Knowledge Approximations in Binary Relation: Granular Computing Approach

The primary goal of this paper is to present a new approximation theory. In classical rough sets, topological concepts, closure, and interior are expressible by elementary sets; hence, they are the elementary knowledge approximations. However, for general binary relations, the corresponding closure and interior cannot always be interpreted by elementary knowledge. The primary results of this paper are to show that the appropriate generalized closure and interior are central knowledge approximations and have many expected nice properties, such as the upper approximation contains lower approximations. Here the central knowledge approximations mean the approximations are expressible by centers sets and the center set denotes the set of points that regard a binary neighborhood of the point p (which is the set of points that are (right) related to p) as its neighborhood. Many examples are used to justify our new view.