A classification study of rough sets generalization

Abstract

In the development of rough set theory, many different interpretations and formulations have been proposed and studied. One can classify the studies of rough sets into algebraic and constructive approaches. While algebraic studies focus on the axiomatization of rough set algebras, the constructive studies concern with the construction of rough set algebras from other well known mathematical concepts and structures. The constructive approaches are particularly useful in the real applications of rough set theory. The main objective of this thesis to provide a systematic review existing works on constructive approaches and to present some additional results. Both constructive and algebraic approaches are first discussed with respect to the classical rough set model. In particular, three equivalent constructive definitions of rough set approximation operators are examined. They are the element based, the equivalence class based, and the subsystem based definitions. Based on the element based and subsystem based definitions, generalized rough set models are reviewed and summarized. One can extend the element based definition by using any binary relations instead of equivalence relations in the classical rough set model. Many classes of rough set models can be established based on the properties of binary relations. The subsystem based definition can be extended in the set-theoretical setting, which leads to rough set models based on Pawlak approximation space, topological space, and closure system. Finally, the connections between the algebraic studies, relation based, and subsystem based formulations are established.