The Development of Fuzzy Rough Sets with the Use of Structures and Algebras of Axiomatic Fuzzy Sets

Abstract

The notion of a rough set was originally proposed by Pawlak underwent a number of extensions and generalizations. Dubois and Prade (1990) introduced fuzzy rough sets which involve the use of rough sets and fuzzy sets within a single framework. Radzikowska and Kerre (2002) proposed a broad family of fuzzy rough sets, referred to as (phi, t)-fuzzy rough sets which are determined by some implication operator (implicator) phi and a certain t-norm. In order to describe the linguistically represented concepts coming from data available in some information system, the concept of fuzzy rough sets are redefined and further studied in the setting of the axiomatic fuzzy set (AFS) theory. Compared with the (phi, t)-fuzzy rough sets, the advantages of AFS fuzzy rough sets are twofold. They can be directly applied to data analysis present in any information system without resorting to the details concerning the choice of the implication phi, t-norm and a similarity relation S. Furthermore such rough approximations of fuzzy concepts come with a well-defined semantics and therefore offer a sound interpretation. Some examples are included to illustrate the effectiveness of the proposed construct. It is shown that the AFS fuzzy rough sets provide a far higher flexibility and effectiveness in comparison with rough sets and some of their generalizations.