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Many-Valued Rough Sets Based on Tied Implications

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Rough Sets and Knowledge Technology (RSKT 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8818))

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Abstract

We investigate a general many-valued rough set theory, based on tied adjointness algebras, from both constructive and axiomatic approaches. The class of tied adjointness algebras constitutes a particularly rich generalization of residuated algebras and deals with implications (on two independently chosen posets (L, ≤  L ) and \(\left( P,\leq_{P}\right) \), interpreting two, possibly different, types of uncertainty) tied by an integral commutative ordered monoid operation on P. We show that this model introduces a flexible extension of rough set theory and covers many fuzzy rough sets models studied in literature. We expound motivations behind the use of two lattices L and P in the definition of the approximation space, as a generalization of the usual one-lattice approach. This new setting increase the number of applications in which rough set theory can be applied.

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References

  1. Abdel-Hamid, A., Morsi, N.: Associatively Tied Implications. Fuzzy Sets and Systems 136, 291–311 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baczyński, M., Jayaram, B.: (S,N)- and R-implications: a state-of-the-art survey. Fuzzy Sets and Systems 159, 1836–1859 (2008)

    Google Scholar 

  3. Baczyński, M., Jayaram, B., Mesiar, R.: R-implications and the exchange principle: The case of border continuous t-norms. Fuzzy Sets and Systems 224, 93–105 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boixander, D., Jacas, J., Recasens, J.: Upper and Lower Approximations of Fuzzy Sets. International Journal of General Systems 29(4), 555–568 (2000)

    Article  MathSciNet  Google Scholar 

  5. Dubois, D., Prade, H.: Rough Fuzzy Sets and Fuzzy Rough Sets. International Journal of General Systems 17, 191–209 (1990)

    Article  MATH  Google Scholar 

  6. Dubois, D., Prade, H.: Putting Fuzzy Sets and Rough Sets Together. In: Słowiński, R. (ed.) Intelligent Decision Support - Handbook of Applications and Advances of the Rough Sets Theory, pp. 203–232. Kluwer Academic Publishers (1992)

    Google Scholar 

  7. Hu, Q., Zhang, L., Chen, D., Pedrycz, W., Yu, D.: Gaussian kernel based fuzzy rough sets: model, uncertainty measures and applications. International Journal of Approximate Reasoning 51, 453–471 (2010)

    Article  MATH  Google Scholar 

  8. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: an algebraic glimpse at substructural logics. Elsevier (2007)

    Google Scholar 

  9. Goguen, J.A.: L-fuzzy sets. Journal of Mathematical Analysis and Applications 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gong, Z., Sun, B., Chen, D.: Rough set theory for the interval-valued fuzzy information systems. Information Sciences 178, 1968–1985 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jayaram, B.: On the law of Importation (x ∧ y) → z ≡ x → (y → z) in Fuzzy Logic. IEEE Transaction on Fuzzy Systems 16, 130–144 (2008)

    Article  Google Scholar 

  12. Mas, M., Monserrat, M., Torrens, J.: The law of importation for discrete implications. Information Sciences 179(24), 4208–4218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mas, M., Monserrat, M., Torrens, J.: A characterization of (U, N), RU,QL and D-implications derived from uninorms satisfying the law of importation. Fuzzy Sets and Systems 161(10), 1369–1387 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Massanet, S., Torrens, J.: The law of importation versus the exchange principle on fuzzy implication. Fuzzy Sets and Systems 168, 47–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Mi, J., Leung, Y., Zhao, H., Feng, T.: Generalized fuzzy rough sets determined by a triangular norm. Information Sciences 178, 3203–3213 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Morsi, N.: Propositional calculus under adjointness. Fuzzy Sets and Systems 132, 91–106 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morsi, N., Roshdy, E.: Issues on adjointness in multiple-valued logics. Information Sciences 176(19), 2886–2909 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morsi, N., Yakout, M.: Axiomatics for Fuzzy Rough Sets. Fuzzy Sets and Systems 100, 327–342 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morsi, N., Lotfallah, W., El-Zekey, M.: The Logic of Tied Implications, Part 1: Properties, Applications and Representation. Fuzzy Sets and Systems 157(5), 647–669 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Morsi, N., Lotfallah, W., El-Zekey, M.: The Logic of Tied Implications, Part 2: Syntax. Fuzzy Sets and Systems 157(15), 2030–2057 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Pei, D.: A generalized model of fuzzy rough sets. International Journal of General Systems 34(5), 603–613 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Radzikowska, A.M., Kerre, E.E.: A Comparative Study of Fuzzy Rough Sets. Fuzzy Sets and Systems 126, 137–155 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Richard, J., Shen, Q.: Fuzzy-rough sets for descriptive dimensionality reduction. In: Proceeding of the IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2002, pp. 29–34 (2002)

    Google Scholar 

  26. Skowron, A., Polkowski, L.: Rough Set in Knowledge Discovery. Springer, Berlin (1998)

    Google Scholar 

  27. Singh, K., Thakur, S., Lal, M.: Vague Rough Set Techniques for Uncertainty Processing in Relational Database Model. Informatica 19(1), 113–134 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Wang, C.Y., Hu, B.Q.: Fuzzy rough sets based on generalized residuated lattices. Information Sciences 248, 31–49 (2013)

    Article  MathSciNet  Google Scholar 

  29. Wu, W., Mi, J., Zhang, W.: Generalized fuzzy rough sets. Information Sciences 151, 263–282 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu, W., Leung, Y., Mi, J.: On characterizations of (I, T)-fuzzy rough approximation operators. Fuzzy Sets and Systems 154, 76–102 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, CE IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Moataz El-Zekey

Authors
  1. Moataz El-Zekey
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Correspondence to Moataz El-Zekey .

Editor information

Editors and Affiliations

  1. Tongji University, Shanghai, China

    Duoqian Miao

  2. Department of Electrical and Computer En, University of Alberta, Edmonton, Alberta, Canada

    Witold Pedrycz

  3. University of Warsaw, Warsaw, Poland

    Dominik Ślȩzak

  4. University of Applied Sciences, München, Germany

    Georg Peters

  5. Tianjin University, Tianjin, China

    Qinghua Hu

  6. Tongji University, Shanghai, China

    Ruizhi Wang

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El-Zekey, M. (2014). Many-Valued Rough Sets Based on Tied Implications. In: Miao, D., Pedrycz, W., Ślȩzak, D., Peters, G., Hu, Q., Wang, R. (eds) Rough Sets and Knowledge Technology. RSKT 2014. Lecture Notes in Computer Science(), vol 8818. Springer, Cham. https://doi.org/10.1007/978-3-319-11740-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-11740-9_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11739-3

  • Online ISBN: 978-3-319-11740-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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