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Constructive and algebraic methods of the theory of rough sets. (English) Zbl 0934.03071
Abstract approximation operators corresponding to a generalization of the classical rough set approach are discussed and compared with constructive methods to define set approximations in the rough set framework. The main results are related to axiomatization of approximation operators and conditions under which different rough algebras are defined. Some characterizations of approximation operators used by different authors are presented.

MSC:
03E72 Theory of fuzzy sets, etc.
68T30 Knowledge representation
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[1] Cattaneo, G., Generalized rough sets: preclusivity fuzzy-intuitionistic (BZ) lattices, Studia logica, 58, 47-77, (1997) · Zbl 0864.03040
[2] Cattaneo, G., Mathematical foundations of roughness and fuzziness, Manuscript, dipartimento di science dell’informazione, univerita di milano, via comelico 39, I-20135, (1997), Milano, Italy
[3] Chellas, B.F., Modal logic: an introduction, (1980), Cambridge University Press Cambridge · Zbl 0431.03009
[4] Cohn, P.M., Universal algebra, (1965), Harper & Row New York · Zbl 0141.01002
[5] Comer, S., An algebraic approach to the approximation of information, Fundamenta informaticae, 14, 492-502, (1991) · Zbl 0727.68114
[6] Comer, S., On connections between information systems, rough sets, and algebraic logic, Algebraic methods in logic and computer science, Banach center publications, 28, 117-124, (1993) · Zbl 0793.03074
[7] Du¨ntsch, I., Rough sets and algebras of relations, (), 95-108
[8] Gehrke, M.; Walker, E., On the structure of rough sets, Bulletin of the Polish Academy of sciences, mathematics, 40, 235-245, (1992) · Zbl 0778.04002
[9] Henkin, L.; Monk, J.D.; Tarski, A., Cylindric algebras I, (1971), North-Holland Amsterdam · Zbl 0214.01302
[10] Hughes, G.E.; Cresswell, M.J., An introduction to modal logic, (1968), Methuen London · Zbl 0205.00503
[11] Iwinski, T.B., Algebraic approach to rough sets, Bulletin of the Polish Academy of sciences, mathematics, 35, 673-683, (1987) · Zbl 0639.68125
[12] Iwinski, T.B., Rough orders and rough concepts, Bulletin of the Polish Academy of sciences, mathematics, 36, 187-192, (1988) · Zbl 0676.06013
[13] Klir, G.J., Multivalued logics versus modal logics: alternative frameworks for uncertainty modelling, (), 3-47
[14] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic, theory and applications, (1995), Prentice Hall Englewood Cliffs, NJ · Zbl 0915.03001
[15] Kortelainen, J., On relationship between modified sets, topological spaces and rough sets, Fuzzy sets and systems, 61, 91-95, (1994) · Zbl 0828.04002
[16] Lemmon, E.J., An extension algebra and the modal system T, Notre dame journal of formal logic, 1, 3-12, (1960) · Zbl 0114.24601
[17] Lemmon, E.J.; Lemmon, E.J., Algebraic semantics for modal logics II, Journal of symbolic logic, Journal of symbolic logic, 31, 191-218, (1966) · Zbl 0147.24805
[18] Lin, T.Y.; Liu, Q., Rough approximate operators: axiomatic rough set theory, (), 256-260 · Zbl 0818.03028
[19] Marchal, B., Modal logic — a brief tutorial, (), 15-23
[20] McKinsey, J.C.C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, The journal of symbolic logic, 6, 117-134, (1941) · JFM 67.0974.01
[21] McKinsey, J.C.C.; Tarski, A., The algebra of topology, Annals of mathematics, 45, 141-191, (1944) · Zbl 0060.06206
[22] Nakamura, A.; Gao, J.M., A logic for fuzzy data analysis, Fuzzy sets and systems, 39, 127-132, (1991) · Zbl 0729.03015
[23] Nakamura, A.; Gao, J.M., On a KTB-modal fuzzy logic, Fuzzy sets and systems, 45, 327-334, (1992) · Zbl 0754.03014
[24] Nguyen, H.T., Intervals in Boolean rings: approximation and logic, Foundations of computer and decision sciences, 17, 131-138, (1992) · Zbl 0781.06011
[25] Orlowska, E., Logic of indiscernibility relations, Bulletin of Polish Academy of science, mathematics, 33, 475-485, (1985) · Zbl 0584.03013
[26] Orlowska, E., Kripke semantics for knowledge representation logics, Studia logica, 49, 255-272, (1990) · Zbl 0726.03023
[27] Orlowska, E., Rough set semantics for non-classical logics, (), 143-148
[28] Pagliani, P., A pure logic-algebraic analysis of rough top and rough bottom equalities, (), 225-241
[29] Pagliani, P., Rough sets theory and logic-algebraic structures, (), 109-190
[30] Pawlak, Z., Information systems, theoretical foundations, Information systems, 6, 205-218, (1981) · Zbl 0462.68078
[31] Pawlak, Z., Rough sets, International journal of computer and information sciences, 11, 341-356, (1982) · Zbl 0501.68053
[32] Pawlak, Z., Rough classification, International journal of man-machine studies, 20, 469-483, (1984) · Zbl 0541.68077
[33] Pawlak, Z.; Wong, S.K.M.; Ziarko, W., Rough sets: probabilistic versus deterministic approach, International journal of man-machine studies, 29, 81-95, (1988) · Zbl 0663.68094
[34] Pomykala, J.A., Approximation operations in approximation space, Bulletin of the Polish Academy of sciences, mathematics, 35, 653-662, (1987) · Zbl 0642.54002
[35] Pomykala, J.A., On definability in the nondeterministic information system, Bulletin of the Polish Academy of sciences, mathematics, 36, 193-210, (1988) · Zbl 0677.68110
[36] Pomykala, J.; Pomykala, J.A., The stone algebra of rough sets, Bulletin of the Polish Academy of sciences, mathematics, 36, 495-508, (1988) · Zbl 0786.04008
[37] Rasiowa, H., An algebraic approach to non-classical logics, (1974), North-Holland Amsterdam · Zbl 0299.02069
[38] Rasiowa, H.; Sikorski, R., Mathematics of metamathematics, (1970), PWN Warsaw · Zbl 0122.24311
[39] Wasilewska, A., Topological rough algebras, (), 411-425 · Zbl 0860.03042
[40] Wiweger, A., On topological rough sets, Bulletin of the Polish Academy of sciences, mathematics, 37, 89-93, (1989) · Zbl 0755.04010
[41] Wong, S.K.M.; Wang, L.S.; Yao, Y.Y., On modeling uncertainty with interval structures, Computational intelligence, 11, 406-426, (1995)
[42] Wybraniec-Skardowska, U., On a generalization of approximation space, Bulletin of the Polish Academy of sciences, mathematics, 37, 51-61, (1989) · Zbl 0755.04011
[43] Wybraniec-Skardowska, U., Unit operations, Zeszytynaukowe wyzszej szkoly pedagogicznej im powstancow slaskich w opolu, matematyka, XXVII, 113-129, (1992)
[44] Yao, Y.Y., Two views of the theory of rough sets in finite universes, International journal of approximate reasoning, 15, 291-317, (1996) · Zbl 0935.03063
[45] Yao, Y.Y., A comparative study of fuzzy sets and rough sets, (1997), Department of Computer Science, Lakehead University Thunder Bay, Ont., Canada P7B 5E1, manuscript · Zbl 0859.04005
[46] Yao, Y.Y.; Lin, T.Y., Generalization of rough sets using modal logic, Intelligent automation and soft computing, an international journal, 2, 103-120, (1996)
[47] Yao, Y.Y.; Wong, S.K.M., A decision theoretic framework for approximating concepts, International journal of man-machine studies, 37, 793-809, (1992)
[48] Zakowski, W., On a concept of rough sets, Demonstratio Mathematica, XV, 1129-1133, (1982) · Zbl 0526.04005
[49] Zakowski, W., Approximations in the space (U,II), Demonstratio Mathematica, XVI, 761-769, (1983) · Zbl 0553.04002
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