## Covering rough sets based on neighborhoods: an approach without using neighborhoods.(English)Zbl 1229.03047

Summary: Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphism is provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.

### MSC:

 03E72 Theory of fuzzy sets, etc. 68T30 Knowledge representation

### Keywords:

approximation; covering; homomorphism; neighborhood; rough set

OEIS
Full Text:

### References:

 [1] Blaszczynski, J.; Greco, S.; Slowinski, R.; Szelag, M., Monotonic variable consistency rough set approaches, Int. J. Approx. Reason., 50, 7, 979-999 (2009) · Zbl 1191.68673 [2] Bonikowski, Z.; Bryniarski, E.; Wybraniec-Skardowska, U., Extensions and intentions in the rough set theory, Inform. Sci., 107, 149-167 (1998) · Zbl 0934.03069 [3] Bryniarski, E., A calculus of rough sets of the first order, Bull. Pol. Acad. Sci., 36, 16, 71-77 (1989) · Zbl 0756.04002 [4] Calegari, S.; Ciucci, D., Granular computing applied to ontologies, Int. J. Approx. Reason., 51, 4, 391-409 (2010) · Zbl 1205.68394 [6] Cattaneo, G.; Ciucci, D., Algebraic structures for rough sets, Lect. Notes Comput. Sci., 3135, 208-252 (2004) · Zbl 1109.68115 [7] Grzymala-Busse, J., Algebraic properties of knowledge representation systems, (Proc. ACM SIGART Int. Symp. Meth. Intell. Syst. (1986), ACM), 432-440 · Zbl 0175.00805 [8] Grzymala-Busse, J., Characteristic relations for incomplete data: a generalization of the indiscernibility relation, Trans. Rough Sets IV: Lect. Notes Comput. Sci., 3700, 58-68 (2005) · Zbl 1136.68531 [9] Grzymala-Busse, J.; Rza¸sa, W., Definability and other properties of approximations for generalized indiscernibility relations, Trans. Rough Sets XI: Lect. Notes Comput. Sci., 5946, 14-39 (2010) · Zbl 1274.68528 [10] Grzymala-Busse, J.; Sedelow, W., On rough sets and information system homomorphism, Bull. Pol. Acad. Sci. Tech. Sci., 36, 3, 233-239 (1988) · Zbl 0691.68093 [11] Hu, Q.; Zhang, L.; Chen, D.; Pedrycz, W.; Yu, D., Gaussian kernel based fuzzy rough sets: model, uncertainty measures and applications, Int. J. Approx. Reason., 51, 4, 453-471 (2010) · Zbl 1205.68424 [12] Inuiguchi, M.; Yoshioka, Y.; Kusunoki, Y., Variable-precision dominance-based rough set approach and attribute reduction, Int. J. Approx. Reason., 50, 8, 1199-1214 (2009) · Zbl 1191.68681 [13] Kondo, M., On the structure of generalized rough sets, Inform. Sci., 176, 5, 589-600 (2005) · Zbl 1096.03065 [14] Kudo, Y.; Murai, T.; Akama, S., A granularity-based framework of deduction, induction, and abduction, Int. J. Approx. Reason., 50, 8, 1215-1226 (2009) · Zbl 1191.68686 [15] Li, D. Y.; Ma, Y., Invariant characters of information systems under some homomorphisms, Inform. Sci., 129, 211-220 (2000) · Zbl 0980.68110 [16] Li, T. J.; Leung, Y.; Zhang, W. X., Generalized fuzzy rough approximation operators based on fuzzy coverings, Int. J. Approx. Reason., 48, 3, 836-856 (2008) · Zbl 1186.68464 [17] (Lin, T. Y.; Cercone, N., Rough Sets and Data Mining (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Boston) · Zbl 0855.00039 [18] Liu, G., The axiomatization of the rough set upper approximation operations, Fund. Inform., 69, 23, 331-342 (2006) · Zbl 1096.68150 [19] Liu, G.; Sai, Y., A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason., 50, 3, 521-528 (2009) · Zbl 1191.68689 [20] Liu, G.; Zhu, W., The algebraic structures of generalized rough set theory, Inform. Sci., 178, 21, 4105-4113 (2008) · Zbl 1162.68667 [21] McKinsey, J. C.C.; Tarski, A., The algebra of topology, Ann. Math., 45, 141-191 (1944) · Zbl 0060.06206 [22] McKinsey, J. C.C.; Tarski, A., Some theorems about the sentential calculi of lewis and heyting, J. Symbolic Logic, 13, 1-15 (1948) · Zbl 0037.29409 [23] Meng, Z.; Shi, Z., A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets, Inform. Sci., 179, 16, 2774-2793 (2009) · Zbl 1191.68667 [24] Pawlak, Z., Rough sets, Int. J. Comput. Inform. Sci., 11, 5, 341-356 (1982) · Zbl 0501.68053 [25] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0758.68054 [26] (Polkowski, L.; Skowron, A., Rough Sets and Current Trends in Computing, vol. 1424 (1998), Springer: Springer Berlin) · Zbl 0891.00026 [27] (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery, vols. 1 and 2 (1998), Physica-Verlag: Physica-Verlag Heidelberg) · Zbl 0910.00028 [28] Pomykala, J. A., Approximation operations in approximation space, Bull. Pol. Acad. Sci., 35, 9-10, 653-662 (1987) · Zbl 0642.54002 [30] Qian, Y.; Liang, J.; Dang, C., Knowledge structure, knowledge granulation and knowledge distance in a knowledge base, Int. J. Approx. Reason., 50, 1, 174-188 (2009) · Zbl 1191.68660 [32] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fund. Inform., 27, 245-253 (1996) · Zbl 0868.68103 [34] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE Trans. Knowl. Data Eng., 12, 2, 331-336 (2000) [35] Tsang, E. C.C.; Chen, D.; Yeung, D. S., Approximations and reducts with covering generalized rough sets, Comput. Math. Appl., 56, 1, 279-289 (2008) · Zbl 1145.68547 [36] Wang, C.; Wu, C.; Chen, D.; Hu, Q.; Wu, C., Communicating between information systems, Inform. Sci., 178, 3228-3239 (2008) · Zbl 1154.68558 [37] Wu, Q. E.; Wang, T.; Huang, Y. X.; Li, J. S., Topology theory on rough sets, IEEE Trans. Syst. Man Cybern. Part B: Cybern., 38, 1, 68-77 (2008) [38] Xu, W. H.; Zhang, W. X., Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets Syst., 158, 22, 2443-2455 (2007) · Zbl 1127.68106 [39] Yang, L.; Xu, L., Algebraic aspects of generalized approximation spaces, Int. J. Approx. Reason., 51, 1, 151-161 (2009) · Zbl 1200.06004 [40] Yang, T.; Li, Q., Reduction about approximation spaces of covering generalized rough sets, Int. J. Approx. Reason., 51, 3, 335-345 (2010) · Zbl 1205.68433 [41] Yao, Y. Y., Constructive and algebraic methods of theory of rough sets, Inform. Sci., 109, 21-47 (1998) · Zbl 0934.03071 [42] Yao, Y. Y., On generalizing pawlak approximation operators, Lect. Notes Artif. Intell., 424, 298-307 (1998) · Zbl 0955.68505 [43] Yao, Y. Y., Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111, 239-259 (1998) · Zbl 0949.68144 [44] Yao, Y. Y., Probabilistic rough set approximations, Int. J. Approx. Reason., 49, 2, 255-271 (2008) · Zbl 1191.68702 [45] Zakowski, W., Approximations in the space $$(u, \pi)$$, Demonstr. Math., 16, 761-769 (1983) · Zbl 0553.04002 [46] Zhai, Y.; Qu, K., On characteristics of information system homomorphisms, Theory Comput. Syst., 44, 3, 414-431 (2009) · Zbl 1176.68235 [47] Zhang, H. Y.; Zhang, W. X.; Wu, W. Z., On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse, Int. J. Approx. Reason., 51, 1, 56-70 (2009) · Zbl 1209.68552 [48] Zhong, N.; Yao, Y. Y.; Ohshima, M., Peculiarity oriented multidatabase mining, IEEE Trans. Knowl. Data Eng., 15, 4, 952-960 (2003) [51] Zhu, P.; Wen, Q., Some improved results on communication between information systems, Inform. Sci., 180, 18, 3521-3531 (2010) · Zbl 1206.68343 [53] Zhu, W., Generalized rough sets based on relations, Inform. Sci., 177, 22, 4997-5011 (2007) · Zbl 1129.68088 [54] Zhu, W., Topological approaches to covering rough sets, Inform. Sci., 177, 6, 1499-1508 (2007) · Zbl 1109.68121 [55] Zhu, W., Relationship among basic concepts in covering-based rough sets, Inform. Sci., 179, 2478-2486 (2009) · Zbl 1178.68579 [56] Zhu, W., Relationship between generalized rough sets based on binary relation and covering, Inform. Sci., 179, 210-225 (2009) · Zbl 1163.68339 [57] Zhu, W.; Wang, F. Y., Reduction and axiomization of covering generalized rough sets, Inform. Sci., 152, 217-230 (2003) · Zbl 1069.68613 [59] Zhu, W.; Wang, F. Y., On three types of covering rough sets, IEEE Trans. Knowl. 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.