Skip to main content

A General Set Theoretic Approximation Framework

  • Conference paper
Advances on Computational Intelligence (IPMU 2012)

Abstract

To approximate sets a number of theories have been appeared for the last decades. Starting up from some general theoretical pre-conditions we give a set of minimum requirements against as the lower and upper approximations. We provide a characterization of them within the proposed general set theoretic approximation framework finding out their compound nature.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Banerjee, M., Chakraborty, M.: Algebras from rough sets. In: Pal, S., Polkowski, L., Skowron, A. (eds.) Rough-Neuro Computing: Techniques for Computing with Words, pp. 157–184. Springer, Berlin (2004)

    Chapter  Google Scholar 

  2. Bonikowski, Z., Bryniarski, E., Wybraniec-Skardowska, U.: Extensions and intensions in the ruogh set theory. Information Sciences 107(1-4), 149–167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1: Methodology and Applications. STUDFUZZ, pp. 59–98. Physica-Verlag, Heidelberg (1997)

    Google Scholar 

  4. Ciucci, D.: A Unifying Abstract Approach for Rough Models. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 371–378. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  5. Ciucci, D.: Approximation algebra and framework. Fundamenta Informaticae 94, 147–161 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Csajbók, Z.: Partial approximative set theory: A generalization of the rough set theory. In: Martin, T., Muda, A.K., Abraham, A., Prade, H., Laurent, A., Laurent, D., Sans, V. (eds.) Proceedings of SoCPaR 2010, Cergy Pontoise / Paris, France, December 7-10, pp. 51–56. IEEE (2010)

    Google Scholar 

  7. Csajbók, Z., Mihálydeák, T.: On the general set theoretical framework of set approximation. In: Proceedings of RST 2011, Milan, Italy, September 14-16, pp. 12–15 (2011)

    Google Scholar 

  8. Csajbók, Z., Mihálydeák, T.: Partial approximative set theory: A generalization of the rough set theory. International Journal of Computer Information Systems and Industrial Management Applications 4, 437–444 (2012)

    Google Scholar 

  9. Csajbók, Z.: Approximation of sets based on partial covering. Theoretical Computer Science 412(42), 5820–5833 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Düntsch, I., Gediga, G.: Approximation Operators in Qualitative Data Analysis. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) TARSKI 2003. LNCS, vol. 2929, pp. 214–230. Springer, Heidelberg (2003)

    Google Scholar 

  11. Greco, S., Matarazzo, B., Slowinski, R.: Rough approximation by dominance relations. International Journal of Intelligent Systems 17, 153–171 (2002)

    Article  MATH  Google Scholar 

  12. Greco, S., Matarazzo, B., Slowinski, R.: Rough sets theory for multicriteria decision analysis. European Journal of Operational Research 129, 1–47 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Inuiguchi, M.: Generalizations of Rough Sets and Rule Extraction. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B.z., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 96–119. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  14. Inuiguchi, M., Tanino, T.: Generalized Rough Sets and Rule Extraction. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 105–112. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11(5), 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  17. Pawlak, Z., Skowron, A.: Rough sets: Some extensions. Information Sciences 177, 28–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Polkowski, L.: Rough Sets: Mathematical Foundations. AISC. Physica-Verlag, A Springer-Verlag Company (2002)

    Google Scholar 

  19. Skowron, A., Stepaniuk, J., Swiniarski, R.: Approximation spaces in rough-granular computing. Fundamenta Informaticae 100(1–4), 141–157 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Skowron, A., Świniarski, R.W., Synak, P.: Approximation Spaces and Information Granulation. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III. LNCS, vol. 3400, pp. 175–189. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  21. Słowiński, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Transactions on Knowledge and Data Engineering 12, 331–336 (2000)

    Article  Google Scholar 

  22. Stepaniuk, J.: Rough–Granular Computing in Knowledge Discovery and Data Mining. SCI. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  23. Yao, Y.Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximation Reasoning 15(4), 291–317 (1996)

    Article  MATH  Google Scholar 

  24. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109(1–4), 21–47 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yao, Y.Y.: On Generalizing Pawlak Approximation Operators. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 298–307. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  26. 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 

  27. Zhu, W.: Topological approaches to covering rough sets. Information Sciences 177(6), 1499–1508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Csajbók, Z., Mihálydeák, T. (2012). A General Set Theoretic Approximation Framework. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances on Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31709-5_61

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31709-5_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31708-8

  • Online ISBN: 978-3-642-31709-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics