On four novel kinds of fuzzy β-covering-based rough sets and their applications to three-way approximations

Abstract

Fuzzy rough sets and three-way decisions, as effective approaches for uncertain knowledge representation, learning and transformation, are widely used in approximate learning of fuzzy concepts. As a new generalization of fuzzy rough sets, fuzzy β-covering-based rough sets can characterize uncertain information flexibly. However, there exist shortcomings of some concepts in fuzzy β-covering, which limits its popularization and application. On the one side, fuzzy β-neighborhood and fuzzy β-co-neighborhood can not guarantee that they have the reflexivity property, which is contrary to the crisp neighborhood operators with reflexivity and has semantic difficulties in understanding. On the other side, most fuzzy β-covering-based rough sets and their extended models cannot satisfy the property that upper approximations contain lower approximations when $\beta \ne 1$, which can not characterize a given objective concept accurately. To break through these defects, we first propose an indiscernible fuzzy β-neighborhood and an indiscernible fuzzy β-co-neighborhood in fuzzy β-covering, both of which fulfill reflexivity. Then, we define four novel kinds of fuzzy β-covering-based rough sets in fuzzy β-covering approximation space, which satisfy the inclusion relationship of the upper and lower approximations. At the same time, some essential properties and the interrelationships between different models are also discussed. Furthermore, we give the topological properties of four novel kinds of fuzzy β-covering-based rough sets. Finally, we present an application of fuzzy β-covering-based rough sets to three-way approximations of fuzzy concepts. The comparative experimental results between the existing fuzzy β-covering-based rough set models and four novel models demonstrate the effectiveness and superiority of the proposed models in approximation learning.