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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 46 (20 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.
Generalization of Rough Sets using Modal Logics
 Intelligent Automation and Soft Computing, an International Journal
"... The theory of rough sets is an extension of set theory with two additional unary settheoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this ..."
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Cited by 43 (20 self)
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The theory of rough sets is an extension of set theory with two additional unary settheoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed. Keywords Rough sets, modal logic, rough set operators, graded rough sets, probabilistic rough sets. 1 Introduction The theory of rough sets is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations [2...
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
Interval structures: a framework for representing uncertain information
 PROCEEDINGS OF THE EIGHTH CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 1992
"... In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the roughset model, the lower and upper bounds of incidence calculus, and the belief and plausibility functions ..."
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Cited by 17 (8 self)
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In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the roughset model, the lower and upper bounds of incidence calculus, and the belief and plausibility functions all obey the axioms of an interval structure. An interval structure can be used to synthesize the decision rules provided by the experts. An efficient algorithm to find the desirable set of rules is developed from a set of sound and complete inference axioms.
Probabilistic approaches to rough sets
 Expert Systems
, 2003
"... This paper reviews probabilistic approaches to rough sets in granulation, approximation, and rule induction. The Shannon entropy function is used to quantitatively characterize partitions of a universe. Both algebraic and probabilistic rough set approximations are studied. The probabilistic approxim ..."
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Cited by 16 (8 self)
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This paper reviews probabilistic approaches to rough sets in granulation, approximation, and rule induction. The Shannon entropy function is used to quantitatively characterize partitions of a universe. Both algebraic and probabilistic rough set approximations are studied. The probabilistic approximations are defined in a decisiontheoretic framework. The problem of rule induction, a major application of rough set theory, is studied in probabilistic and informationtheoretic terms. Two types of rules are analyzed, the local, low order rules, and the global, high order rules. 1
Data Mining: Research Trends, Challenges, and Applications
 in Roughs Sets and Data Mining: Analysis of Imprecise Data
, 1997
"... Data mining is an interdisciplinary research area spanning severals disciplines such as database systems, machine learning, intelligent information systems, statistics, and expert systems. Data mining has evolved into an important and active area of research because of theoretical challenges and pra ..."
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Cited by 16 (7 self)
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Data mining is an interdisciplinary research area spanning severals disciplines such as database systems, machine learning, intelligent information systems, statistics, and expert systems. Data mining has evolved into an important and active area of research because of theoretical challenges and practical applications associated with the problem of discovering (or extracting) interesting and previously unknown knowledge from very large realworld databases. Many aspects of data mining have been investigated in several related fields. A unique but important aspect of the problem lies in the significance of needs to extend these studies to include the nature of the contents of the realworld databases. In this chapter, we discuss the theory and foundational issues in data mining, describe data mining methods and algorithms, and review data mining applications. Since a major focus of this book is on rough sets and its applications to database mining, one full section is devoted to summari...
Intervalset algebra for qualitative knowledge representation
 Proceedings of the 5th International Conference on Computing and Information, IEEE Computer
, 1994
"... The notion of interval sets is introduced as a new kind of sets, represented by a pair of sets, namely, the lower and upper bounds. The intervalset algebra may be regarded as a counterpart of the intervalnumber algebra. It provides a useful tool to represent qualitative information. Operations on ..."
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Cited by 14 (10 self)
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The notion of interval sets is introduced as a new kind of sets, represented by a pair of sets, namely, the lower and upper bounds. The intervalset algebra may be regarded as a counterpart of the intervalnumber algebra. It provides a useful tool to represent qualitative information. Operations on interval sets are also defined, based on the corresponding settheoretic operations on their members. In addition, basic properties of intervalset algebra are examined, and the relationships between interval sets, rough sets and fuzzy sets are analyzed. 1
On modeling uncertainty with interval structures
 Computational Intelligence
, 1995
"... In this paper, we introduce the notion of interval structures in an attempt to establish a unified framework for representing uncertain information. Two views are suggested for the interpretation of an interval structure. A typical example using the compatibility view is the roughset model in which ..."
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Cited by 12 (7 self)
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In this paper, we introduce the notion of interval structures in an attempt to establish a unified framework for representing uncertain information. Two views are suggested for the interpretation of an interval structure. A typical example using the compatibility view is the roughset model in which the lower and upper approximations form an interval structure. Incidence calculus adopts the allocation view in which an interval structure is defined by the tightest lower and upper incidence bounds. The relationship between interval structures and intervalbased numeric belief and plausibility functions is also examined. As an application of the proposed model, an algorithm is developed for computing the tightest incidence bounds.
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