On an Open Problem for Definable Subsets of Covering Approximation Spaces

Authors: Mei He, Ying Ge, Jingyu Qian

Abstract:

Let (U;D) be a Gr-covering approximation space (U; C) with covering lower approximation operator D and covering upper approximation operator D. For a subset X of U, this paper investigates the following three conditions: (1) X is a definable subset of (U;D); (2) X is an inner definable subset of (U;D); (3) X is an outer definable subset of (U;D). It is proved that if one of the above three conditions holds, then the others hold. These results give a positive answer of an open problem for definable subsets of covering approximation spaces.

Keywords: Covering approximation space, covering approximation operator, definable subset, inner definable subset, outer definable subset.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333078

References:

[1] Z. Bonikowski, E. Bryniarski and U. Wybraniec, Extensions and intentions in the rough set theory, Information Sciences, 107(1998), 149-167.
[2] X. Ge, Further investigations on higher mathematics scores for Chinese university students, International Journal of Computational and Mathematical Sciences, 3(2009), 181-185.
[3] X. Ge, An application of covering approximation spaces on network security, Computers and Mathematics with Applications, 60(2010), 1191- 1199.
[4] X. Ge, J. Li and Y. Ge, Some separations in covering approximation, International Journal of Computational and Mathematical Sciences, 4(2010), 156-160.
[5] X. Ge and Z. Li, Definable subsets in covering approximation spaces, International Journal of Computational and Mathematical Sciences, 5(2011), 31-34.
[6] X. Ge and J. Qian, Some investigations on higher mathematics scores for Chinese university students, International Journal of Computer and Information Engineering, 3(2009), 209-212.
[7] E. Lashin, A. Kozae, A. Khadra and T. Medhat, Rough set theory for topological spaces, International Journal of Approximate Reasoning, 40(2005), 35-43.
[8] Z. Pawlak, Rough sets, International journal of computer and information science, 11(1982), 341-356.
[9] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, 1991.
[10] Z. Pawlak, Rough classification, International journal of Humancomputer Studies, 51(1999), 369-383.
[11] Z. Pawlak and A. Skowron, Rudiments of rough sets, Information Sciences, 177(2007), 3-27.
[12] Z. Pawlak and A. Skowron, Rough sets: Some extensions, Information Sciences, 177(2007), 28-40.
[13] Z. Pawlak and A. Skowron, Rough sets and Boolean reasoning, Information Sciences, 177(2007), 41-73.
[14] D. Pei, On definable concepts of rough set models, Information Sciences, 177(2007), 4230-4239.
[15] J. A. Pomykala, Approximation operations in approximation spaces, Bull. Pol. Acad. Sci., 35(1987), 653-662.
[16] K. Qin, Y. Gao and Z. Pei, On covering rough sets, Lecture Notes in Artificial Intelligence, 4481(2007), 34-41.
[17] P. Samanta and M. Chakraborty, Covering based approaches to rough sets and implication lattices, Lecture Notes in Artificial Intelligence, 5908(2009), 127-134.
[18] Y. Yao, Views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning, 15(1996), 291-317.
[19] Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111(1998), 239-259.
[20] Y. Yao, On generalizing rough set theory, Lecture Notes in AI, 2639(2003), 44-51.
[21] Z. Yun, Xun Ge and X. Bai, Axiomatization and conditions for neighborhoods in a covering to form a partition, Information Sciences, (2011), doi:10.1016/j.ins.2011.01.013.
[22] W. Zakowski, Approximations in the Space (u, Π), Demonstratio Mathematica, 16(1983), 761-769.
[23] W. Zhu, Topological approaches to covering rough sets, Information Sciences, 177(2007), 1499-1508.
[24] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Information Sciences, 179(2009), 210-225.
[25] W. Zhu and F. Wang, Reduction and axiomatization of covering generalized rough sets, Information Sciences, 152(2003), 217-230.
[26] W. Zhu and F. Wang, On Three Types of Covering Rough Sets, IEEE Transactions on Knowledge and Data Engineering, 19(2007), 1131-1144.