×

Generalization of Pawlak’s rough approximation spaces by using \(\delta \beta \)-open sets. (English) Zbl 1264.54005

Summary: The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of objects using an equivalence relation on the universe of his approximation space. This paper extends Pawlak’s rough set theory to a topological model where the set approximations are defined using the topological notion \(\delta \beta \)-open sets. A number of important results using the topological notion \(\delta \beta \)-open set are obtained. We also, proved that some of the properties of Pawlak’s rough set model are special instances of those of topological generalizations. Moreover, several important measures, related to the new model, such as accuracy measure and quality of approximation are presented.

MSC:

54A40 Fuzzy topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abu-Donia, H. M., Comparison between different kinds of approximations by using a family of binary relations, Knowl. Based Syst., 21, 911-919 (2008)
[2] Abu-Donia, H. M.; Salama, A. S., \(β\)-approximation spaces, J. Hybrid Comput. Res., 1, 2 (2008) · Zbl 1264.54005
[3] Abu-Donia, H. M.; Nasef, A. A.; Marei, E. A., Finite information systems, Appl. Math. Inform. Sci., 1, 1, 13-21 (2007) · Zbl 1138.94335
[4] Abd El-Monsef, M. E.; El-Deeb, S. N.; Mahmoud, R. A., \(β\)-open sets and \(β\)-continuous mappings, Bull. Fac. Assiut Uni., 12, 1, 77-90 (1983) · Zbl 0577.54008
[5] Andrijevic, D., Semi-preopen sets, Mat. Vesnik, 38, 24-32 (1986) · Zbl 0604.54002
[6] Calegari, S.; Ciucci, D., Granular computing applied to ontologies, Int. J. Approx. Reason., 51, 391-409 (2010) · Zbl 1205.68394
[7] Chuchro, M., On rough sets in topological Boolean algebras, Rough sets and Knowledge Discovery Baff AB, 157-160 (1993) · Zbl 0822.68106
[8] Ciucci, D.; Flaminio, Tommaso, Generalized rough approximations in \(Ł \prod \frac{1}{2} \), Int. J. Approx. Reason, 48, 544-558 (2008) · Zbl 1189.03054
[9] Diker, M., Definability and textures, Int. J. Approx. Reason., 53, 558-572 (2012) · Zbl 1258.03081
[10] El-Tayar, N. E.; Tsai, R. S.; Carruptand, P. A.; Testa, B., J. Chem. Soc. Perkin Trans., 2, 79-84 (1992)
[11] Hatir, E.; Noiri, T., Decompositions of continuity and complete continuity, Acta Math. Hungary, 11, 3, 281-287 (2006) · Zbl 1121.54026
[12] Hatir, E.; Noiri, T., On δ β-continuous functions, Chaos, Solitons Fractals, 42, 205-211 (2009) · Zbl 1198.54040
[13] Huang, K. Y.; Chang, Ting-Hua; Chang, Ting-Cheng, Determination of the threshold value \(β\) of variable precision rough set by fuzzy algorithms, Int. J. Approx. Reason., 52, 7, 1056-1072 (2011)
[14] J. James, Alpigini, F. James, peters, Andrzej skowron and Ning zhong, Rough set elements, Rough sets and Current trends in compuring, \(5^{th}\); J. James, Alpigini, F. James, peters, Andrzej skowron and Ning zhong, Rough set elements, Rough sets and Current trends in compuring, \(5^{th}\)
[15] Kelley, J., General Topology (1955), Van Nostrand Company · Zbl 0066.16604
[16] Kryszkiewicz, M., Rough set approach to incomplete information systems, Inform. Sci., 112, 39-49 (1998) · Zbl 0951.68548
[17] Levine, N., Semi open sets and semi continuity topological spaces, Amer. Math. Monthly, 70, 24-32 (1963)
[18] Liu, G.; Sai, Y., A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason., 50, 521-528 (2009) · Zbl 1191.68689
[19] J.J. Li, Topological Methods on the Theory of Covering Generalized Rough Sets Pattern Recognition and Artificial Intelligence 17(1) (2004) 7-10.; J.J. Li, Topological Methods on the Theory of Covering Generalized Rough Sets Pattern Recognition and Artificial Intelligence 17(1) (2004) 7-10.
[20] Mashhour, A. S.; Abd El-Monsef, M. E.; El-Deeb, S. N., On pre-continuous and week pre-continuous mappings, Proc. Math. Phys. Soc. Egypt, 53, 47-53 (1982) · Zbl 0571.54011
[21] Mousavi, A.; Maralani, P. J., Relative sets and rough sets, Int. J. Appl. Math. Comput. Sci., 11, 3, 637-654 (2001) · Zbl 0986.03042
[22] Ma, L., On some types of neighborhood-related covering rough sets, Int. J. Approx. Reason. (2012) · Zbl 1246.03068
[23] Njasted, O., On some classes of nearly open sets, Pro. J. Math., 15, 961-970 (1965) · Zbl 0137.41903
[24] Novotny, M.; Pawlak, Z., Characterization of rough top equalities and rough bottom equalities, Bull. Polish. Acad. of Sci., Math., 33, 92-97 (1985) · Zbl 0569.68083
[25] Novotny, M.; Pawlak, Z., On rough equalities, Bull. Polish. Acad. Sci. Math., 33, 99-104 (1985) · Zbl 0569.68084
[26] Li, X.; Liu, S., Matroidal approaches to rough sets via closure operators, Int. J. Approx. Reason., 53, 513-527 (2012) · Zbl 1246.68233
[27] Pawlak, Z., Rough sets, Int. J. Comput. Inform. Sci., 11, 341-356 (1982) · Zbl 0501.68053
[28] Pawlak, Z., Rough Sets, Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic: Kluwer Academic Boston · Zbl 0758.68054
[29] Pei, Z.; Pei, D.; Zheng, Li, Topology vs generalized rough sets, Int. J. Approx. Reason., 52, 2, 231-239 (2011) · Zbl 1232.03044
[30] Qian, Y. H.; Liang, J. Y.; Yao, Y. Y.; Dang, C. Y., MGRS: A multi-granulation rough set, Inform. Sci., 180, 949-970 (2010) · Zbl 1185.68695
[31] Qin, K.; Yang, J.; Pei, Z., Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 178, 4138-4141 (2008) · Zbl 1153.03316
[32] Salama, A. S., Topological solution of missing attribute values problem in incomplete information tables, Inform. Sci., 180, 631-639 (2010)
[33] Shen, Y.; Wang, Faxing, Rough approximations of vague sets in fuzzy approximation space, Int. J. Approx. Reason., 52, 2, 281-296 (2011) · Zbl 1232.03045
[34] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fund. Inform., 27, 245-253 (1996) · Zbl 0868.68103
[35] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE Trans. Knowledge Data Eng., 12, 331-336 (2000)
[36] Stone, M. H., Applications of the theory of Boolean rings to general topology, TAMS, 41, 375-810 (1937) · Zbl 0017.13502
[37] Velicko, N. V., H-closed topological spaces, Am. Math. Soc. Transl., 2, 78, 103-118 (1968) · Zbl 0183.27302
[38] Walczak, B.; Massart, D. L., Tutorial Rough sets theory, Chemometr. Intell. Lab. Syst., 47, 1-16 (1999)
[39] Wiweger, A., On topological rough sets, Bull. Pol. Acad. Math., 37, 89-93 (1989) · Zbl 0755.04010
[40] Wybraniec-Skardowska, U., On a generalization of approximation space, Bull. Pol. Acad. Sci., 37, 51-61 (1989) · Zbl 0755.04011
[41] Yao, Y. Y., Three-way decisions with probabilistic rough sets, Inform. Sci., 180, 3, 341-353 (2010)
[42] Yao, Y. Y.; Lin, T. Y., Generalization of rough sets using modal logic, Intell. Automat. Soft Comput., 2, 103-120 (1996)
[43] Yao, Y. Y., Two views of the theory of rough sets in finite Universes, Int. J. Approx. Reason., 15, 291-317 (1997) · Zbl 0935.03063
[44] Yao, Y. Y., Generalized rough set models, (Rough Sets in Knowledge Discovery (1998), physica-verlag: physica-verlag Hidelberg), 286-318 · Zbl 0946.68137
[45] Zhu, W., Topological approaches to covering rough sets, Inform. Sci., 177, 1499-1508 (2007) · Zbl 1109.68121
[46] Zhang, H. P.; Ouyang, Y.; Wangc, Z., Note on Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 179, 471-473 (2009) · Zbl 1159.03328
[47] Zhu, W., Relationship between generalized rough sets based on binary relation and covering, Inform. Sci., 179, 210-225 (2009) · Zbl 1163.68339
[48] Zhu, W., Generalized rough sets based on relations, Inform. Sci., 177, 4997-5011 (2007) · Zbl 1129.68088
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.