Covering based rough set approximations.

*(English)*Zbl 1248.68496Summary: We propose a framework for the study of covering based rough set approximations. Three equivalent formulations of the classical rough sets are examined by using equivalence relations, partitions, and \(\sigma \)-algebras, respectively. They suggest the element based, the granule based and the subsystem based definitions of approximation operators. Covering based rough sets are systematically investigated by generalizing these formulations and definitions. A covering of universe of objects is used to generate different neighborhood operators, neighborhood systems, coverings, and subsystems of the power set of the universe. They are in turn used to define different types of generalized approximation operators. Within the proposed framework, we review and discuss covering based approximation operators according to the element, granule, and subsystem based definitions.

##### MSC:

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

PDF
BibTeX
XML
Cite

\textit{Y. Yao} and \textit{B. Yao}, Inf. Sci. 200, 91--107 (2012; Zbl 1248.68496)

Full Text:
DOI

##### References:

[1] | Abo-Tabl, E.A., A comparison of two kinds of definitions of rough approximations based on a similarity relation, Information sciences, 181, 2587-2596, (2011) · Zbl 1216.68289 |

[2] | Abu-Donia, H.M., Comparison between different kinds of approximations by using a family of binary relations, Knowledge-based systems, 21, 911-919, (2008) |

[3] | Bartol, W.; Miró, J.; Pióro, K.; Rosselló, F., On the coverings by tolerance classes, Information sciences, 166, 193-211, (2004) · Zbl 1101.68863 |

[4] | D. Bianucci, G. Cattaneo, Information entropy and granulation co-entropy of partitions and coverings: a summary, in: LNCS Transactions on Rough Sets X, LNCS 5656, 2009, pp. 15-66. · Zbl 1248.94038 |

[5] | Bianucci, D.; Cattaneo, G.; Ciucci, D., Entropies and co-entropies of coverings with application to incomplete information systems, Fundamenta informaticae, 75, 77-105, (2007) · Zbl 1108.68112 |

[6] | Bonikowski, Z.; Bryniarski, E.; Wybraniec-Skardowska, U., Extensions and intensions in the rough set theory, Information sciences, 107, 149-167, (1998) · Zbl 0934.03069 |

[7] | Bryniarski, E., A calculus of rough sets of the first order, Bulletin of the Polish Academy of science, mathematics, 16, 71-78, (1989) · Zbl 0756.04002 |

[8] | Chen, D.G.; Wang, C.Z.; Hu, Q.H., A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets, Information sciences, 177, 3500-3518, (2007) · Zbl 1122.68131 |

[9] | D.J. Chen, Y. Liu, K.T. Wu, K.Y. Qin, New operators in covering rough set theory, in: Proceedings of the Third International Conference on Intelligent System and Knowledge Engineering, 2008, pp. 920-924. |

[10] | D. Ciucci, Temporal dynamics in rough sets based on coverings, in: Proceedings of the Fifth International Conference on Rough Sets and Knowledge Technology, LNCS(LNAI) 6401, 2010, pp. 126-133. |

[11] | Cohn, P.M., Universal algebra, (1965), Harper and Row Publishers New York · Zbl 0141.01002 |

[12] | Couso, I.; Dubois, D., Rough sets, coverings and incomplete information, Fundamenta informaticae, 108, 223-247, (2011) · Zbl 1279.68308 |

[13] | Deng, T.Q.; Chen, Y.M.; Xu, W.L.; Dai, Q.H., A novel approach to fuzzy rough sets based on a fuzzy covering, Information sciences, 177, 2308-2316, (2007) |

[14] | Diker, M.; Uğur, A.A., Textures and covering based rough sets, Information sciences, 184, 44-63, (2012) · Zbl 1239.03032 |

[15] | Du, Y.; Hu, Q.H.; Zhu, P.F.; Ma, P.J., Rule learning for classification based on neighborhood covering reduction, Information sciences, 181, 5457-5467, (2011) |

[16] | T. Feng, J.S. Mi, W.Z. Wu, Covering based generalized rough fuzzy sets, in: Proceedings of the First International Conference on Rough Set and Knowledge Technology, LNCS(LNAI) 4062, 2006, pp. 208-215. · Zbl 1196.03072 |

[17] | Feng, T.; Zhang, S.P.; Mi, J.S., The reduction and fusion of fuzzy covering systems based on the evidence theory, International journal of approximate reasoning, 53, 87-103, (2012) · Zbl 1242.68326 |

[18] | Feynman, R., The character of physical law, (2001), The MIT Press Cambridge, Massachusetts |

[19] | Ge, X.; Li, J.J.; Ge, Y., Some separations in covering approximation spaces, International journal of computational and mathematical sciences, 4, 156-160, (2010) |

[20] | J.W. Grzymala-Busse, Characteristic relations for incomplete date: a generalization of the indiecernibility relation, in: Transactions on Rough Sets IV, LNCS 3700, 2005, pp. 58-68. · Zbl 1136.68531 |

[21] | Leung, Y.; Li, D.Y., Maximal consistent block technique for rule acquisition in incomplete information systems, Information sciences, 153, 85-106, (2003) · Zbl 1069.68605 |

[22] | Li, F.; Yin, Y.Q., Approaches to knowledge reduction of covering decision systems based on information theory, Information sciences, 179, 1704-1794, (2009) · Zbl 1179.68193 |

[23] | Li, J.J., Topological methods on the theory of covering generalized rough sets (in Chinese), Pattern recognition and artificial intelligence, 17, 7-10, (2004) |

[24] | T.J. Li, Rough approximation operators in covering approximation spaces, in: Proceedings of the Fifth International Conference on Rough Sets and Current Trends in Computing, LNCS(LNAI) 4259, 2006, pp. 174-182. · Zbl 1162.68693 |

[25] | Li, T.J.; Leung, Y.; Zhang, W.X., Generalized fuzzy rough approximation operators based on fuzzy coverings, International journal of approximate reasoning, 48, 836-856, (2008) · Zbl 1186.68464 |

[26] | Li, T.J.; Wu, W.Z., Attribute reduction in formal contexts: a covering rough set approach, Fundamenta informaticae, 111, 15-32, (2011) · Zbl 1237.68214 |

[27] | X.A. Li, S.Y. Liu, Matroidal approaches to rough sets via closure operators, International Journal of Approximate Reasoning, doi:10.1016/j.ijar.2011.12.005. · Zbl 1246.68233 |

[28] | Y. Li, T. Feng, S.P. Zhang, Z.W. Li, A generalized model of covering rough sets and its application in medical diagnosis, in: Proceedings of 2010 International Conference on Machine Learning and Cybernetics, 2010, pp. 145-150. |

[29] | G.P. Lin, J.J., Li, A covering-based pessimistic multigranulation rough set, in: Proceedings of the Seventh International Conference on Intelligent Computing, LNBI 6840, 2012, pp. 673-680. |

[30] | C.H. Liu, D.Q. Miao, Covering rough set model based on multi-granulations, in: Proceedings of the Thirteenth International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, LNCS(LNAI) 6743, 2011, pp. 87-90. |

[31] | Liu, G.L.; Sai, Y., A comparison of two types of rough sets induced by coverings, International journal of approximate reasoning, 50, 521-528, (2009) · Zbl 1191.68689 |

[32] | Mohanty, D., Rough set on generalized covering approximation spaces, International journal of computer science and research, 1, 43-49, (2010) |

[33] | Mordeson, J.N., Rough set theory applied to (fuzzy) ideal theory, Fuzzy sets and systems, 121, 315-324, (2001) · Zbl 1030.68085 |

[34] | Orłowska, E., Semantics of nondeterministic possible worlds, Bulletin of the Polish Academy of sciences, mathematics, 33, 453-458, (1985) · Zbl 0584.03012 |

[35] | Orłowska, E., Semantics analysis of inductive reasoning, Theoretical computer science, 43, 81-89, (1986) · Zbl 0601.68059 |

[36] | Pawlak, Z., Rough sets, International journal of computer and information sciences, 11, 341-356, (1982) · Zbl 0501.68053 |

[37] | Pawlak, Z., Rough set, Theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers Boston · Zbl 0758.68054 |

[38] | Pomykała, J.A., Approximation operators in approximation space, Bulletin of the Polish Academy of science, mathematics, 35, 653-662, (1987) · Zbl 0642.54002 |

[39] | Pomykała, J.A., On definability in the nondeterministic information system, Bulletin of the Polish Academy of science, mathematics, 36, 193-210, (1988) · Zbl 0677.68110 |

[40] | K.Y. Qin, Y. Gao, Z. Pei, On covering rough sets, in: Proceedings of the Second International Conference on Rough Sets and Knowledge Technology, LNCS(LNAI) 4481, 2007, pp. 34-41. |

[41] | P. Samanta, M.K. Chakraborty, Covering based approaches to rough sets and implication lattices, in: Proceedings of the Twelfth International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, LNCS(LNAI) 5908, 2009, pp. 127-134. |

[42] | Shi, Z.H.; Gong, Z.T., The further investigation of covering-based rough sets: uncertainty characterization, similarity measure and generalized models, Information sciences, 180, 3745-3763, (2010) · Zbl 1205.68430 |

[43] | W. Sierpiński, General Topology (C. Krieger, Trans.), University of Toronto, Toronto, 1956. |

[44] | D. Ślęzak, P. Wasilewski, Granular sets -foundations and case study of tolerance spaces, in: Proceedings of the Eleventh International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, LNCS(LNAI) 4482, 2007, pp. 435-442. · Zbl 1198.68245 |

[45] | Słowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE transactions on knowledge and data engineering, 12, 331-336, (2000) |

[46] | E.C.C Tsang, D.G. Chen, J.W.T Lee, D.S. Yeung, On the upper approximations of covering generalized rough sets, in: Proceedings of the Third International Conference on Machine Learning and Cybernetics, 2004, pp. 4200-4203. |

[47] | Wang, J.; Dai, D.; Zhou, Z., Fuzzy covering generalized rough sets (in Chinese), Journal of zhoukou teachers college, 21, 2, 20-22, (2004) |

[48] | S.P. Wang, F. Min, W. Zhu, Covering numbers in covering-based rough sets, in: Proceedings of the Thirteenth International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, LNCS(LNAI) 6743, 2011, pp. 72-78. |

[49] | S.P. Wang, W. Zhu, F. Min, Transversal and function matroidal structures of covering-based rough sets, in: Proceedings of the Sixth International Conference on Rough Sets and Knowledge Technology, LNCS(LNAI) 6954, 2011, pp. 146-155. |

[50] | S.P. Wang, W. Zhu, P.Y. Zhu, Poset approaches to covering-based rough sets, in: Proceedings of the Fifth International Conference on Rough Sets and Knowledge Technology, LNCS(LNAI) 6401, 2010, pp. 25-29. |

[51] | Wasilewska, A., Conditional knowledge representation systems – model for an implementation, Bulletin of the Polish Academy of sciences, mathematics, 37, 63-69, (1987) · Zbl 0753.68088 |

[52] | P. Wasilewski, D. Ślęzak, Foundations of rough sets from vagueness perspective, in: A.E. Hassanien, Z. Suraj, D. Ślęzak, P. Lingras (Eds.), Rough Computing: Theory, Techniques, and Applications, Information Science Reference, Hershey, 2008, pp. 1-37. |

[53] | M.F. Wu, X.W. Wu, C.G. Shen, A new type of covering approximation operators, in: Proceedings of International Conference on Electronic Computer Technology, 2009, pp. 334-338. |

[54] | Wu, W.Z.; Zhang, W.X., Neighborhood operator systems and approximations, Information sciences, 144, 201-207, (2002) |

[55] | Wybraniec-Skardowska, U., On a generalization of approximation space, Bulletin of the Polish Academy of science, mathematics, 37, 51-61, (1989) · Zbl 0755.04011 |

[56] | Xu, W.H.; Zhang, W.X., Measuring roughness of generalized rough sets induced by a covering, Fuzzy sets and systems, 158, 2443-2455, (2007) · Zbl 1127.68106 |

[57] | Xu, Z.; Wang, Q., On the properties of covering rough sets model (in Chinese), Journal of henan normal university, 33, 1, 130-132, (2005) · Zbl 1091.03509 |

[58] | Yang, T.; Li, Q.G., Reduction about approximation spaces of covering generalized rough sets, International journal of approximate reasoning, 51, 335-345, (2010) · Zbl 1205.68433 |

[59] | Yao, Y.Y., Two views of the theory of rough sets in finite universes, International journal of approximate reasoning, 15, 291-317, (1996) · Zbl 0935.03063 |

[60] | Yao, Y.Y., Constructive and algebraic methods of the theory of rough sets, Information sciences, 109, 21-47, (1998) · Zbl 0934.03071 |

[61] | Yao, Y.Y., Relational interpretations of neighborhood operators and rough set approximation operators, Information sciences, 101, 239-259, (1998) · Zbl 0949.68144 |

[62] | Y.Y. Yao, On generalizing Pawlak approximation operators, in: Proceedings of the First International Conference on Rough Sets and Current Trends in Computing, LNCS(LNAI) 1424, 1998, pp. 298-307. · Zbl 0955.68505 |

[63] | Yao, Y.Y., Generalized rough set models, (), 286-318 · Zbl 0946.68137 |

[64] | Yao, Y.Y., Information granulation and rough set approximation, International journal of intelligent systems, 16, 87-104, (2001) · Zbl 0969.68079 |

[65] | Y.Y. Yao, On generalizing rough set theory, in: Proceedings of the Ninth International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, LNCS(LNAI) 2639, 2003, pp. 44-51. · Zbl 1026.68669 |

[66] | Yao, Y.Y., Neighborhood systems and approximation retrieval, Information sciences, 176, 3431-3452, (2006) · Zbl 1119.68074 |

[67] | Y.Y. Yao, A note on definability and approximations, in: LNCS Transactions on Rough Sets VII, LNCS 4400, 2007, pp. 274-282. · Zbl 1187.68617 |

[68] | Y.Y. Yao, Y.H. Chen, Subsystem based generalizations of rough set approximations, in: Proceedings of the Fifteenth International Symposium on Foundations of Intelligent Systems, LNCS(LNAI) 3488, 2005, pp. 210-218. · Zbl 1132.68760 |

[69] | Yao, Y.Y.; Lin, T.Y., Generalization of rough sets using modal logics, Intelligent automation and computing, 2, 103-120, (1996) |

[70] | Yun, Z.Q.; Ge, X.; Bai, X.L., Axiomatization and conditions for neighborhoods in a covering to form a partition, Information sciences, 181, 1735-1740, (2011) · Zbl 1216.68299 |

[71] | Żakowski, W., Approximations in the space (U, π), Demonstratio Mathematica, IXV, 761-769, (1983) · Zbl 0553.04002 |

[72] | Zhang, Y.L.; Luo, M.K., On minimization of axiom sets characterizing covering-based approximation operators, Information sciences, 181, 3032-3042, (2011) · Zbl 1216.68300 |

[73] | Zhang, Y.L.; Li, J.J.; Wu, W.Z., On axiomatic characterizations of three pairs of covering based three approximation operators, Information sciences, 180, 274-287, (2010) · Zbl 1186.68470 |

[74] | X.W. Zheng, J.H. Dai, A variable precision covering generalized rough set model, in: Proceedings of the Sixth International Conference on Rough Sets and Knowledge Technology, LNCS(LNAI) 6954, 2011, pp. 120-125. |

[75] | Zhu, P., Covering rough sets based on neighborhoods: an approach without using neighborhoods, International journal of approximate reasoning, 52, 461-472, (2011) · Zbl 1229.03047 |

[76] | P. Zhu, Q.Y. Wen, Entropy and co-entropy of a covering approximation space, International Journal of Approximate Reasoning, doi:10.1016/j.ijar.2011.12.004. · Zbl 1246.68237 |

[77] | W. Zhu, Properties of the second type of covering-based rough sets, in: Proceedings of the 2006 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology (WI-IAT 2006 Workshops), 2006, pp. 494-497. |

[78] | W. Zhu, Properties of the fourth type of covering-based rough sets, in: Proceedings of the Sixth International Conference on Hybrid Intelligent Systems, 2006, p. 43. |

[79] | Zhu, W., Topological approaches to covering rough sets, Information sciences, 177, 1499-1508, (2007) · Zbl 1109.68121 |

[80] | Zhu, W., Relationship between generalized rough sets and based on binary relation and covering rough sets, Information sciences, 179, 210-225, (2009) · Zbl 1163.68339 |

[81] | Zhu, W., Relationship among basic concepts in covering-based rough sets, Information sciences, 179, 2478-2486, (2009) · Zbl 1178.68579 |

[82] | Zhu, W.; Wang, F.Y., Some results on covering generalized rough sets (in Chinese), Pattern recognition and artificial intelligence, 15, 6-13, (2002) |

[83] | Zhu, W.; Wang, F.Y., Reduction and axiomization of covering generalized rough sets, Information sciences, 152, 217-230, (2003) · Zbl 1069.68613 |

[84] | W. Zhu, F.Y. Wang, Properties of the first type of covering-based rough sets, in: Proceedings of the Sixth IEEE International Conference on Data Mining-Workshops (ICDMW’06), 2006, pp. 407-411. |

[85] | W. Zhu, F.Y. Wang, A new type of covering rough sets, in: Proceedings of the Third International IEEE Conference on Intelligent Systems, 2006, pp. 444-449. |

[86] | W. Zhu, F.Y. Wang, Relationships among three types of covering rough sets, in: Proceedings of the 2006 IEEE International Conference on Granular Computing, 2006, pp. 43-48. |

[87] | Zhu, W.; Wang, F.Y., On three types of covering rough sets, IEEE transactions on knowledge and data engineering, 19, 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.