Characteristic numbers and approximation operators in generalized rough approximation system

Abstract

Covering rough set theory and binary relation based rough set theory are regarded as two important different generalizations of Z. Pawlak's classical rough set theory. In fact, when serial relations are considered, they are the same thing. If R is a serial relation from a universe U to another universe V, then {{R−1(y)}:y∈V} is a covering of U and {{R(x)}:x∈U} is a covering of V. In a covering approximation space (U,C), the relation R defined as R(x)={C∈C:x∈C} is just a serial relation from U to V=C. So U and V have the same status and (U,V,R) is indeed a whole, then (U,V,R) is called an Approximation System in order to take a comprehensive study from a new and higher perspective. By introducing nthl neighborhoods and kthr neighborhoods recursively, a new set of upper and lower approximation operators is established, and the Approximation System is further referred to as the Generalized Rough Approximation System (GRAS). In order to describe some stability of the Approximate System, characteristic numbers such as l-cycle number, r-cycle number, link number and connect number are also introduced. From there, partitions of GRAS can be constructed and sub-GRAS can be defined. This makes the whole of GRAS modularized, which is a big simplification when it comes to solving real problems. In addition, some applications of characteristic numbers and approximation operators are illustrated by examples, and the Boolean matrix calculation formulas of all approximations are proved. Finally, the important special GRAS (U,U,R) is studied in detail and an application example is given.