Abstract
Theory of covering rough sets is one kind of effective methods for knowledge discovery. In Bonikowski covering approximation spaces, all definable sets on the universe form a knowledge space. This paper focuses on the theoretic study of knowledge spaces of covering approximation spaces. One kind of dependence relations among covering approximation spaces is introduced, the relationship between the dependence relation and lower and upper covering approximation operators are discussed in detail, and knowledge spaces of covering approximation spaces are well characterized by them. By exploring the dependence relation between a covering approximation space and its sub-spaces, the notion of the reduction of covering approximation spaces is induced, and the properties of the reductions are investigated.
References
Bonikowski, Z., Bryniarski, E., Wybraniec, U.: Extensions and intentions in the rough set theory. Inf. Sci. 107, 149–167 (1998)
Caspard, N., Monjardet, B.: The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey. Discrete Appl. Math. 127, 241–269 (2003)
Chen, D., Wang, C., Hu, Q.: A new approach to attributes reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf. Sci. 177, 3500–3518 (2007)
Doignon, J.P., Falmagne, J.C.: Knowledge Spaces. Springer, Heidelberg (1999)
Kortelainen, J.: On the relationship between modified sets, topological spaces and rough sets. Fuzzy Sets Syst. 61, 91–95 (1994)
Liu, C., Miao, D.Q., Qian, J.: On multi-granulation covering rough sets. Int. J. Approx. Reason. 44, 1404–1418 (2015)
Li, T.J., Leung, Y., Zhang, W.X.: Generalized fuzzy rough approximation operators based on fuzzy coverings. Int. J. Approx. Reason. 48, 836–856 (2008)
Restrepo, M., Cornelis, C., Gomez, J.: Partial order relation for approximation operators in covering based rough sets. Inf. Sci. 284, 44–59 (2014)
Monjardet, B.: The presence of lattice theory in discrete problems of mathematical social sciences. Why. Math. Soc. Sci. 46, 103–144 (2003)
Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)
Qian, Y.H., Liang, J.Y., Pedrycz, W., Dang, C.Y.: Positive approximation: an accelerator for attribute reduction in rough set theory. Artif. Intell. 174, 597–618 (2010)
Slowinski, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. Knowl. Data Eng. 12, 331–336 (2000)
Wang, L., Yang, X., Yang, J.: Relationships among generalized rough sets in six coverings and pure reflexive neighborhood system. Inf. Sci. 207, 66–78 (2012)
Xu, W.H., Zhang, W.X.: Measuring roughness of generalized rough sets induced by a covering. Fuzzy Sets Syst. 158, 2443–2455 (2007)
Yang, T., Li, Q.: Reduction about approximation spaces of covering generalized rough sets. Int. J. Approx. Reason. 51, 335–345 (2010)
Yao, Y.Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16, 87–104 (2001)
Yao, Y.Y.: The superiority of three-way decisions in probabilistic rough set models. Inf. Sci. 181, 1080–1096 (2011)
Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)
Yue, X.D., Miao, D.Q., Zhang, N., Cao, L.B., Wu, Q.: Multiscale roughness measure for color image segmentation. Inf. Sci. 216, 93–112 (2012)
Zakowski, W.: Approximations in the space \((u,\prod )\). Demonstratio Mathematica 16, 761–769 (1983)
Zhang, J.B., Li, T.R., Chen, H.M.: Composite rough sets for dynamic data mining. Inf. Sci. 257, 81–100 (2014)
Zhu, W., Wang, F.-Y.: Covering based granular computing for conflict analysis. In: Mehrotra, S., Zeng, D.D., Chen, H., Thuraisingham, B., Wang, F.-Y. (eds.) ISI 2006. LNCS, vol. 3975, pp. 566–571. Springer, Heidelberg (2006)
Zhu, W., Wang, F.Y.: Reduction and axiomization of covering generalized rough sets. Inf. Sci. 152, 217–230 (2003)
Zhu, W.: Relationship between generalized rough sets based on binary relation and covering. Inf. Sci. 179, 210–225 (2009)
Zhu, W., Wang, F.Y.: The fourth type of covering-based rough sets. Inf. Sci. 201, 80–92 (2012)
Acknowledgements
This work was supported by grants from the National Natural Science Foundation of China (Nos. 11071284, 61075120, 61272021, 61202206) and the Zhejiang Provincial Natural Science Foundation of China (Nos. LY14F030001, LZ12F03002, LY12F02021).
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Li, TJ., Gu, SM., Wu, WZ. (2015). Knowledge Spaces and Reduction of Covering Approximation Spaces. In: Ciucci, D., Wang, G., Mitra, S., Wu, WZ. (eds) Rough Sets and Knowledge Technology. RSKT 2015. Lecture Notes in Computer Science(), vol 9436. Springer, Cham. https://doi.org/10.1007/978-3-319-25754-9_15
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