A unified framework for characterizing rough sets with evidence theory in various approximation spaces

Different rough set models may be numerically characterized with evidence theory by adopting different methodologies. That is to say, the evidence theory-based characteristics of rough sets vary with different approximation spaces and different rough set approximations. To address this issue, this paper proposes a theoretic framework within which rough set theory can be unifiedly characterized by evidence theory no matter what types of rough set approximations and approximation spaces they are. The main works are presented as follows. First, we declare a principle that in any given approximation space, a belief structure can be constructed, based on which the lower and upper approximation operators can be measured by the belief and plausibility functions if and only if a basic condition is satisfied. Second, in a given decision approximation space, a belief structure is also derived, based on which the lower and upper approximation operators can be always measured by the belief and plausibility functions without any additional condition. As theoretical applications, we employ the new principles to examine the evidence theory-based characteristics of four types of substantial rough sets, including the covering rough sets in covering approximation spaces, the decision-theoretic rough sets in Pawlak approximation spaces, the Pawlak rough sets in Pawlak decision approximation spaces, and the multigranulation rough sets in Pawlak decision approximation spaces, respectively. The proposed framework is applicable to various approximation spaces and various rough set approximations, and hence can make us clear the basic principles for relating rough set theory with evidence theory.