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A Note on Definability and Approximations

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Transactions on Rough Sets VII

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 4400))

Abstract

Definability and approximations are two important notions of the theory of rough sets. In many studies, one is used to define the other. There is a lack of an explicit interpretation of the physical meaning of definability. In this paper, the definability is used as a more primitive notion, interpreted in terms of formulas of a logic language. A set is definable if there is a formula that defines the set, i.e., the set consists of all those elements satisfying the formula. As a derived notion, the lower and upper approximations of a set are two definable sets that approximate the set from below and above, respectively. This formulation may be more natural, bringing new insights into our understanding of rough set approximations.

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References

  1. Banerjee, M., Chakraborty, M.K.: Algebras from rough sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing, pp. 157–184. Springer, Heidelberg (2004)

    Google Scholar 

  2. Buszkowski, W.: Approximation spaces and definability for incomplete information systems. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 115–122. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  3. Grzymala-Busse, J.W.: Incomplete data and generalization of indiscernibility relation, definability, and approximations. In: Ślęzak, D., et al. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 244–253. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  4. Hobbs, J.R.: Granularity. In: Proceedings of the 9th International Joint Conference on Artificial Intelligence, pp. 432–435 (1985)

    Google Scholar 

  5. Järvinen, J., Kortelainen, J.: A note on definability in rough set theory. In: De Baets, B., et al. (eds.) Current Issues in Data and Knowledge Engineering, pp. 272–277. Akademicka Oficyna Wydawnicza EXIT, Warsaw (2004)

    Google Scholar 

  6. Kreisel, G., Krivine, J.L.: Elements of Mathematical Logic. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  7. Kunen, K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  8. Mi, J.S., Zhang, W.X.: An axiomatic characterization of a fuzzy generalization of rough sets. Information Sciences 160, 235–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Orlowska, E.: Logical aspects of learning concepts. International Journal of Approximate Reasoning 2, 349–364 (1988)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Pawlak, Z., et al.: Rough sets. Communications of the ACM 38, 89–95 (1995)

    Article  Google Scholar 

  12. Pawlak, Z., Skowron, A.: Rudiments of rough sets. Information Sciences 177, 3–27 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning. Information Sciences 177, 41–73 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pomykala, J.A.: On definability in the nondeterministic information system. Bulletin of the Polish Academy of Sciences: Mathematics 36, 193–210 (1987)

    MathSciNet  Google Scholar 

  16. Smith, E.E.: Concepts and induction. In: Posner, M.I. (ed.) Foundations of Cognitive Science, pp. 501–526. The MIT Press, Cambridge (1989)

    Google Scholar 

  17. Sowa, J.F.: Conceptual Structures, Information Processing in Mind and Machine. Addison-Wesley, Reading (1984)

    MATH  Google Scholar 

  18. Van Mechelen, I., et al. (eds.): Categories and Concepts, Theoretical Views and Inductive Data Analysis. Academic Press, New York (1993)

    Google Scholar 

  19. Wasilewska, A.: Definable ses in knowledge representation systems. Bulletin of the Polish Academy of Sciences: Mathematics 35, 629–635 (1987)

    MATH  MathSciNet  Google Scholar 

  20. Wu, W.Z., Zhang, W.X.: Constructive and axiomatic approaches of fuzzy approximation operators. Information Sciences 159, 233–254 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yao, J.T., Yao, Y.Y., Zhao, Y.: Foundations of classification. In: Lin, T.Y., et al. (eds.) Foundations and Novel Approaches in Data Mining, pp. 75–97. Springer, Berlin (2006)

    Google Scholar 

  22. Yao, Y.Y.: A comparative study of formal concept analysis and rough set theory in data analysis. In: Tsumoto, S., et al. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 59–68. Springer, Heidelberg (2004)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111, 239–259 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zhu, W.: Topological approaches to covering rough sets. Information Sciences, in press

    Google Scholar 

  26. Zhu, W., Wang, F.Y.: Reduction and axiomization of covering generalized rough sets. Information Sciences 152, 217–230 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

    Yiyu Yao

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  1. Yiyu Yao
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Editor information

James F. Peters Andrzej Skowron Victor W. Marek Ewa Orłowska Roman Słowiński Wojciech Ziarko

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Yao, Y. (2007). A Note on Definability and Approximations. In: Peters, J.F., Skowron, A., Marek, V.W., Orłowska, E., Słowiński, R., Ziarko, W. (eds) Transactions on Rough Sets VII. Lecture Notes in Computer Science, vol 4400. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71663-1_17

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  • DOI: https://doi.org/10.1007/978-3-540-71663-1_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-71662-4

  • Online ISBN: 978-3-540-71663-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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