Abstract
This paper consists of an extensive survey of various generalized approaches to the lower and upper approximations of a set, the two approximations being first defined by Pawlak while introducing rough set theory. Particularly, relational, covering based and operator based approaches are considered. Categorization of various approaches in terms of implication lattices is shown. Significance of this categorization in rough logics is briefly mentioned.
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References
Banerjee, M., Chakraborty, M.K.: Rough Sets Through Algebraic Logic. Fundam. Inform. 28(3-4), 211–221 (1996)
Benerjee, M., Chakraborty, M.K.: Algebras from Rough Sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing - Techniques for Computing with Words, Springer, Heidelberg (2004)
Banerjee, M., Yao, Y.: A Categorial Basis for Granular Computing. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 427–434. Springer, Heidelberg (2007)
Bonikowski, Z.: A Certain Copnception of the Calculus of Rough Sets. Notre Dame J. Formal Logic 33, 412–421 (1992)
Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intentions in the rough set theory. Journal of Information Sciences 107, 149–167 (1998)
Bunder, M.W., Banerjee, M., Chakraborty, M.K.: Some Rough Consequence Logics and their Interrelations. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets VIII. LNCS, vol. 5084, pp. 1–20. Springer, Heidelberg (2008)
Grzymała-Busse, J.W., Grzymala-Busse, W.J.: An Experimental Comparison of Three Rough Set Approaches to Missing Attribute Values. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 31–50. Springer, Heidelberg (2007)
Cattaneo, G.: Foundational and Mathematical Investigation of Roughness Theory (preprint)
Cattaneo, G., Ciucci, D.: Lattices with interior and closure operators and abstract approximation spaces. In: Peters, J.F., Skowron, A., Wolski, M., Chakraborty, M.K., Wu, W.-Z. (eds.) Transactions on Rough Sets X. LNCS, vol. 5656, pp. 67–116. Springer, Heidelberg (2009)
Chakraborty, M.K., Banerjee, M.: Rough dialogue and implication lattices. Fundamenta Informaticae 75(1-4), 123–139 (2007)
Chakraborty, M.K., Samanta, P.: Consistency-Degree Between Knowledges. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 133–141. Springer, Heidelberg (2007)
Jarinen, J.: Approximations and Rough Sets Based on Tolerances. In: Ziarko, W.P., Yao, Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, pp. 182–189. Springer, Heidelberg (2001)
Li, T.-J.: Rough Approximation Operators in Covering Approximation Spaces. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 174–182. Springer, Heidelberg (2006)
Lin, T.Y.: Granular Computing on Binary Relations I: Data Mining and Neighbourhood Systems. In: Skowron, A., Polkowski, L. (eds.) Rough Sets in Knowledge Discovery, pp. 107–121. Physica-Verlag, Berlin (1998)
Lin, T.Y.: Granular Computing on Binary Relations II: Rough Set Representations on Belief Functions. In: Skowron, A., Polkowski, L. (eds.) Rough Sets in Knowledge Discovery, pp. 122–140. Physica-Verlag, Berlin (1998)
Lin, T.Y.: What is Granular Computing. Journal of Latex Class Files 1(11) (November 2002)
Liu, J., Liao, Z.: The sixth type of covering-based rough sets. In: IEEE International Conference on Granular Computing, GrC 2008, pp. 438–441 (2008)
Orlowska, E. (ed.): Incomplete Information: Rough Set Analysis. Physcia-Verlag, Heidelberg (1997)
Pal, S.K., Polkowski, L., Skowron, A. (eds.): Rough-Neural Computing. Springer, Heidelberg (2004)
Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences 11(5) (1982)
Pawlak, Z.: ROUGH SETS - Theoritical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordrecht (1991)
Pagliani, P.: Pre-topologies and Dynamic Spaces. In: Wang, G., Liu, Q., Yao, Y., Skowron, A. (eds.) RSFDGrC 2003. LNCS (LNAI), vol. 2639, pp. 146–155. Springer, Heidelberg (2003); Extended Version in Fundamenta Informatica 59(2-3), 221–239 (2004)
Pagliani, P., Chakraborty, M.K.: A geometry of Approximation - Rough Set theory: Logic, Algebra and Topology of Conceptual Patterns. Springer Science + Business Media B.V (2008)
Pomykala, J.A.: Approximation operations in approximation space. Bulletin of the Polish Academy of Sciences: Mathematics 35, 653–662 (1987)
Pomykala, J.A.: Approximation, Similarity and Rough Constructions. ILLC Prepublication Series for Computation and Complexity Theory CT-93-07, University of Amsterdam (1993)
Pomykala, J., Pomykala, J.A.: The Stone Algebra of Rough Sets. Bull. Polish Acad. Sci. Math. 36(7-8), 495–508 (1988)
Qin, K., Gao, Y., Pei, Z.: On Covering Rough Sets. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślęzak, D. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 34–41. Springer, Heidelberg (2007)
Samanta, P., Chakraborty, M.K.: On Extension of Dependency and Consistency Degrees of Two Knowledges Represented by Covering. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 351–364. Springer, Heidelberg (2008)
Samanta, P., Chakraborty, M.K.: Covering Based Approaches to Rough Sets and Implication Lattices. In: Sakai, H., Chakraborty, M.K., Hassanien, A.E., Ślęzak, D., Zhu, W. (eds.) RSFDGrC 2009. LNCS (LNAI), vol. 5908, pp. 127–134. Springer, Heidelberg (2009)
Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)
Slezak, D., Wasilewski, P.: Granular Sets – Foundations and Case Study of Tolerance Spaces. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 435–442. Springer, Heidelberg (2007)
Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Journal of Information Sciences 109, 21–47 (1998)
Yao, Y.Y.: On generalizing rough set theory. In: Wang, G., Liu, Q., Yao, Y., Skowron, A. (eds.) RSFDGrC 2003. LNCS (LNAI), vol. 2639, pp. 44–51. Springer, Heidelberg (2003)
Yun, Z., Ge, X: Some notes on covering-based rough sets (preprint)
Zhu, W.: Topological approaches to covering rough sets. ScienceDirect, Information Sciences 177, 1499–1508 (2007)
Zhu, W.: Relationship between generalized rough sets based on binary relation and covering. Information Sciences 179, 210–225 (2009)
Zhu, W.: 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)(WI-IATW 2006) (2006)
Zhu, W., Wang, F.-Y.: Relationship among Three Types of Covering Rough Sets. In: Proc. IEEE Int’l Conf. Grannuler Computing (GrC 2006), May 2006, pp. 43–48 (2006)
Zhu, W., Wang, F.-Y.: Properties of the First Type of Covering-Based Rough Sets. In: Sixth IEEE International Conference on Data Mining - Workshops, ICDMW 2006 (2006)
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Samanta, P., Chakraborty, M.K. (2011). Generalized Rough Sets and Implication Lattices. In: Peters, J.F., et al. Transactions on Rough Sets XIV. Lecture Notes in Computer Science, vol 6600. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21563-6_10
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DOI: https://doi.org/10.1007/978-3-642-21563-6_10
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