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Generalized Rough Sets and Implication Lattices

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Transactions on Rough Sets XIV

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 6600))

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

  1. Banerjee, M., Chakraborty, M.K.: Rough Sets Through Algebraic Logic. Fundam. Inform. 28(3-4), 211–221 (1996)

    MathSciNet  MATH  Google Scholar 

  2. 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)

    Google Scholar 

  3. 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)

    Chapter  Google Scholar 

  4. Bonikowski, Z.: A Certain Copnception of the Calculus of Rough Sets. Notre Dame J. Formal Logic 33, 412–421 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intentions in the rough set theory. Journal of Information Sciences 107, 149–167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. 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)

    Chapter  Google Scholar 

  8. Cattaneo, G.: Foundational and Mathematical Investigation of Roughness Theory (preprint)

    Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. Chakraborty, M.K., Banerjee, M.: Rough dialogue and implication lattices. Fundamenta Informaticae 75(1-4), 123–139 (2007)

    MathSciNet  MATH  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. 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)

    Chapter  Google Scholar 

  13. 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)

    Chapter  Google Scholar 

  14. 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)

    Google Scholar 

  15. 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)

    Google Scholar 

  16. Lin, T.Y.: What is Granular Computing. Journal of Latex Class Files 1(11) (November 2002)

    Google Scholar 

  17. 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)

    Google Scholar 

  18. Orlowska, E. (ed.): Incomplete Information: Rough Set Analysis. Physcia-Verlag, Heidelberg (1997)

    MATH  Google Scholar 

  19. Pal, S.K., Polkowski, L., Skowron, A. (eds.): Rough-Neural Computing. Springer, Heidelberg (2004)

    Book  MATH  Google Scholar 

  20. Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences 11(5) (1982)

    Google Scholar 

  21. Pawlak, Z.: ROUGH SETS - Theoritical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    Google Scholar 

  24. Pomykala, J.A.: Approximation operations in approximation space. Bulletin of the Polish Academy of Sciences: Mathematics 35, 653–662 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Pomykala, J.A.: Approximation, Similarity and Rough Constructions. ILLC Prepublication Series for Computation and Complexity Theory CT-93-07, University of Amsterdam (1993)

    Google Scholar 

  26. Pomykala, J., Pomykala, J.A.: The Stone Algebra of Rough Sets. Bull. Polish Acad. Sci. Math. 36(7-8), 495–508 (1988)

    MathSciNet  MATH  Google Scholar 

  27. 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)

    Chapter  Google Scholar 

  28. 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)

    Chapter  Google Scholar 

  29. 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)

    Chapter  Google Scholar 

  30. Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)

    MathSciNet  MATH  Google Scholar 

  31. 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)

    Chapter  Google Scholar 

  32. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Journal of Information Sciences 109, 21–47 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Chapter  Google Scholar 

  34. Yun, Z., Ge, X: Some notes on covering-based rough sets (preprint)

    Google Scholar 

  35. Zhu, W.: Topological approaches to covering rough sets. ScienceDirect, Information Sciences 177, 1499–1508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhu, W.: Relationship between generalized rough sets based on binary relation and covering. Information Sciences 179, 210–225 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Google Scholar 

  38. 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)

    Google Scholar 

  39. 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)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Katwa College, Katwa, Burdwan, West Bengal, India

    Pulak Samanta

  2. Center for Soft Computing Research, Indian Statistical Institute, Kolkata, India

    Mihir Kumar Chakraborty

  3. Centre for Cognitive Science, Jadavpur University, Kolkata, India

    Mihir Kumar Chakraborty

Authors
  1. Pulak Samanta
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  2. Mihir Kumar Chakraborty
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Editor information

Editors and Affiliations

  1. University of Manitoba, R3T 5V6, Winnipeg, MB, Canada

    James F. Peters

  2. University of Warsaw, 02-097, Warsaw, Poland

    Andrzej Skowron  & Dominik Slezak  & 

  3. Kyushu Institute of Technology, 804, Tobata, Kitakyushu, Japan

    Hiroshi Sakai

  4. Jadavpur University, Kolkata, Indian Statistical Institute, Kolkata, India

    Mihir Kumar Chakraborty

  5. Cairo University, Orman, Giza, Egypt

    Aboul Ella Hassanien

  6. Zhangzhou Normal University, Fujian, China

    William Zhu

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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

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21562-9

  • Online ISBN: 978-3-642-21563-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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