Since introduction of the theory of rough set in early eighties, considerable work has been done on the development and application of this new theory. The paper provides a review of the Pawlak rough set model and its extensions, with emphasis on the formulation, characterization, and interpretation of various rough set models. 1

This paper presents and compares two views of the theory of rough sets. The operator-oriented 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 set-oriented 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 operator-oriented and set-oriented views are useful in the understanding and application of the theory of rough sets.

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

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

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 many-valued logic and the other is based on modal logic. Two views of rough sets are presented, set-oriented view and operator-oriented view. Rough sets under set-oriented view are closely related to fuzzy sets, which leads to non-truth-functional fuzzy set operators. Both of them may be considered as deviations of classical set algebra. In contrast, rough sets under operator-oriented 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, many-valued logic, possible-world semantics, product systems, rough sets. 1

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 decision-theoretic 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 decision-theoretic 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

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

We extend traditional Description Logics (DL) with a simple mechanism to handle approximate concept definitions in a qualitative way. Often, for example in medical applications, concepts are not definable in a crisp way but can fairly exhaustively be constrained through a particular sub- and a particular super-concept. We introduce such lower and upper approximations based on rough-set semantics, and show that reasoning in these languages can be reduced to standard DL satisfiability. This allows us to apply Rough Description Logics in a study of medical trials about sepsis patients, which is a typical application for precise modeling of vague knowledge. The study shows that Rough DL-based reasoning can be done in a realistic use case and that modeling vague knowledge helps to answer important questions in the design of clinical trials. 1

Abstract. A basic notion shared by rough set analysis and formal concept analysis is the definability of a set of objects based on a set of properties. The two theories can be compared, combined and applied to each other based on definability. In this paper, the notion of rough set approximations is introduced into formal concept analysis. Rough set approximations are defined by using a system of definable sets. The similar idea can be used in formal concept analysis. The families of the sets of objects and the sets of properties established in formal concept analysis are viewed as two systems of definable sets. The approximation operators are then formulated with respect to the systems. Two types of approximation operators, with respect to lattice-theoretic and set-theoretic interpretations, are studied. The results provide a better understanding of data analysis using rough set analysis and formal concept analysis. 1

Abstract. This paper summarizes various formulations of the standard rough set theory. It demonstrates how those formulations can be adopted to develop different generalized rough set theories. The relationships between rough set theory and other theories are discussed. 1 Formulations of Standard Rough Sets The theory of rough sets can be developed in at least two different manners, the constructive and algebraic methods [16–20, 25, 29]. The constructive methods define rough set approximation operators using equivalence relations or their induced partitions and subsystems; the algebraic methods treat approximation operators as abstract operators. 1.1 Constructive methods Suppose U is a finite and nonempty set called the universe. Let E ⊆ U × U be an equivalence relation on U. The pair apr = (U, E) is called an approximation space [6, 7]. A few definitions of rough set approximations can be given based on different representations of an equivalence relation.