On the connections between multigranulation rough sets and generalized rough sets

Abstract

Abstract. This note makes a comparative study of multigranulation rough set model and generalized rough set model. It is pointed out that the pessimistic multigranulation rough set model (PMGRS) is a kind of generalized rough set model, it is induced by a reflexive and symmetric relation on the universe. On the contrary, the optimistic multigranulation rough set model (OMGRS) is not a generalized rough set model. In other words, OMGRS can not be induced by a binary relation in general. Furthermore, some mistakes in the existing papers are pointed out. Pessimistic multigranulation rough setsRough set theory, proposed by Pawlak [1], is a well-established mechanism for dealing with vagueness and uncertainty in data analysis. Rough set model is constructed on the basis of an approximation space ( , ) U R , where U is a non-empty set of objects (also called the universe of discourse) and R is an equivalence relation imposed upon U . An arbitrary subset of the universe (or target concept) is approximated by a pair of lower and upper approximations constructed by the equivalence classes associated with the equivalence relation R . The concepts of upper and lower approximations in rough set theory allow us to discover the knowledge hidden in information systems and express it in the form of decision rules.In Pawlaks rough set model, equivalence relation is a key and primitive notion. The equivalence classes are building blocks for constructing the lower and upper approximations. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. To solve this problem, Pawlak's rough set model has been generalized to arbitrary binary relation based rough set model, covering based rough set model, fuzzy rough set model, etc, to meet the needs of some real application problems. Yao [2] proposed generalized rough set model and investigated the constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation on the universe and defines a pair of lower and upper approximation operators using the binary relation. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. The axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators.Definition 1 [1] Let U be a finite universe and R an equivalence relation on U . The pair ( , ) U R is referred to as a Pawlak approximation space. For any X U ⊆ , the lower approximation