On Generalizing Rough Set Theory

This paper summarizes various formulations of the standard rough set theory. It demonstrates how those formulations can be adopted to develop different generalized rough set theories. The relationships between rough set theory and other theories are discussed. 1 Formulations of Standard Rough Sets The theory of rough sets can be developed in at least two different manners, the constructive and algebraic methods [16–20, 25, 29]. The constructive methods define rough set approximation operators using equivalence relations or their induced partitions and subsystems; the algebraic methods treat approximation operators as abstract operators. 1.1 Constructive methods Suppose U is a finite and nonempty set called the universe. Let E ⊆ U × U be an equivalence relation on U . The pair apr = (U,E) is called an approximation space [6, 7]. A few definitions of rough set approximations can be given based on different representations of an equivalence relation. An equivalence relation E can be conveniently represented by a mapping from U to 2 , where 2 is the power set of U . More specifically, the mapping [·]E is given by: [x]E = {y ∈ U | xEy}. (1) The subset [x]E is the equivalence class containing x. The family of all equivalence classes is commonly known as the quotient set and is denoted by U/E = {[x]E | x ∈ U}. It defines a partition of the universe, namely, a family of pairwise disjoint subsets whose union is the universe. A new family of subsets, denoted by σ(U/E), can be obtained from U/E by adding the empty set ∅ and making it closed under set union, which is a subsystem of 2 . In fact, it is an σ-algebra of subsets of U and a sub-Boolean algebra of the Boolean algebra (2 , ,∩,∪). Furthermore, σ(U/E) defines uniquely a topological space (U, σ(U/E)), in which σ(U/E) is the family of all open and closed sets [6]. Under the equivalence relation, we only have a coarsened view of the universe. Each equivalence class is considered as a whole granule instead of many individuals [21]. They are considered as the basic or elementary definable, observable, or measurable subsets of the universe [7, 20]. By the construction of Yao, Y.Y., On generalizing rough set theory, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of the 9th International Conference (RSFDGrC 2003), LNAI 2639, pp. 44-51, 2003.