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As two important methods used to deal with uncertainty, the rough set theory and the evidence theory have close connections with each other. The purpose of this paper is to examine relationships between the relation-based rough set theory and the evidence theory, and to present interpretations of belief structures in relation-based rough set algebras. The probabilities of relation lower and upper approximations from a serial relation yield a pair of belief and plausibility functions and its belief structure. Properties of the belief structures induced by different relation-based rough set algebras are explored in this paper. The belief structure induced from a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation is reflexive (transitive, symmetric, Euclidean, respectively). Conversely, for a reflexive (transitive, symmetric, Euclidean, respectively) belief structure, there exist a probability and a reflexive (serial and transitive, serial and symmetric, serial and Euclidean, respectively) relation such that the belief and plausibility functions defined by the known belief structure are, respectively, the belief and plausibility functions induced by the relation approximation operators. Then, necessary and sufficient conditions for a belief structure to be the belief structure induced by the relation approximation operators from different binary relations are presented. [ABSTRACT FROM AUTHOR]

Rough set theory is an important tool to deal with uncertainty, granularity and incompleteness of knowledge. The covering-based rough sets is one of the most important extensions of the classical Pawlak rough sets. In covering-based rough set theory, many types of rough sets were already defined in the literature. To find out the relationships among different types of covering-based rough sets and give an axiomatic definition is a very important issue. The generalized rough sets in neighborhood system is another important extension of the classical Pawlak rough sets. In this paper, we investigate relationships between covering-based rough sets and generalized rough sets in neighborhood system. Then, twenty-three types of the element based, the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems. Moreover, properties of the generalized approximation operators in neighborhood systems are presented. [ABSTRACT FROM AUTHOR]

Abstract: Rough set theory is an important technique to deal with vagueness and granularity in information systems. In rough set theory, relation-based rough sets and covering-based rough sets are two important extensions of the classical rough sets. This paper investigates relationships between relation-based rough sets and covering-based rough sets. Firstly, we unify five types of covering-based rough sets under a framework of a type of relation-based rough sets, and establish the equivalences between four types of covering-based rough sets and the type of relation-based rough sets, respectively. Then, within the proposed framework, conditions required for two different coverings to generate the same approximation operators are investigated. We also present some sufficient and necessary conditions for different types of covering-based rough sets to be equal. Furthermore, two examples are taken to illustrate applications of the discussion about relationships between covering-based rough sets and relation-based rough sets. [Copyright &y& Elsevier]

Abstract: Rough set theory was proposed by Pawlak to deal with the vagueness and granularity in information systems. The classical relation-based Pawlak rough set theory has been extended to covering-based generalized rough set theory. The rough set axiom system is the foundation of the covering-based generalized rough set theory, because the axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators. In this paper, the equivalent characterizations for the independent axiom sets of four types of covering-based generalized rough sets are investigated, and more refined axiom sets are presented. [Copyright &y& Elsevier]

Abstract: Rough set theory is a useful tool for dealing with inexact, uncertain or vague knowledge in information systems. The classical rough set theory is based on equivalence relations and has been extended to covering based generalized rough set theory. This paper investigates three types of covering generalized rough sets within an axiomatic approach. Concepts and basic properties of each type of covering based approximation operators are first reviewed. Axiomatic systems of the covering based approximation operators are then established. The independence of axiom set for characterizing each type of covering based approximation operators is also examined. As a result, two open problems about axiomatic characterizations of covering based approximation operators proposed by Zhu and Wang in (IEEE Transactions on Knowledge and Data Engineering 19(8) (2007) 1131–1144, Proceedings of the Third IEEE International Conference on Intelligent Systems, 2006, pp. 444–449) are solved. [Copyright &y& Elsevier]