Study of definable subsets in covering approximation spaces of rough sets

Abstract

Intelligent decision systems often need to deal with vague and uncertain data. Several approaches are commonly used to address this problem, such as statistical methods, machine learning, and fuzzy set. Overlapping with but different from the fuzzy set theory, rough set theory is a relatively new mathematical approach to vague data analysis. A rough set is basically an approximation representation of the given data. The representation is expressed in two subsets defined on the data set: the upper and lower approximations. The main difference between the rough set theory and other approaches is that it does not rely on preliminary information about the data such as membership probabilities of the data items required for fuzzy set. One of the open questions in rough set is to decide if a subset of a covering approximation space is definable. In this paper, we answer this question by investigating the approximation operator and conclude the relation of the inner definable, outer definable, and definable subsets of a covering approximation space under certain conditions. keyword: Rough set, covering, approximation space, definable subset.

Relative Reduction of Neighborhood-Covering Pessimistic Multigranulation Rough Set Based on Evidence Theory

“…In this section, we review some basic concepts related to covering rough sets, multigranulation rough sets and evidence theory. More details can be found in [9,11,19,20,28,29].…”

“…(neighborhood-covering information systems). [28] Let (U, C) be a covering approximation space. For X ⊆ U, N = {(x) C | x ∈ U}, we call (U, N) the neighborhood-covering information systems induced by (U, C).…”

Approximation Operator Based on Neighborhood Systems

“…Let U be a universe, C a covering of U , then we call U together with covering C a covering approximation space, denoted by (U, C). Definition 2.3 (The neighborhood of a point x [24]). Let (U, C) be a Covering approximation space.…”

“…For example, it does not need any additional information about data in the process of uncertain data. From then on, many researchers have made many significant contributions to developing the rough theory [11,13,15,17,18,19,20,21,22,23,24,25,26]. However, a problem with Pawlak's rough set theory is that partition or equivalence relation is explicitly used in the definition of the lower and upper approximations.…”

The characterizations of upper approximation operators based on special coverings

“…It has also been applied successfully in process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis, and so on. Many researchers have made some significant contributions to developing the rough theory [3,4,[7][8]12,[14][15][17][18][19][20][21][22][23][24][25][26]. However, a problem with Pawlak's rough set theory is that partition or equivalence relation is explicitly used in the definition of the lower and upper approximations.…”