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Three-Valued Logic for Reasoning about Covering-Based Rough Sets

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Book cover Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 42))

Abstract

In the chapter we present a tool for reasoning about covering-based rough sets in the form of three-valued logic in which the value t corresponds to the positive region of a set, the value f — to the negative region and the undefined value u — to the boundary of a given set. Atomic formulas of the logic represent either membership of objects of the universe in rough sets or the subordination relation between objects generated by the covering underlying the approximation space, and complex formulas are built out of the atomic ones using three-valued Kleene connectives. We give a strongly sound sequent calculus for the logic defined in this way and prove its strong completeness for a subset of its language.

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References

  1. Avron, A.: Classical Gentzen-type methods in propositional many-valued logics. In: Fitting, M., Orłowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic. STUDFUZZ, vol. 114, pp. 117–155. Physica Verlag, Heidelberg (2003)

    Google Scholar 

  2. Avron, A.: Logical Non-Determinism as a Tool for Logical Modularity: An Introduction. In: Artemov, S., Barringer, H., d’Avila Garcez, A.S., Lamb, L.C., Woods, J. (eds.) We Will Show Them: Essays in Honor of Dov Gabbay, vol. 1, pp. 105–124. College Publications (2005)

    Google Scholar 

  3. Avron, A., Konikowska, B.: Multi-valued calculi for Logics based on Non-determinism. Logic Journal of the IGPL 13, 365–387 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Avron, A., Konikowska, B.: Rough sets and 3-valued Logics. Studia Logica 90(1), 69–92 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avron, A., Lev, I.: Canonical Propositional Gentzen-Type Systems. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 529–544. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  6. Avron, A., Lev, I.: Non-deterministic multiple-valued structures. Journal of Logic and Computation 15, 241–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balbiani, P., Vakarelov, D.: A modal Logic for Indiscernibility and Complementarity in Information Systems. Fundamenta Informaticae 45, 173–194 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Banjeeri, M.: Rough sets and 3-valued Lukasiewicz logic. Fundamenta Informaticae 32, 213–220 (1997)

    Google Scholar 

  9. Demri, S., Konikowska, B.: Relative Similarity Logics are Decidable: Reduction to FO2 with Equality. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 279–293. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  10. Demri, S., Orłowska, E., Vakarelov, D.: Indiscernibility and complementarity relations in information systems. In: Gerbrandy, J., Marx, M., de Rijke, M., Venema, Y. (eds.) JFAK: Esays Dedicated to Johan van Benthem on the Ocasion of his 50-th Birthday. Amsterdam University Press (1999)

    Google Scholar 

  11. Deneva, A., Vakarelov, D.: Modal logics for local and global similarity relations. Fundamenta Informaticae 31(3-4), 295–304 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Duentsch, I., Konikowska, B.: A multimodal logic for reasoning about complementarity. Journal for Applied Non-Classical Logics 10(3-4), 273–302 (2000)

    Article  MATH  Google Scholar 

  13. Iturrioz, L.: Rough Sets and Three-valued Structures. In: Orłowska, E. (ed.) Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa. STUDFUZZ, vol. 24, pp. 596–603. Physica-Verlag, Heidelberg (1999)

    Google Scholar 

  14. Kleene, S.C.: Introduction to Metamathematics, D. van Nostrad Co. (1952)

    Google Scholar 

  15. Konikowska, B.: A logic for reasoning about relative similarity. In: Orłowska, E., Rasiowa, H. (eds.) Special Issue of Studia Logica, Reasoning with Incomplete Information. Studia Logica, vol. 58, pp. 185–226 (1997)

    Google Scholar 

  16. Łukasiewicz, J.: On 3-valued Logic, 1920. In: McCall, S. (ed.) Polish Logic. Oxford University Press (1967)

    Google Scholar 

  17. Lin, T.Y., Cercone, N. (eds.): Rough sets and Data Mining. Analysis of Imprecise Data. Kluwer, Dordrecht (1997)

    MATH  Google Scholar 

  18. Øhrn, A., Komorowski, J., Skowron, A., Synak, P.: The design and implementation of a knowledge discovery toolkit based on rough sets — The ROSETTA system. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1. Methodology and Applications, pp. 376–399. Physica Verlag, Heidelberg (1998)

    Google Scholar 

  19. Øhrn, A., Komorowski, J., Skowron, A., Synak, P.: The ROSETTA software system. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 2. Applications, Case Studies and Software Systems, pp. 572–576. Physica Verlag, Heidelberg (1998)

    Google Scholar 

  20. Orłowska, E.: Reasoning with Incomplete Information: Rough Set Based Information Logics. In: Proceedings of SOFTEKS Workshop on Incompleteness and Uncertainty in Information Systems, pp. 16–33 (1993)

    Google Scholar 

  21. Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis. STUDFUZZ, vol. 13, pp. 109–190. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  22. Pawlak, Z.: Rough sets. Intern. J. Comp. Inform. Sci. 11, 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  24. Pawlak, Z.: Rough set approach to knowledge-based decision support. European Journal of Operational Research 29(3), 1–10 (1997)

    Google Scholar 

  25. Pawlak, Z.: Rough sets theory and its applications to data analysis. Cybernetics and Systems 29, 661–688 (1998)

    Article  MATH  Google Scholar 

  26. Pomykała, J.A.: Approximation operations in approximation space. Bull. Pol. Acad. Sci. 35(9-10), 653–662 (1987)

    MATH  Google Scholar 

  27. Sen, J., Chakraborty, M.K.: A study of intenconnections between rough and 3-valued Lukasiewicz logics. Fundamenta Informaticae 51, 311–324 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vakarelov, D.: Information systems, similarity relations and modal logics. In: Orlowska, E. (ed.) Incomplete Information: Rough Set Analysis. STUDFUZZ, pp. 492–550. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  29. Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111(1-4), 239–259 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Google Scholar 

  31. Zakowski, W.: On a concept of rough sets. Demonstratio Mathematica XV, 1129–1133 (1982)

    MathSciNet  Google Scholar 

  32. Zhang, Y.-L., Li, J.J., Wu, W.-Z.: On axiomatic characterizations of three types of covering-based approximation operators. Information Sciences 180, 174–187 (2010)

    MathSciNet  Google Scholar 

  33. Zhu, W., Wang, F.-Y.: On three types of covering-based rough sets. IEEE Transactions on Knowledge and Data Engineering 19(8), 1131–1144 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland

    Beata Konikowska

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  1. Beata Konikowska
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Correspondence to Beata Konikowska .

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Editors and Affiliations

  1. Institute of Mathematics, Warsaw University, Banacha 2, Warsaw, 02097, Poland

    Andrzej Skowron

  2. Institute of Computer Science, University of Rzeszów, ul. Dekerta 2, Rzeszów, 35-030, Poland

    Zbigniew Suraj

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Konikowska, B. (2013). Three-Valued Logic for Reasoning about Covering-Based Rough Sets. In: Skowron, A., Suraj, Z. (eds) Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam. Intelligent Systems Reference Library, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30344-9_16

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  • DOI: https://doi.org/10.1007/978-3-642-30344-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30343-2

  • Online ISBN: 978-3-642-30344-9

  • eBook Packages: EngineeringEngineering (R0)

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