Generalization of Rough Sets using Modal Logics
TL;DR: A number of extended rough set models are proposed and examined based on the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, which correspond to different modal logic systems.
Abstract: The theory of rough sets is an extension of set theory with two additional unary set-theoretic 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.
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TL;DR: 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, and introduces a special class of neighborhood system, called 1-neighborhood systems.
967 citations
TL;DR: This paper reviews and compares constructive and algebraic approaches in the study of rough set algebras and states axioms that must be satisfied by the operators.
772 citations
TL;DR: This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used and the connections between fuzzy relations and fuzzy rough approximation operators are examined.
568 citations
TL;DR: This paper presents and compares two views of the theory of rough sets: the operator-oriented and set-oriented views, which interprets rough set theory as an extension of set theory with two additional unary operators.
562 citations
TL;DR: The granulation structures used by standard rough set theory and the corresponding approximation structures are reviewed and the notion of neighborhood systems is also explored.
Abstract: 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 coarse-grained 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 multi-level granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored.
515 citations
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TL;DR: A generalized model of rough sets called variable precision model (VP-model), aimed at modelling classification problems involving uncertain or imprecise information, is presented and the main concepts are introduced formally and illustrated with simple examples.
1,975 citations
"Generalization of Rough Sets using ..." refers background in this paper
...It is therefore not surprising that similar e orts have been attempted in both rough sets and modal logics to incorporate such information [5, 6, 7, 8, 9, 10, 42, 44]....
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Book•
01 Jul 1992TL;DR: Partial table of contents:Issues in the MANAGEMENT of UNCERTAINty A Survey of Uncertain and Approximate Inference.
Abstract: Partial table of contents: ISSUES IN THE MANAGEMENT OF UNCERTAINTY A Survey of Uncertain and Approximate Inference (R. Neapolitan) Rough Sets: A New Approach to Vagueness (Z. Pawlak) ASPECTS OF FUZZY LOGIC: THEORY AND IMPLEMENTATIONS LT-Fuzzy Logics (H. Rasiowa & N. Cat Ho) On Fuzzy Intuitionistic Logic (E. Turunen) On Modifier Logic (J. Mattila) FUZZY LOGIC FOR APPROXIMATE REASONING Presumption, Prejudice, and Regularity in Fuzzy Material Implication (T. Whalen & B. Schott) Inference for Information Systems Containing Probabilistic and Fuzzy Uncertainties (J. Baldwin) FUZZY LOGIC FOR KNOWLEDGE REPRESENTATION AND ELICITATION Approximate Reasoning in Diagnosis, Therapy, and Prognosis (A. Rocha, et al.) Elementary Learning in a Fuzzy Expert System (J. Buckley) KNOWLEDGE-BASED SYSTEMS USING FUZZY LOGIC Structured Local Fuzzy Logics in MILORD (J. Agustm, et al.) The Validation of Fuzzy Knowledge-Based Systems (A. Chang & L. Hall) FUZZY LOGIC FOR INTELLIGENT DATABASE MANAGEMENT SYSTEMS Fuzzy Querying in Conventional Databases (P. Bosc & O. Pivert) Index.
714 citations
01 Dec 1992-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: This paper shows that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained, and can derive both the algebraic and probabilistic rough set approximations.
Abstract: This paper explores the implications of approximating a concept based on the Bayesian decision procedure, which provides a plausible unification of the fuzzy set and rough set approaches for approximating a concept. We show that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained. On the other hand, if a given concept is approximated by two sets, we can derive both the algebraic and probabilistic rough set approximations. Moreover, based on the well known principle of maximum (minimum) entropy, we give a useful interpretation of fuzzy intersection and union. Our results enhance the understanding and broaden the applications of both fuzzy and rough sets.
572 citations
"Generalization of Rough Sets using ..." refers background in this paper
...It is therefore not surprising that similar e orts have been attempted in both rough sets and modal logics to incorporate such information [5, 6, 7, 8, 9, 10, 42, 44]....
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TL;DR: In this article, the authors introduced the notion of an approximation apace as the pair A = (U,R), where U denotes an a rb i t ry non-empty set and S denotes some equivalence r e l a t i o n on U, and the best upper approximation of X in A is defined by AX.
Abstract: Introduction In [l] Z. Pawlak introduced the notion of an approximation apace as the pair A = (U,R), where U denotes an a rb i t ra ry non-empty set and S denotes some equivalence r e l a t i o n on U, ca l led here i n d i s c e r n i b i l l t y r e l a t i o n . Equivalence c l a s s e s of R are ca l l ed elementary s e t s in A. Every union of elementary s e t s in A and an empty set are ca l l ed composed s e t s in A. I f X c U , then the l e a s t composed set in A containing X w i l l be ca l led the best upper approximation of X in A, and w i l l be denoted by AX. The g rea te s t composed set in A contained in A w i l l be ca l l ed the best lower approximation of X in A, and w i l l be denoted by AX. A d e f i n i t i o n of these two notions that we gave in [2] i s based on a system of axioms f o r approximations and i s d i f f e r e n t that given in [1]. In [1] and [3] [4] Z. Pawlak introduced a l so the notions of rough equa l i ty , rough inc lus ion , rough r e l a t i o n and the notion of the approximation of funct ion in the space A. Basic idea of a l l these notions i s connected with the f a c t that in some app l i ca t ions we are unable to say f o r sure whether some element belongs to the set X or not . Theory of approximations in a sense of the papers [ l ] [4 ] i s a mathematical method f o r approximate c l a s s i f i c a t i o n of o b j e c t s . In many branches of computer science these problems are of primary concern. This theory can be viewed as an a l t e r native to the theory of fuzzy s e t s [ 5 ] , and theory of to le rance space [6] , however there are some e s s e n t i a l d i f f e r e n c e s between these three theor ie s .
448 citations
"Generalization of Rough Sets using ..." refers background in this paper
..., a re exive and symmetric relation) instead of an equivalence relation [43]....
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...[43] Zakowski, W., \Approximations in the space (U; )," Demonstratio Mathemat-ica, Vol. XVI, pp. 761-769, 1983....
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...If < is a compatibility relation, generalized rough set operators are di erent from the ones proposed by Pomykala [31] and Zakowski [43]....
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01 Jul 1988-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: Gains and Boose as discussed by the authors, Machine Learning and Uncertain Reasoning 3, pages 227-242, 1990; see also: International Journal of Man Machine Studies 29 (1988) 81-85
Abstract: W: B. Gains and J. Boose, editors, Machine Learning and Uncertain Reasoning 3, pages 227-242. Academic Press, New York, NY, 1990. see also: International Journal of Man Machine Studies 29 (1988) 81-85
431 citations