Boundary region-based rough sets and uncertainty measures in the approximation space

The approximation operators play an important role in rough set theory, which are mainly defined by means of neighborhood systems. In this paper, firstly, we try to propose a class of novel definitions of the approximation operators via a predefined boundary region based on a binary relation. Then we compare the proposed concepts with the originals, the necessary and sufficient conditions of their equipollence are investigated. Secondly, we give the definitions of boundary region based on a covering. By employing the boundary, a class of novel definitions of the approximation operators based on a covering are proposed. It is shown that the proposed operators are equivalent to a class of covering approximation operators introduced by Zakowski. Thirdly, the relationship between general binary relations and coverings based approximation operators is investigated with the aid of the novel boundary region. Finally, the more reasonable characterizations of the accuracy and the roughness are proposed by employing the boundary operators. Meanwhile, we study uncertainty measures of approximation spaces based on a partition and a covering.