Fuzzy Approximation Space Research Articles

Rough approximation operators on two universes of discourse and their fuzzy extensions

Based on a crisp binary relation between two universes of discourse, one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper. Based on these induced notions, four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse. Furthermore, basic properties of the new approximation operators are investigated, and the relationships among the operators are also examined. Subsequently, conditions under which some or all of these approximation operators are equivalent are obtained. On the other hand, the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator I and a triangular norm T . Consequently, four pairs of fuzzy rough approximation operators are constructed. Basic properties and comparison of the fuzzy rough approximation operators are discussed. If I is an R-implicator with respect to a t-norm T , then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy T -partition.

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.

Rough sets on fuzzy approximation spaces and applications to distributed knowledge systems

A fuzzy relation is an extension of crisp relation on any set U. Fuzzy proximity relations on U are much more general and abundant than equivalence relations. The fuzzy approximation space which depends upon a fuzzy proximity relation defined on a set U is a generalisation of the concept of knowledge base. So, rough sets defined on fuzzy approximation spaces extend the concept of rough sets on knowledge bases. The results of the present paper extend the basic properties of rough sets and results involving set theoretic operations on types of rough sets that have been established in Tripathy and Mitra (2008).

Fuzzy probabilistic approximation spaces and their information measures

Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work

An axiomatic characterization of a fuzzy generalization of rough sets

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find the assumptions which permit a given fuzzy set-theoretic operator to represent a upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.