# zbMATH — the first resource for mathematics

Relationship between generalized rough sets based on binary relation and covering. (English) Zbl 1163.68339
Summary: Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering-based rough sets and the generalized rough sets based on binary relation. Firstly, we present basic concepts and properties of this kind of rough sets. Then we investigate the relationships between this type of generalized rough sets and other five types of covering-based rough sets. The major contribution in this paper is that we establish the equivalency between this type of covering-based rough sets and a type of binary relation based rough sets. Through existing results in binary relation based rough sets, we present axiomatic systems for this type of covering-based lower and upper approximation operations. In addition, we explore the relationships among several important concepts such as minimal description, reduction, representative covering, exact covering, and unary covering in covering-based rough sets. Investigation of this type of covering-based will benefit to our understanding of other types of rough sets based on covering and binary relation.

##### MSC:
 68T30 Knowledge representation 68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text:
##### References:
 [1] Bargiela, A.; Pedrycz, W., Granular computing: an introduction, (2002), Kluwer Academic Publishers Boston [2] Bartol, W.; Miro, J.; Pioro, K.; Rossello, F., On the coverings by tolerance classes, Inform. sci., 166, 1-4, 193-211, (2004) · Zbl 1101.68863 [3] Bonikowski, Z.; Bryniarski, E.; Wybraniec-Skardowska, U., Extensions and intentions in the rough set theory, Inform. sci., 107, 149-167, (1998) · Zbl 0934.03069 [4] Bryniarski, E., A calculus of rough sets of the first order, Bull. Pol. acad. sci., 36, 16, 71-77, (1989) · Zbl 0756.04002 [5] G. Cattaneo, D. Ciucci, Algebraic structures for rough sets, in: LNCS, vol. 3135, 2004, pp. 208-252. · Zbl 1109.68115 [6] Dai, J., Rough 3-valued algebras, Inform. sci., 178, 8, 1986-1996, (2008) · Zbl 1134.06008 [7] T. Deng, Y. Chen, On reduction of morphological covering rough sets, in: FSKD 2006, LNAI, vol. 4223, 2006, pp. 266-275. [8] Deng, T.; Chen, Y.; Xu, W.; Dai, Q., A novel approach to fuzzy rough sets based on a fuzzy covering, Inform. sci., 177, 2308-2326, (2007) · Zbl 1119.03051 [9] T. Feng, J. Mi, W. Wu, Covering-based generalized rough fuzzy sets, in: RSKT 2006, LNAI, vol. 4062, 2006, pp. 208-215. · Zbl 1196.03072 [10] Li, J., Topological methods on the theory of covering generalized rough sets, Pattern recogn. artif. intell., 17, 1, 7-10, (2004), (in Chinese) [11] W. Li, N. Zhong, Y.-Y. Yao, J. Liu, C. Liu, Developing intelligent applications in social e-mail networks, in: RSCTC 2006, LNAI, vol. 4259, 2006, pp. 776-785. · Zbl 1162.68724 [12] Lin, T.Y.; Liu, Q., Rough approximate operators: axiomatic rough set theory, (), 256-260 · Zbl 0818.03028 [13] T.Y. Lin, Granular computing – structures, representations, and applications, in: LNAI, vol. 2639, 2003, pp. 16-24. · Zbl 1026.68636 [14] Liu, G.L., The axiomatization of the rough set upper approximation operations, Fundam. inform., 69, 23, 331-342, (2006) · Zbl 1096.68150 [15] Liu, G.L., Generalized rough sets over fuzzy lattices, Inform. sci., 178, 6, 1651-1662, (2008) · Zbl 1136.03328 [16] Liu, G.L.; Zhu, W., The algebraic structures of generalized rough set theory, Inform. sci., 178, 21, 4105-4113, (2008) · Zbl 1162.68667 [17] Mordeson, J., Rough set theory applied to (fuzzy) ideal theory, Fuzzy sets syst., 121, 315-324, (2001) · Zbl 1030.68085 [18] Pal, S.K.; Shankar, B.U.; Mitra, P., Granular computing, rough entropy and object extraction, Pattern recogn. lett., 26, 2509-2517, (2005) [19] Pawlak, Z., Rough sets, Int. J. comput. inform. sci., 11, 341-356, (1982) · Zbl 0501.68053 [20] Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers Boston · Zbl 0758.68054 [21] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Inform. sci., 177, 1, 3-27, (2007) · Zbl 1142.68549 [22] Pawlak, Z.; Skowron, A., Rough sets and Boolean reasoning, Inform. sci., 177, 1, 41-73, (2007) · Zbl 1142.68551 [23] Pawlak, Z.; Skowron, A., Rough sets: some extensions, Inform. sci., 177, 1, 28-40, (2007) · Zbl 1142.68550 [24] () [25] Pomykala, J.A., Approximation operations in approximation space, Bull. Pol. acad. sci., 35, 9-10, 653-662, (1987) · Zbl 0642.54002 [26] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundam. inform., 27, 245-253, (1996) · Zbl 0868.68103 [27] A. Skowron, J.F. Peters, Rough sets: trends and challenges, in: RSFDGrC 2003, LNAI, vol. 2639, 2003, pp. 25-34. · Zbl 1026.68653 [28] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE trans. knowledge data eng., 12, 2, 331-336, (2000) [29] E. Tsang, D. Cheng, J. Lee, D. Yeung, On the upper approximations of covering generalized rough sets, in: Proceedings of the 3rd International Conference Machine Learning and Cybernetics, 2004, pp. 4200-4203. [30] Wang, F.Y., On the abstraction of conventional dynamic systems: from numerical analysis to linguistic analysis, Inform. sci., 171, 233-259, (2005) · Zbl 1068.93010 [31] Wang, J.; Dai, D.; Zhou, Z., Fuzzy covering generalized rough sets, J. zhoukou teachers college, 21, 2, 20-22, (2004) [32] Xu, Z.; Wang, Q., On the properties of covering rough sets model, J. henan normal univ. (nat. sci.), 33, 1, 130-132, (2005) · Zbl 1091.03509 [33] J.T. Yao, Y.Y. Yao, Induction of classification rules by granular computing, in: Rough Sets and Current Trends in Computing, 2002, pp. 331-338. · Zbl 1013.68514 [34] J.T. Yao, W.-N. Liu, The STP model for solving imprecise problems, in: IEEE GrC 2006, 2006, pp. 683-687. [35] Y.Y. Yao, On generalizing pawlak approximation operators, in: LNAI, vol. 1424, 1998, pp. 298-307. · Zbl 0955.68505 [36] Yao, Y.Y., Relational interpretations of neighborhood operators and rough set approximation operators, Inform. sci., 111, 1-4, 239-259, (1998) · Zbl 0949.68144 [37] Yao, Y.Y., Constructive and algebraic methods of theory of rough sets, Inform. sci., 109, 21-47, (1998) · Zbl 0934.03071 [38] Y.Y. Yao, N. Zhong, Potential applications of granular computing in knowledge discovery and data mining, in: Proceedings of World Multiconference on Systemics, Cybernetics and Informatics, 1999, pp. 573-580. [39] Y.Y. Yao, Granular computing: basic issues and possible solutions, in: Proceedings of the Fifth Joint Conference on Information Sciences, vol. 1, 2000, pp. 186-189. [40] Y.Y. Yao, A partition model of granular computing, in: LNCS, vol. 3100, 2004, pp. 232-253. · Zbl 1104.68776 [41] Yao, Y.Y., Neighborhood systems and approximate retrieval, Inform. sci., 176, 3431-3452, (2007) · Zbl 1119.68074 [42] Yeung, D.; Chen, D.; Tsang, E.; Lee, J.; Xizhao, W., On the generalization of fuzzy rough sets, IEEE trans. fuzzy syst., 13, 3, 343-361, (2005) [43] Zadeh, L.A., Fuzzy sets, Inform. control, 8, 338-353, (1965) · Zbl 0139.24606 [44] Zadeh, L., Toward a generalized theory of uncertainty (GTU): an outline, Inform. sci., 172, 1-2, 1-40, (2005) · Zbl 1074.94021 [45] Zakowski, W., Approximations in the space $$(u, \pi)$$, Demonstratio math., 16, 761-769, (1983) · Zbl 0553.04002 [46] Zhong, N.; Dong, J.Z.; Ohsuga, S., Using rough sets with heuristics to feature selection, J. intell. inform. syst., 16, 3, 199-214, (2001) · Zbl 0994.68149 [47] Zhong, N.; Yao, Y.; Ohshima, M., Peculiarity oriented multidatabase mining, IEEE trans. knowledge data eng., 15, 4, 952-960, (2003) [48] Zhong, N., Rough sets in knowledge discovery and data mining, J. jpn. soc. fuzzy theory syst., 13, 6, 581-591, (2001) [49] Zhu, F.; He, H.-C., The axiomization of the rough set, Chin. J. comput., 23, 3, 330-333, (2000) [50] Zhu, W.; Wang, F.-Y., Reduction and axiomization of covering generalized rough sets, Inform. sci., 152, 1, 217-230, (2003) · Zbl 1069.68613 [51] W. Zhu, F.-Y. Wang, Binary relation based rough set, in: IEEE FSKD 2006, LNAI, vol. 4223, 2006, pp. 276-285. [52] W. Zhu, F.-Y. Wang, Relationships among three types of covering rough sets, in: IEEE GrC 2006, pp. 43-48. [53] W. Zhu, F.-Y. Wang, A new type of covering rough sets, in: IEEE IS 2006, London, 4-6 September 2006, pp. 444-449. [54] W. Zhu, F.-Y. Wang, Properties of the first type of covering-based rough sets, in: Proceedings of DM Workshop 06, ICDM 06, Hong Kong, China, December 18, 2006, pp. 407-411. [55] W. Zhu, Properties of the second type of covering-based rough sets, in: Workshop Proceedings of GrC&BI 06, IEEE WI 06, Hong Kong, China, December 18, 2006, pp. 494-497. [56] W. Zhu, Properties of the fourth type of covering-based rough sets, in: HIS’06, AUT Technology Park, Auckland, New Zealand, December 13-15, 2006, p. 43. [57] W. Zhu, F.-Y. Wang, Covering based granular computing for analysis of conflict, in: LNCS, vol. 3975, 2006, pp. 566-571. [58] Zhu, W., Topological approaches to covering rough sets, Inform. sci., 177, 6, 1499-1508, (2007) · Zbl 1109.68121 [59] W. Zhu, Basic concepts in covering-based rough sets, in: ICNC’07, 2007, vol. 5, pp. 283-286. [60] W. Zhu, F.-Y. Wang, Topological properties in covering-based rough sets, in: FSKD 2007, China, vol. 1, 24-27 August 2007, pp. 289-293. [61] Zhu, W., Generalized rough sets based on relations, Inform. sci., 177, 22, 4997-5011, (2007) · Zbl 1129.68088 [62] Zhu, W.; Wang, F.-Y., On three types of covering rough sets, IEEE trans. knowledge data eng., 19, 8, 1131-1144, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.