A Survey on Axiomatic Characterizations of Rough Approximation Operators
WU Weizhi
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316022 Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhejiang Ocean University, Zhoushan 316022
Abstract Lower and upper approximation operators are the foundation in the study of theoretic aspect of rough set theory as well as its practical applications. One of the main directions of the theoretic study of rough sets is the axiomatic characterization of rough approximation operators. Based on various binary relations, constructive definitions of classical rough approximation operators, rough fuzzy approximation operators, and fuzzy rough approximation operators are firstly introduced. Axiomatic characterizations of these approximation operators are then summarized and analyzed. Finally, perspectives and comparison of rough set approximation operators with other mathematical structures are discussed.
Fund:Supported by National Natural Science Foundation of China(No.41631179,61573321,61272021)
About author:: (WU Weizhi, born in 1964, Ph.D., professor. His research interests include rough set, granular computing, concept lattice, approximate reasoning.)
[1] PAWLAK Z. Rough Sets. International Journal of Computer and Information Science, 1982, 11(5): 341-356. [2] PAWLAK Z. Rough Sets: Theoretical Aspects of Reasoning about Data. Boston, USA: Kluwer Academic Publishers, 1991. [3] YAO Y Y. Constructive and Algebraic Methods of the Theory of Rough Sets. Information Sciences, 1998, 109(1/2/3/4): 21-47. [4] LIN T Y, LIU Q. Rough Approximate Operators: Axiomatic Rough Set Theory // ZIARKO W, ed. Rough Sets, Fuzzy Sets and Know-ledge Discovery. Berlin, Germany: Springer, 1994: 256-260. [5] 祝 峰,何华灿.粗集的公理化.计算机学报, 2000, 33(3): 330-333. (ZHU F, HE H C. The Axiomatization of the Rough Set. Chinese Journal of Computers, 2000, 33(3): 330-333.) [6] YAO Y Y. Generalized Rough Set Model // POLKOWSKI L, SKOWRON A, eds. Rough Sets in Knowledge Discovery 1. Methodology and Applications. Heidelberg, Germany: Physica-Verlag, 1998: 286-318. [7] YAO Y Y. Two Views of the Theory of Rough Sets in Finite Universes. International Journal of Approximate Reasoning, 1996, 15(4): 291-317. [8] THIELE H. On Axiomatic Characterisations of Crisp Approximation Operators. Information Sciences, 2000, 129(1/2/3/4): 221-226. [9] YANG X P, LI T J. The Minimization of Axiom Sets Characterizing Generalized Approximation Operators. Information Sciences, 2006, 176(7): 887-899. [10] WU W Z, MI J S. Some Mathematical Structures of Generalized Rough Sets in Infinite Universes of Discourse // PETERS J F, SKOWRON A, eds. Transactions on Rough Sets XIII. Berlin, Germany: Springer-Verlag, 2011, 6499: 175-206. [11] LI T J, YANG X P. An Axiomatic Characterization of Probabilistic Rough Sets. International Journal of Approximate Reasoning, 2014, 55(1): 130-141. [12] ZHANG Y L, LI J J, WU W Z. On Axiomatic Characterizations of Three Pairs of Covering Based Approximation Operators. Information Sciences, 2010, 180(2): 274-287. [13] ZHANG Y L, LUO M K. On Minimization of Axiom Sets Characterizing Covering-Based Approximation Operators. Information Sciences, 2011, 181(14): 3032-3042. [14] ZHANG X H, ZHANG Y N, XUE Z O, et al. T-Rough Approximation Pairs and Covering Based Rough Sets. Fundamenta Informaticae, 2015, 142(1/2/3/4): 195-212 [15] ZHU W, WANG F Y. Reduction and Axiomization of Covering Generalized Rough Sets. Information Sciences, 2003, 152(1): 217-230. [16] MORSI N N, YAKOUT M M. Axiomatics for Fuzzy Rough Sets. Fuzzy Sets and Systems, 1998, 100(1/2/3): 327-342. [17] THIELE H. On Axiomatic Characterisation of Fuzzy Approximation Operators I, the Fuzzy Rough Set Based Case // Proc of the 31st IEEE International Symposium on Multiple-Valued Logic. Washington, USA: IEEE, 2001: 239-247. [18] THIELE H. On Axiomatic Characterisation of Fuzzy Approximation Operators II, the Rough Fuzzy Set Based Case // Proc of the 31st IEEE International Symposium on Multiple-Valued Logic. Wa-shington, USA: IEEE, 2001: 330-335. [19] 米据生,吴伟志,张文修.粗糙集的构造与公理化方法.模式识别与人工智能, 2002, 15(3): 280-284. (MI J S, WU W Z, ZHANG W X. Constructive and Axiomatic Approaches of the Theory of Rough Sets. Pattern Recognition and Artificial Intelligence, 2002, 15(3): 280-284.) [20] 吴伟志,张文修,徐宗本.粗糙模糊集的构造与公理化方法.计算机学报, 2004, 27(2): 197-203. (WU W Z, ZHANG W X, XU Z B. Characterizating Rough Fuzzy Sets in Constructive and Axiomatic Approaches. Chinese Journal of Computers, 2004, 27(2): 197-203.) [21] WU W Z, MI J S, ZHANG W X. Generalized Fuzzy Rough Sets. Information Sciences, 2003, 151: 263-282. [22] WU W Z, ZHANG W X. Constructive and Axiomatic Approaches of Fuzzy Approximation Operators. Information Sciences, 2004, 159(3/4): 233-254. [23] YANG X P. Minimization of Axiom Sets on Fuzzy Approximation Operators. Information Sciences, 2007, 177(8): 3840-3854. [24] MI J S, LEUNG Y, ZHAO H Y, et al. Generalized Fuzzy Rough Sets Determined by a Triangular Norm. Information Sciences, 2008, 178(6): 3203-3213. [25] WU W Z. On Some Mathematical Structures of T-Fuzzy Rough Set Algebras in Infinite Universes of Discourse. Fundamenta Informaticae, 2011, 108(3/4): 337-369. [26] RADZIKOWSKA A M, KERRE E E. A Comparative Study of Fuzzy Rough Sets. Fuzzy Sets and Systems, 2002, 126: 137-155. [27] WU W Z, LEUNG Y, MI J S. On Characterizations of (I,T)-Fuzzy Rough Approximation Operators. Fuzzy Sets and Systems, 2005, 15(1): 76-102. [28] MI J S, ZHANG W X. An Axiomatic Characterization of a Fuzzy Generalization of Rough Sets. Information Sciences, 2004, 160(1/2/3/4): 235-249. [29] YEUNG D S, CHEN D G, TSANG E C C, et al. On the Generalization of Fuzzy Rough Sets. IEEE Transactions on Fuzzy Systems, 2005, 13(3): 343-361. [30] WU W Z, LEUNG Y, SHAO M W. Generalized Fuzzy Rough Approximation Operators Determined by Fuzzy Implicators. International Journal of Approximate Reasoning, 2013, 54(9): 1388-1409. [31] HOOSHMANDSL M R, KARIMI A, ALMBARDAR M, et al. Axiomatic Systems for Rough Set-Valued Homomorphisms of Associative Rings. International Journal of Approximate Reasoning, 2013, 54(2): 297-306. [32] LIU G L. Generalized Rough Sets over Fuzzy Lattices. Information Sciences, 2008, 178(6): 1651-1662. [33] LIU X D, PEDRYCZ W, CHAI T Y, et al. The Development of Fuzzy Rough Sets with the Use of Structures and Algebras of Axiomatic Fuzzy Sets. IEEE Transactions on Knowledge and Data Engineering, 2009, 21(3): 443-462. [34] SHE Y H, WANG G J. An Axiomatic Approach of Fuzzy Rough Sets Based on Residuated Lattices. Computers and Mathematics with Applications, 2009, 58(1): 189-201. [35] WANG C Y, HU B Q. Fuzzy Rough Sets Based on Generalized Residuated Lattices. Information Sciences, 2013, 248: 31-49. [36] WANG L D, LIU X D, QIU W R. Nearness Approximation Space Based on Axiomatic Fuzzy Sets. International Journal of Approximate Reasoning, 2012, 53(2): 200-211. [37] YIN Y Q, LI H J, JUN Y B. On Algebraic Structure of Intuitionistic Fuzzy Soft Sets. Computers and Mathematics with Applications, 2012, 64(9): 2896-2911. [38] ZHANG X H, ZHOU B, LI P. A General Frame for Intuitionistic Fuzzy Rough Sets. Information Sciences, 2012, 216: 34-49. [39] ZHOU L, WU W Z. On Generalized Intuitionistic Fuzzy Approximation Operators. Information Sciences, 2008, 178(11): 2448-2465. [40] ZHOU L, WU W Z. Characterization of Rough Set Approximations in Atanassov Intuitionistic Fuzzy Set Theory. Computers and Mathe-matics with Applications, 2011, 62(1): 282-296. [41] ZHOU L, WU W Z, ZHANG W X. On Characterization of Intuitionistic Fuzzy Rough Sets Based on Intuitionistic Fuzzy Implicators. Information Sciences, 2009, 179(7): 883-898. [42] 张红英,张文修.一类广义区间值模糊粗糙集的公理化刻画. 高校应用数学学报, 2009, 24(2): 239-248. (ZHANG H Y, ZHANG W X. An Axiomatic Characterization of Generalized Interval-Valued Fuzzy Rough Sets. Applied Mathema-tics: A Journal of Chinese Universities, 2009, 24(2): 239-248.) [43] 俞育才,张小红.可换双剩余格上的广义模糊粗糙集及其公理化.计算机科学与探索, 2012, 6(2): 175-182. (YU Y C, ZHANG X H. General Fuzzy Rough Sets Based on Commutative Double Residuated Lattices with Axiomatic Systems. Journal of Frontiers of Computer Science and Technology, 2012, 6(2): 175-182.) [44] MA Z M, LI J J, MI J S. Some Minimal Axiom Sets of Rough Sets. Information Sciences, 2015, 312: 40-54. [45] YANG X P, YANG Y. Independence of Axiom Sets on Intuitionistic Fuzzy Rough Approximation Operators. International Journal of Machine Learning and Cybernetics, 2013, 4(5): 505-513. [46] LIU G L. Axiomatic Systems for Rough Sets and Fuzzy Rough Sets. International Journal of Approximate Reasoning, 2008, 48(3): 857-867. [47] LIU G L. Using One Axiom to Characterize Rough Set and Fuzzy Rough Set Approximations. Information Sciences, 2013, 223: 285-296. [48] WU W Z, XU Y H, SHAO M W, et al. Axiomatic Characterizations of (S,T)-Fuzzy Rough Approximation Operators. Information Sciences, 2016, 334: 17-43. [49] WU W Z, LI T J, GU S M. Using One Axiom to Characterize Fuzzy Rough Approximation Operators Determined by a Fuzzy Implication Operator. Fundamenta Informaticae, 2015, 142(1/2/3/4): 87-104.
2017, Vol. 30 
