Generalization of Rough Sets using Modal Logics

The theory of rough sets is an extension of set theory with two additional unary set-theoretic operators deened based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satissed by a binary relation, such as serial, reeexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to diierent modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.