Granular rough sets and granular shadowed sets: Three-way approximations in Pawlak approximation spaces

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and, at the same time, a single granular object in the quotient space. The new two-space view leads to more insights into and a deeper understanding of rough set theory. In this paper, we revisit results from rough sets from the two-space perspective and introduce the notions of granular rough sets and probabilistic granular rough sets in the quotient space, as three-way approximations of sets in the ground space. We propose a concept of granular shadowed sets in the quotient space, as three-way approximations of fuzzy sets in the ground space. We formulate a cost-sensitive method to construct a granular shadowed set from a fuzzy set. We show that, when the costs satisfy some conditions, the three granular approximations become the same for the special case where a fuzzy set is in fact a set. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper introduces the concept of Pawlak approximation space, explores the impact of the two-space view on rough set theory, proposes the notions of granular rough sets and probabilistic granular rough sets in the quotient space, and discusses the construction method and properties of granular shadowed sets.