Using One Axiom to Characterize Fuzzy Rough Approximation Operators Determined by a Fuzzy Implication Operator

• Autorzy: Wu W.-Z. , Li T-J. , Gu S.-M.

Identyfikatory

DOI

10.3233/FI-2015-1285

• Języki publikacji: EN

Abstrakty

EN

Axiomatic characterizations of approximation operators are important in the study of rough set theory. In this paper, axiomatic characterizations of relation-based fuzzy rough approximation operators determined by a fuzzy implication operator I are investigated. We first review the constructive definitions and properties of lower and upper I-fuzzy rough approximation operators. We then propose an operator-oriented characterization of I-fuzzy rough sets. We show that the lower and upper I-fuzzy rough approximation operators generated by an arbitrary fuzzy relation can be described by single axioms. We further examine that I-fuzzy rough approximation operators corresponding to some special types of fuzzy relations, such as serial, reflexive, and T -transitive ones, can also be characterized by single axioms.

Słowa kluczowe

EN

approximation operators; fuzzy implication operators; fuzzy rough sets; rough sets; triangular norms

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2015
• Tom: Vol. 142, nr 1-4
• Strony: 87--104
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Wu W.-Z.

• School of Mathematics, Physics and Information Science Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China
• Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China

autor

Li T-J.

• School of Mathematics, Physics and Information Science Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China
• Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China

autor

Gu S.-M.

• School of Mathematics, Physics and Information Science Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China
• Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province Zhejiang Ocean University Zhoushan, Zhejiang 316022, P. R. China

Bibliografia

• [1] Abdel-Hamid, A.A., Morsi, N.N.: On the relationship of extended necessity measures to implication operators on the unit interval, Information Sciences, 82, 1995, 129–145.
• [2] Baczynski,M., Jayaram, B.: Fuzzy implications, Studies in Fuzziness and Soft Computing, vol. 231, Springer, Berlin, 2008.
• [3] Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application, International Journal of Approximate Reasoning, 35, 2004, 55–95.
• [4] Hooshmandasl, M.R., Karimi, A., Almbardar, M., Davvaz, B.: Axiomatic systems for rough set-valued homomorphisms of associative rings, International Journal of Approximate Reasoning, 54, 2013, 297–306.
• [5] Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, Trends in Logic, vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
• [6] Klir, G.J., Yuan, B.: Fuzzy Logic: Theory and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1995.
• [7] Lin, T.Y., Liu, Q.: Rough approximate operators: axiomatic rough set theory, in: Ziarko, W. (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer, Berlin, 1994, pp. 256–260.
• [8] Liu, G.L.: Generalized rough set over fuzzy lattices, Information Sciences, 178, 2008, 1651–1662.
• [9] Liu, G.L.: Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximate Reasoning, 48, 2008, 857–867.
• [10] Liu, G.L.: Using one axiom to characterize rough set and fuzzy rough set approximations, Information Sciences, 223, 2013, 285–296.
• [11] Liu, X.D., Pedrycz, W., Chai, T.Y., Song, M.L.: The development of fuzzy rough sets with the use of structures and algebras of axiomatic fuzzy sets, IEEE Transactions on Knowledge and Data Engineering, 21, 2009, 443–462.
• [12] Mas, M., Monserrat,M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions, IEEE Transactions on Fuzzy Systems, 15, 2007, 1107–1121.
• [13] Mi, J.-S., Leung, Y., Zhao, H.-Y., Feng, T.: Generalized fuzzy rough sets determined by a triangular norm, Information Sciences, 178, 2008, 3203–3213.
• [14] Mi, J.-S., Zhang, W.-X.: An axiomatic characterization of a fuzzy generalization of rough sets, Information Sciences, 160, 2004, 235–249.
• [15] Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems, 100, 1998, 327–342.
• [16] Ouyang, Y., Wang, Z.D., Zhang, H.-P.: On fuzzy rough sets based on tolerance relations, Information Sciences, 180, 2010, 532–542.
• [17] Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126, 2002, 137–155.
• [18] Ruan, D., Kerre, E.E.: Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems, 54, 1993, 23–37.
• [19] She, Y.H., Wang, G.J.: An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications, 58, 2009, 189–201.
• [20] Thiele, H.: On axiomatic characterisation of crisp approximation operators, Information Sciences, 129, 2000, 221–226.
• [21] Thiele, H.: On axiomatic characterisation of fuzzy approximation operators I, the fuzzy rough set based case, RSCTC 2000, Banff Park Lodge, Bariff, Canada, 2000, Conference Proceedings, pp. 239–247.
• [22] Thiele, H.: On axiomatic characterisation of fuzzy approximation operators II, the rough fuzzy set based case, Proceeding of the 31st IEEE International Symposium on Multiple-Valued Logic, 2001, pp. 330–335.
• [23] Wang, L.D., Liu, X.D., Qiu, W.R.: Nearness approximation space based on axiomatic fuzzy sets, International Journal of Approximate Reasoning, 53, 2012, 200–211.
• [24] Wu,W.-Z.: On some mathematical structures of T -fuzzy rough set algebras in infinite universes of discourse, Fundamenta Informaticae, 108, 2011, 337–369.
• [25] Wu, W.-Z., Leung, Y., Mi, J.-S.: On characterizations of (I, T)-fuzzy rough approximation operators, Fuzzy Sets and Systems, 15, 2005, 76–102.
• [26] Wu, W.-Z., Leung, Y., Shao, M.-W.: Generalized fuzzy rough approximation operators determined by fuzzy implicators, International Journal of Approximate Reasoning, 54, 2013, 1388–1409.
• [27] Wu, W.-Z., Mi, J.-S.: Some mathematical structures of generalized rough sets in infinite universes of discourse, LNCS Transactions on Rough Sets, XIII, 2011, 175–206.
• [28] Wu, W.-Z., Mi, J.-S., Zhang, W.-X.: Generalized fuzzy rough sets, Information Sciences, 151, 2003, 263–282.
• [29] Wu, W.-Z., Zhang, W.-X.: Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences, 159, 2004, 233–254.
• [30] Yang, X.-P.: Minimization of axiom sets on fuzzy approximation operators, Information Sciences, 177, 2007, 3840–3854.
• [31] Yang, X.-P., Li, T.-J.: The minimization of axiom sets characterizing generalized approximation operators, Information Sciences, 176, 2006, 887–899.
• [32] Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets, Journal of Information Sciences, 109, 1998, 21–47.
• [33] Yao, Y.Y.: Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning, 15, 1996, 291–317.
• [34] Yao, Y.Y., Lin, T. Y.: Generalization of rough sets using modal logic, Intelligent Automation and Soft Computing, 2, 1996, 103–120.
• [35] Yeung, D.S., Chen, D.G., Tsang, E.C.C., Lee, J.W.T.,Wang, X.Z.: On the generalization of fuzzy rough sets, IEEE Transactions on Fuzzy Systems, 13, 2005, 343–361.
• [36] Zhang, Y.L., Li, J.J., Wu, W.-Z.: On axiomatic characterizations of three pairs of covering based approximation operators, Information Sciences, 180, 2010, 274–287.
• [37] Zhang, Y.L., Luo, M.K.: On minimization of axiom sets characterizing covering-based approximation operators, Information Sciences, 181, 2011, 3032–3042.
• [38] Zhu,W., Wang, F.-Y.: Reduction and axiomization of covering generalized rough sets, Information Sciences, 152, 2003, 217–230.