Generalizing the Explicit Concept of Rough Set on the Basis of Modal Logic
Helmut Thiele · 2001
In usual publications on rough set theory, rough sets are very often defined (or are described) as ordered pairs of their lower and upper approximations with respect to a fixed equivalence relation on the universe considered. In contrast to this approach we are of the opinion that a rough set is an unknown (or non-deterministic) set which generates a certain lower and upper approximation. Now, one is faced with the fact that different sets can generate the same lower and upper approximations (the fact of “non-determinism”). Following Z. Pawlak and other authors this difficulty can be overcome by defining a rough set as the system of all sets which generate the same lower and the same upper approximation. For such systems of sets (“non-deterministic” sets) we define a partial order relation and operations like intersection and union as for usual crisp sets or fuzzy sets and investigate the obtained lattice. Furthermore, we can state that the lower and the upper approximation can be interpreted by the modal box and diamond operator, respectively, where the reachability relation used is an equivalence relation on the universe considered. Now, we replace this equivalence relation by an arbitrary binary relation on the universe. Using this relation as the reachability relation in defining the box and diamond operator, we get a generalization of the lower and upper approximation. So, we obtain a generalization of the “classical” rough set theory in Pawlak ’s sense. In particular, we can elaborate this approach using concepts of modal logic.