A decision procedure for rough set equalities

We use the notion of the rough set diagram introduced by A. Wasilewska and L. Vigneron (1998) 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, simplified, and full rough diagrams and show that there are 2^S(n)-1 topologically different simplified rough diagrams over n sets, where S(n) is the number of different simple rough diagram configurations, which is equal to the number of essential cases of models of simple rough Boolean algebra for n set variables (S(1)=3, S(2)=15, S(3)=255, …)