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31
Two views of the theory of rough sets in finite universes
 International Journal of Approximate Reasoning
, 1996
"... This paper presents and compares two views of the theory of rough sets. The operatororiented view interprets rough set theory as an extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are related to the interior and closure operators in top ..."
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Cited by 45 (19 self)
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This paper presents and compares two views of the theory of rough sets. The operatororiented view interprets rough set theory as an extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are related to the interior and closure operators in topological spaces, the necessity and possibility operators in modal logic, and lower and upper approximations in interval structures. The setoriented view focuses on the interpretation and characterization of members of rough sets. Iwinski type rough sets are formed by pairs of definable (composed) sets, which are related to the notion of interval sets. Pawlak type rough sets are defined based on equivalence classes of an equivalence relation on the power set. The relation is defined by the lower and upper approximations. In both cases, rough sets may be interpreted, or related to, families of subsets of the universe, i.e., elements of a rough set are subsets of the universe. Alternatively, rough sets may be interpreted using elements of the universe based on the notion of rough membership functions. Both operatororiented and setoriented views are useful in the understanding and application of the theory of rough sets.
Relational interpretations of neighborhood operators and rough set approximation operators
 Information Sciences
, 1998
"... This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations. A special class of neighborhood systems, called 1neighborhood systems, is introduced. Three extensions of Pawl ..."
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Cited by 37 (16 self)
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This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations. A special class of neighborhood systems, called 1neighborhood systems, is introduced. Three extensions of Pawlak approximation operators are analyzed. Properties of neighborhood and approximation operators are studied, and their connections are examined.
Information granulation and rough set approximation
 International Journal of Intelligent Systems
, 2001
"... Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarsegrained views of the universe. Approximation of concepts, represented by subsets of the universe, d ..."
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Cited by 32 (15 self)
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Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarsegrained views of the universe. Approximation of concepts, represented by subsets of the universe, deals with the descriptions of concepts using granules. In the context of rough set theory, this paper examines the two related issues. The granulation structures used by standard rough set theory and the corresponding approximation structures are reviewed. Hierarchical granulation and approximation structures are studied, which results in stratified rough set approximations. A nested sequence of granulations induced by a set of nested equivalence relations leads to a nested sequence of rough set approximations. A multilevel granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored. 1
Constructive and algebraic methods of the theory of rough sets
 Information Sciences
, 1998
"... This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtai ..."
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Cited by 21 (4 self)
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This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtained from different types of binary relations. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Various classes of rough set algebras are characterized by different sets of axioms. Axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators. 1
Granular Computing on Binary Relations ii: Rough set representations and belief functions
 Rough Sets In Knowledge Discovery
, 1998
"... This is a continuation of [13]. Let us quote few words from it. "Granulation:::appears:::in di erent names, such as chunking, clustering, data compression, divide and conquer, information hiding, interval computations, and rough set theory, justto name a few. " " the computing theory ..."
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Cited by 17 (6 self)
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This is a continuation of [13]. Let us quote few words from it. "Granulation:::appears:::in di erent names, such as chunking, clustering, data compression, divide and conquer, information hiding, interval computations, and rough set theory, justto name a few. " " the computing theory on information granulation
A comparative study of fuzzy sets and rough sets
 Information Sciences
, 1998
"... This paper reviews and compares theories of fuzzy sets and rough sets. Two approaches for the formulation of fuzzy sets are reviewed, one is based on manyvalued logic and the other is based on modal logic. Two views of rough sets are presented, setoriented view and operatororiented view. Rough se ..."
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Cited by 16 (2 self)
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This paper reviews and compares theories of fuzzy sets and rough sets. Two approaches for the formulation of fuzzy sets are reviewed, one is based on manyvalued logic and the other is based on modal logic. Two views of rough sets are presented, setoriented view and operatororiented view. Rough sets under setoriented view are closely related to fuzzy sets, which leads to nontruthfunctional fuzzy set operators. Both of them may be considered as deviations of classical set algebra. In contrast, rough sets under operatororiented view are different from fuzzy sets, and may be regarded as an extension of classical set algebra. Key words: approximation operators, fuzzy sets, interval fuzzy sets, modal logic, manyvalued logic, possibleworld semantics, product systems, rough sets. 1
Granular Computing Using Information Tables
 In: Data Mining, Rough Sets and Granular Computing
, 2002
"... Abstract. A simple and more concrete granular computing model may be developed using the notion of information tables. In this framework, each object in a finite nonempty universe is described by a finite set of attributes. Based on attribute values of objects, one may decompose the universe into pa ..."
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Cited by 14 (10 self)
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Abstract. A simple and more concrete granular computing model may be developed using the notion of information tables. In this framework, each object in a finite nonempty universe is described by a finite set of attributes. Based on attribute values of objects, one may decompose the universe into parts called granules. Objects in each granule share the same or similar description in terms of their attribute values. Studies along this line have been carried out in the theories of rough sets and databases. Within the proposed model, this paper reviews the pertinent existing results and presents their generalizations and applications. 1
Neighborhood Systems A Qualitative Theory for Rough and Fuzzy
 Sets”, Workshop on Rough Set Theory, Proceedings of Second Annual Joint Conference on Information Science, Wrightsville
, 1995
"... The theory of neighborhood systems is abstracted from the geometric notion of "near " or "negligible distances. " It is a "new " theory of the classical concept of neighborhood systems within the context of advanced computing. By definition neighborhood systems include both rough sets and topologica ..."
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Cited by 13 (7 self)
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The theory of neighborhood systems is abstracted from the geometric notion of "near " or "negligible distances. " It is a "new " theory of the classical concept of neighborhood systems within the context of advanced computing. By definition neighborhood systems include both rough sets and topological spaces as special cases. The deeper and more interesting part is in its interactions with fuzzy sets: Intuitively, qualitative fuzzy sets should be characterized by "elastic" membership functions that can tolerate “a small amount of continuous stretching with limited number of broken points. ” Based on neighborhood systems we develop a theory for such qualitative fuzzy sets. As illustrations fuzzy inferences and Lyapunov stability are discussed.
Information granulation and approximation in a decisiontheoretical model of rough sets
, 2003
"... Summary. Granulation of the universe and approximation of concepts in the granulated universe are two related fundamental issues in the theory of rough sets. Many proposals dealing with the two issues have been made and studied extensively. We present a critical review of results from existing studi ..."
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Cited by 12 (8 self)
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Summary. Granulation of the universe and approximation of concepts in the granulated universe are two related fundamental issues in the theory of rough sets. Many proposals dealing with the two issues have been made and studied extensively. We present a critical review of results from existing studies that are relevant to a decisiontheoretic modeling of rough sets. Two granulation structures are studied, one is a partition induced by an equivalence relation and the other is a covering induced by a reflexive relation. With respect to the two granulated views of the universe, element oriented and granule oriented definitions and interpretations of lower and upper approximation operators are examined. The structures of the families of fixed points of approximation operators are investigated. We start with the notions of rough membership functions and graded set inclusion defined by conditional probability. This enables us to examine different granulation structures and the induced approximations in a decisiontheoretic setting. By reviewing and combining results from existing studies, we attempt to establish a solid foundation for rough sets and to provide a systematic way for determining the required parameters in defining approximation operators. 1
On generalizing Pawlak approximation operators
 Proceedings of the First International Conference, RSCTC’98, LNAI 1424
, 1998
"... Abstract. This paper reviews and discusses generalizations of Pawlak rough set approximation operators in mathematical systems, such as topological spaces, closure systems, lattices, and posets. The structures of generalized approximation spaces and the properties of approximation operators are anal ..."
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Cited by 12 (4 self)
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Abstract. This paper reviews and discusses generalizations of Pawlak rough set approximation operators in mathematical systems, such as topological spaces, closure systems, lattices, and posets. The structures of generalized approximation spaces and the properties of approximation operators are analyzed. 1