Relationships between relation-based rough sets and belief structures

Abstract

As two important methods used to deal with uncertainty, the rough set theory and the evidence theory have close connections with each other. The purpose of this paper is to examine relationships between the relation-based rough set theory and the evidence theory, and to present interpretations of belief structures in relation-based rough set algebras. The probabilities of relation lower and upper approximations from a serial relation yield a pair of belief and plausibility functions and its belief structure. Properties of the belief structures induced by different relation-based rough set algebras are explored in this paper. The belief structure induced from a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation is reflexive (transitive, symmetric, Euclidean, respectively). Conversely, for a reflexive (transitive, symmetric, Euclidean, respectively) belief structure, there exist a probability and a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation such that the belief and plausibility functions defined by the known belief structure are, respectively, the belief and plausibility functions induced by the relation approximation operators. Then, necessary and sufficient conditions for a belief structure to be the belief structure induced by the relation approximation operators from different binary relations are presented.