Study of Covering Based Multi Granular Rough Sets and Their Topological Properties

Abstract

Abstract-The notions of basic rough sets introduced by Pawlak as a model of uncertainty, which depends upon a single equivalence relation has been extended in many directions. Over the years, several extensions to this rough set model have been proposed to improve its modeling capabilities. From the granular computing point of view these models are single granulations only. This single granulation model has been extended to multi-granulation set up by taking more than one equivalence relations simultaneously. This led to the notions of optimistic and pessimistic multi-granulation. One direction of extension of the basic rough set model is dependent upon covers of universes instead of partitions and has better modeling power as in many real life scenario objects cannot be grouped into partitions but into covers, which are general notions of partitions. So, multigranulations basing on covers called covering based multi-granulation rough sets (CBMGRS) were introduced. In the literature four types of CBMGRSs have been introduced. The first two types of CBMGRS are based on minimal descriptor and the other two are based on maximal descriptor. In this paper all these four types of CBMGRS are studied from their topological characterizations point of view. It is well known that there are four kinds of basic rough sets from the topological characterisation point of view. We introduce similar characterisation for CBMGRSs and obtained the kinds of the complement, union, and intersection of such sets. These results along with the accuracy measures of CBMGRSs are supposed to be applicable in real life situations. We provide proofs and counter examples as per the necessity of the situations to establish our claims.