A set theory for soft computing: a unified view of fuzzy sets via neighborboods

Abstract

The notion of fuzzy is context dependent, so for each context very often there is a fuzzy theory. Present papers use the notion of neighborhood systems to unify them. A neighborhood system is an association that assigns to each datum a list of data (a neighborhood). Rough sets and topological spaces are special cases. A real world fuzzy set should allow small amount of perturbation, so it should have an elastic membership function. Mathematically, such an elastic membership function can be expressed by a highly structured subset of membership function space. Structured sets can be singletons, equivalence classes, neighborhoods, or their fuzzified versions. This paper proposed that fuzzy sets should be abstractly defined by such structures and are termed soft sets (sofsets). Based on such structures, W-sofset, F-sofset, P-sofset, B-sofset, C-sofset, N-sofset, FP-sofset, and FF-sofsets have been identified. In this sequence, a predecessor is always a special case of a successor. Each type represents some implicit form of classical fuzzy theory. It is hoped that such a unified view will provide a useful set theory for soft computing.