A semantically sound approach to Pawlak rough sets and covering-based rough sets

In this paper, we discuss a semantically sound approach to covering-based rough sets. We recall and elaborate on a conceptual approach to Pawlak's rough set model, in which we consider a two-part descriptive language. The first part of the language is used to describe conjunctive concepts, while in the second part disjunctions are allowed as well. Given the language, we discuss its elementary and definable sets, and we study how the approximation operators can be seen as derived notions of the family of definable sets, which is represented by a Boolean algebra over a partition. Furthermore, we generalize the two parts of the language in order to describe concepts of covering-based rough sets. Unfortunately, the family of definable sets will no longer be represented by a Boolean algebra over a partition, but by the union-closure of a covering. Therefore, only the derived covering-based lower approximations of sets are definable for the generalized language. In addition, it is discussed how the two-part languages are used to construct decision rules, which are used in data mining and machine learning.