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A comparative study of fuzzy sets and rough sets. (English) Zbl 0932.03064

The paper reviews and compares theories of fuzzy sets and of rough sets. Two formulations of fuzzy sets, viz., in terms of many-valued and of modal logics are considered. In both cases, the theory of fuzzy sets may be viewed as a deviation from classical set theory. Rough sets are considered from the operator-oriented standpoint, based on the notions of lower and upper approximations, and from the set-oriented one, where the notion of rough membership function is fundamental. In the latter case, fuzzy and rough sets are closely related since rough membership functions are special fuzzy membership functions. In the operator-oriented approach, the theory of rough sets is an extension of classical set algebra with a pair of additional operators. Finally, combinations of fuzzy and rough sets are considered. Among others, interval fuzzy sets are defined both for modal-logic-based fuzzy sets and the set-oriented view of rough sets.

MSC:

03E72 Theory of fuzzy sets, etc.
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[1] Cattaneo, G.; Nistico´, G., Brouwer-Zadeh poset and three-valued Lukasiewicz posets, Fuzzy Sets and Systems, 33, 165-190 (1989) · Zbl 0682.03036
[2] Chanas, S.; Kuchta, D., Further remarks on the relation between rough and fuzzy sets, Fuzzy Sets and Systems, 47, 391-394 (1992) · Zbl 0755.04008
[3] Chellas, B. F., Modal Logic: An Introduction (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0431.03009
[4] Dubois, D.; Prade, H., A review of fuzzy set aggregation connectives, Information Sciences, 36, 85-121 (1985) · Zbl 0582.03040
[5] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-209 (1990) · Zbl 0715.04006
[6] Haack, S., Philosophy of Logics (1978), Cambridge University Press: Cambridge University Press Cambridge
[7] Klir, G. J., Multivalued logics versus modal logics: alternative frameworks for uncertainty modelling, (Wang, P. P., Advances in Fuzzy Theory and Technology (1994), Department of Electrical Engineering, Duke University: Department of Electrical Engineering, Duke University Durham, North Carolina), 3-47
[8] Klir, G. J.; Harmanec, D., On modal logic interpretation of possibility theory, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 2, 237-245 (1994) · Zbl 1232.68114
[9] Klir, G. J.; Yuan, B., Fuzzy Sets and Fuzzy Logic, Theory and Applications (1995), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0915.03001
[10] Kortelainen, J., On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems, 61, 91-95 (1994) · Zbl 0828.04002
[11] Kruse, R.; Schwecke, E.; Heinsohn, J., Uncertainty and Vagueness in Knowledge Based Systems (1991), Springer: Springer Berlin · Zbl 0755.68129
[12] Lemmon, E. J., Algebraic semantics for modal logics I, II, Journal of Symbolic Logic, 31, 191-218 (1966) · Zbl 0147.24805
[13] Lin, T. Y., Topological and fuzzy rough sets, (Slowinski, R., Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Boston), 287-304 · Zbl 0820.68001
[14] Lin, T. Y., Neighborhood systems: a qualitative theory for fuzzy and rough sets, (Wang, P. P., Advances in Machine Intelligence and Soft-Computing (1997), Department of Electrical Engineering, Duke University: Department of Electrical Engineering, Duke University Durham, North Carolina), 132-155
[15] Lin, T. Y.; Liu, Q., Rough approximate operators, (Ziarko, W. P., Rough Sets, Fuzzy Sets and Knowledge Discovery (1994), Springer: Springer London), 256-260 · Zbl 0818.03028
[16] Nakamura, A.; Gao, J. M., On a KTB-modal fuzzy logic, Fuzzy Sets and Systems, 45, 327-334 (1992) · Zbl 0754.03014
[17] Negoiţă, C. V.; Ralescu, D. A., Applications of Fuzzy Sets to Systems Analysis (1975), Birkha¨user: Birkha¨user Basel · Zbl 0326.94002
[18] Negoiţă, C. V.; Ralescu, D. A., Representation theorems for fuzzy concepts, Kybernetes, 4, 169-174 (1975) · Zbl 0352.02044
[19] Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences, 11, 341-356 (1982) · Zbl 0501.68053
[20] Pawlak, Z., Rough sets and fuzzy sets, Fuzzy Sets and Systems, 17, 99-102 (1985) · Zbl 0588.04004
[21] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0758.68054
[22] Pawlak, Z.; Skowron, A., Rough membership functions, (Zadeh, L. A.; Kacprzyk, J., Fuzzy Logic for the Management of Uncertainty (1994), Wiley: Wiley New York), 251-271 · Zbl 0794.03045
[23] Rasiowa, H., An Algebraic Approach to Non-classical Logics (1974), North-Holland: North-Holland Amsterdam · Zbl 0299.02069
[24] Rescher, N., Many-Valued Logic (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0248.02023
[25] Weber, S., A general concept of fuzzy connectives, negation, and implications based on \(t\)-norms and \(t\)-conorms, Fuzzy Sets and Systems, 11, 115-134 (1983) · Zbl 0543.03013
[26] Wong, S. K.M.; Ziarko, W., Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy Sets and Systems, 21, 357-362 (1987) · Zbl 0618.60002
[27] Wygralak, M., Rough sets and fuzzy sets — Some remarks on interrelations, Fuzzy Sets and Systems, 29, 241-243 (1989) · Zbl 0664.04010
[28] Yao, Y. Y., Combination of rough and fuzzy sets based on α-level sets, (Cercone, N., Rough Sets and Data Mining: Analysis for Imprecise Data (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Boston), 301-321, T.Y. Lin · Zbl 0859.04005
[29] Yao, Y. Y., Two views of the theory of rough sets in finite universes, International Journal of Approximation Reasoning, 15, 291-317 (1996) · Zbl 0935.03063
[30] 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)
[31] Yao, Y. Y.; Lingras, P. J., Interpretations of belief functions in the theory of rough sets, Information Sciences, 104, 81-106 (1998) · Zbl 0923.04007
[32] Yao, Y. Y.; Wong, S. K.M., Generalized probabilistic rough set models, (Chen, Y. Y.; Hirota, K.; Yen, J. Y., Soft Computing in Intelligent Systems and Information Processing, Proceedings of 1996 Asian Fuzzy Systems Symposium. Soft Computing in Intelligent Systems and Information Processing, Proceedings of 1996 Asian Fuzzy Systems Symposium, Kenting, Taiwan (1996), IEEE Press: IEEE Press New York), 158-163
[33] Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
[34] Zadeh, L. A., Toward a restructuring of the foundations of fuzzy logic (FL), (Abstract of BISC Seminar (1997), Computer Science Division, EECS, University of California: Computer Science Division, EECS, University of California Berkeley)
[35] Zadeh, L. A., Forward, (Orlowska, E., Incomplete Information: Rough Set Analysis (1998), Physica-Verlag: Physica-Verlag Heidelberg), v-vi
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