for simple rough boolean algebra, which is to construct and consider all essential cases of models. The decision technique also gives us insights into the structure of rough diagrams: we introduce the notions of simple, simpliied, and full rough diagrams and show that there are 2 S (n) ?1 topologically diierent simpliied rough diagrams over n sets, where S (n) is the number of diierent simple rough diagram conngurations, which is equal to the number of essential cases of models of simple rough boolean algebra for n set variables The idea of rough set was proposed by Pawlak in 1]. The assumption is that initially we have some information about the elements of the universe, according to which some elements can be "seen" as the same. This assumption leads to the following deenitions 2]: A topological space is a pair hU; Ii, where U is a set (the universe) and I is the interior function such that for any A; B U we have I A A, I (A \ B) = I A \ I B, I I A = I A, I U = U. For any A U the set I (A) (is the complement of a set with respect to U) is denoted by C A and called the closure of A. A topological space satisses clopen property if C I A = I A and I C A = C A for any A U. In this case hU; Ii is called generalized rough set 6]. A A A I C Figure 1. Rough diagram.