Semantics of Fuzzy Sets in Rough Set Theory

Abstract

Abstract. The objective of this chapter is to provide a semantic framework for fuzzy sets in the theory of rough sets. Rough membership functions are viewed as a special type of fuzzy membership functions interpretable using conditional probabilities. The relationships between fuzzy membership functions and rough membership functions, between core and support of fuzzy set theory and lower and upper approximation of rough set theory, are investigated. It is demonstrated that both theories share the same qualitative properties. Interpretations of fuzzy sets in rough set theory lead to constraints on membership values. Two types of constraints on membership values are studies, namely, constraints on membership values of related elements and constraints on membership values of related sets. The classical rough set model and generalized rough set models are discussed.

Constructive Fuzzy Sets with Similarity Semantics

“…The following restrictions on abstract fuzzy set-theoretic operations can be obtained from the set-theoretic operations of the cores and supports [21]:…”

“…A constructive fuzzy set model can be expressed as a triplet (U, K, M ) [21]. A similar structure was used on the study of a granular computing approach for problem solving [27].…”

“…M refers to a method by which fuzzy sets are constructed. Since the construction of fuzzy sets is an inseparable part of the theory, we may provide a semantic interpretation of fuzzy sets [21].…”

“…It is argued that fuzzy sets can be defined by primitive notions with semantic interpretations, such as distance, frequency and cost [21]. Equivalence classes can be treated as primitive notions and they can be defined and explained using an information table [17], [22].…”

“…A fuzzy set can be constructed by equivalence classes. The membership value of an object in the derived fuzzy set is interpreted as the conditional probability that the object belongs to a prototype set given that the object belongs to some equivalence class [16], [21].…”