ROUGH FUZZY SETS AND FUZZY ROUGH SETS*

TLDR

Rough set theory, introduced by Pawlak, has frequently been compared to fuzzy set theory, with debates over which framework is more general or useful. This paper argues that rough sets and fuzzy sets serve distinct purposes and seeks to clarify their differences while unifying related works across various settings. The authors combine vagueness and coarseness by deriving upper and lower approximations of fuzzy sets under coarsened reference scales, extending Shafer’s coarsened belief functions, employing fuzzy similarity relations, and using fuzzy granules to form fuzzy partitions. Their combined framework converges to a formulation close to Caianiello’s C‑calculus. Index terms include fuzzy sets, rough sets, C‑calculus, random sets, belief functions, and similarity relations; the paper originates from a 1988 conference talk.

Abstract

Abstract The notion of a rough set introduced by Pawlak has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other. In this paper we argue that both notions aim to different purposes. Seen this way, it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems. First, one may think of deriving the upper and lower approximations of a fuzzy set, when a reference scale is coarsened by means of an equivalence relation. We then come close to Caianiello's C-calculus. Shafer's concept of coarsened belief functions also belongs to the same line of thought. Another idea is to turn the equivalence relation into a fuzzy similarity relation, for the modeling of coarseness, as already proposed by Farinas del Cerro and Prade. Instead of using a similarity relation, we can start with fuzzy granules which make a fuzzy partition of the reference scale. The main contribution of the paper is to clarify the difference between fuzzy sets and rough sets, and unify several independent works which deal with similar ideas in different settings or notations. INDEX TERMS: Fuzzy setsrough setsC-calculusrandom setsbelief functionssimilarity relations Notes *This paper is based on a talk at the International Conference on Fuzzy Sets in Informatics, Moscow, September 20-23, 1988.