Study of definable subsets in covering approximation spaces of rough sets

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.