A comparative study of fuzzy rough sets. (English) Zbl 1004.03043

Radzikowska, Anna Maria; Kerre, Etienne E.

Fuzzy Sets Syst. 126, No. 2, 137-155 (2002).

A new approach to the fuzzification of well-known Pawlak’s rough sets is discussed. New sets, called \(({\mathcal I},{\mathcal T})\)-fuzzy sets, are introduced, where \({\mathcal I}\) is an implicator, i.e., a function \({\mathcal I}:[0,1]^2 \to[0,1]\) satisfying the conditions: \({\mathcal I}(1,0)=0\), \({\mathcal I}(0,0)={\mathcal I}(0,1)={\mathcal I}(1,1)=1\), and \({\mathcal T}\) is a triangular norm, i.e., an increasing, associative and commutative mapping \({\mathcal T}:[0,1]^2\to [0,1]\) that satisfies the boundary condition for all \(x\in[0,1]:{\mathcal T}(x,1)=x\). The basic properties of the new sets are given.

Reviewer: Krassimir Atanassov (Sofia)

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

fuzzification of rough sets; implicator; triangular norm

References:

[1]	Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, Internat. J. General Systems, 17, 2-3, 191-209 (1990) · Zbl 0715.04006
[2]	Dubois, D.; Prade, H., Putting fuzzy sets and rough sets together, (Słowiński, R., Intelligent Decision Support (1992), Kluwer Academic: Kluwer Academic Dordrecht), 203-232
[3]	E.E. Kerre (Ed.), Introduction to the Basic Principles of Fuzzy Set Theory and Some of Its Applications, Communication & Cognition, Gent, 1993.; E.E. Kerre (Ed.), Introduction to the Basic Principles of Fuzzy Set Theory and Some of Its Applications, Communication & Cognition, Gent, 1993.
[4]	Klir, G. J.; Yuan, B., Fuzzy Logic: Theory and Applications (1995), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001
[5]	T.Y. Lin (Ed.), Proceedings of the CSC’96 Workshop on Rough Sets and Database Mining, San Jose State University, 1995.; T.Y. Lin (Ed.), Proceedings of the CSC’96 Workshop on Rough Sets and Database Mining, San Jose State University, 1995.
[6]	Lin, T. Y.; Liu, Q., Rough approximate operatorsaxiomatic rough set theory, (Ziarko, W. P., Rough Sets, Fuzzy Sets and Knowledge Discovery (1994), Springer: Springer Berlin) · Zbl 0818.03028
[7]	Nachtegael, M.; Kerre, E. E., The dizzy number of fuzzy implication operators on finite chains, (Ruan, D.; etal., Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry (1998), World Scientific: World Scientific Singapore), 29-35 · Zbl 0957.03033
[8]	Nakamura, A., Fuzzy rough sets, Note Multiple-Valued Logic Japan, 9, 8, 1-8 (1988)
[9]	E. Orłowska (Ed.), Incomplete Information: Rough Set Analysis, in Studies in Fuzziness and Soft Computing, Springer, Berlin, 1998.; E. Orłowska (Ed.), Incomplete Information: Rough Set Analysis, in Studies in Fuzziness and Soft Computing, Springer, Berlin, 1998. · Zbl 0886.68127
[10]	Orłowska, E., Studying incompleteness of informationa class of information logics, (Kijania-Placek, K.; Woleński, J., The Lvov-Warsaw School and Contemporary Philosophy (1998), Kluwer Academic Press: Kluwer Academic Press Dordrecht), 283-300 · Zbl 0968.03020
[11]	Orłowska, E., Many-valuedness and uncertainty, Proc. 27th Internat. Symp. on Multiple-Valued Logics, Antigonish, Canada, May 1997; also in Multiple Valued Logic—Internat. J., 4, 207-227 (1999) · Zbl 0941.03024
[12]	Pawlak, Z., Rough sets, Internat. J. Comput. Inform. Sci., 11, 5, 341-356 (1982) · Zbl 0501.68053
[13]	Pawlak, Z., Rough Sets—Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0758.68054
[14]	Pawlak, Z., Hard and soft sets, (Ziarko, W. P., Rough Sets, Fuzzy Sets and Knowledge Discovery (1994), Springer: Springer London), 130-135 · Zbl 0819.04008
[15]	Ruan, D.; Kerre, E. E., Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems, 54, 23-37 (1993) · Zbl 0784.68078
[16]	Skowron, A.; Polkowski, L., Rough Sets in Knowledge Discovery, vols. 1, 2 (1998), Springer: Springer Berlin · Zbl 0910.00028
[17]	R. Słowiński (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Kluwer Academic Publishers, Boston, 1992.; R. Słowiński (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Kluwer Academic Publishers, Boston, 1992. · Zbl 0820.68001
[18]	Smets, Ph.; Magrez, P., Implication in fuzzy logic, Internat. J. Approx. Reasoning, 1, 327-347 (1987) · Zbl 0643.03018
[19]	H. Thiele, On the definition of modal operators in fuzzy logic, Proc. ISMVL-93, Sacramento, California, 1993, pp. 62-67.; H. Thiele, On the definition of modal operators in fuzzy logic, Proc. ISMVL-93, Sacramento, California, 1993, pp. 62-67.
[20]	H. Thiele, Fuzzy rough sets versus rough fuzzy sets—an interpretation and a comparative study using concepts of modal logics, University of Dortmund, Tech. Report No CI-30/98, 1998; also in Proc. EUFIT’97, pp. 159-167.; H. Thiele, Fuzzy rough sets versus rough fuzzy sets—an interpretation and a comparative study using concepts of modal logics, University of Dortmund, Tech. Report No CI-30/98, 1998; also in Proc. EUFIT’97, pp. 159-167.
[21]	H. Thiele, Generalizing the explicit concept of rough set on the basis of modal logic, in: B. Reusch (Ed.), Computational Intelligence in Theory and Practice, Advances in Soft Computing, Springer, Heidelberg, to appear.; H. Thiele, Generalizing the explicit concept of rough set on the basis of modal logic, in: B. Reusch (Ed.), Computational Intelligence in Theory and Practice, Advances in Soft Computing, Springer, Heidelberg, to appear. · Zbl 1002.68168
[22]	Vakarelov, D., Information systems, similarity relations and modal logics, (Orłowska, E., Incomplete Information: Rough Set Analysis (1998), Springer: Springer Berlin), 492-550
[23]	Ziarko (Ed.), W. P., Rough Sets, Fuzzy Sets and Knowledge Discovery, Workshop in Computing (1994), Springer: Springer London · Zbl 0812.00038