Covering Based Pessimistic Multigranular Approximate Rough Equalities and Their Properties

Introduction

Data in real life are mostly imprecise in nature and so the conventional tools for formal modeling, reasoning and computing, which are crisp, deterministic and precise in characteristics, are inadequate to handle them. The notion of Rough set introduced by Pawlak (Pawlak, 1982) is one of the most efficient models to handle such imprecision in data. Applications as realistic as done in (Sharmila et al., 2016) and (Vashist, 2014) provide enough evidence. However, its definition depends upon equivalence relations, which are relatively rare in nature. So, many extensions of basic rough sets have been proposed in the literature and covering based rough sets, which are defined by using covers instead of partitions over universes are some such extensions Yao et al., 2012). An extensive account of its extension to the context of fuzzy approximation space and intuitionistic fuzzy approximation spaces is provided in (Tripathy, 2009a). Rough set theory is useful in the field of data mining as is shown in (Rana et al., 2016). Probabilistic flavour is added to the uncertainty based model of rough sets to enhance its applicability as is done in (Haldar, 2014).

The topic of fuzzy information granulation was first proposed and discussed by L.A. Zadeh in 1979 (Zadeh, 1979). It did not attend much attention till he revived it in a seminal paper in 1997 (Zadeh, 1997). Granulation of an object ‘A’ leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity or functionality (Yao, 2005). The theory of fuzzy information granulation (TFIG) is inspired by the ways in which humans granulate information and reason with it. Granular computing is a superset of the theory of fuzzy information granulation, rough set theory and interval computations, and is a subset of granular mathematics. An underlying idea of granular computing is the use of groups, classes or clusters of elements called granules. From a philosophical and theoretical point of view, many authors argued that information granulation is very essential to human problem solving and hence has a very significant impact on the design and implementation of intelligent systems.

From the point of view of granular computing, basic rough set theory deals with single granulation (Qian et al., 2006). However, in some application areas we need to handle more than one granulation at a time and this necessitated the development of multi-granular rough sets (MGRS), where at least two equivalence relations are taken for granulation of a universe. This concept is further extended by considering covers and this led to the development of covering based multi granular rough sets (CBMGRS) (Lin et al., 2011; Liu et al., 2011). Four types of CBMGRS are defined and their properties are established.