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Initial Comparison of Formal Approaches to Fuzzy and Rough Sets

  • Adam Grabowski
  • Takashi Mitsuishi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9119)

Abstract

Fuzzy sets and rough sets are well-known approaches to incomplete or imprecise data. In the paper we compare two formalizations of these sets within one of the largest repositories of computer-checked mathematical knowledge – the Mizar Mathematical Library. Although the motivation was quite similar in both developments, these approaches – proposed by us – vary significantly. Paradoxically, it appeared that fuzzy sets are much closer to the set theory implemented within the Mizar library, while in order to make more feasible view for rough sets we had to choose relational structures as a basic framework. The formal development, although counting approximately 15 thousand lines of source code, is by no means closed – it allows both for further generalizations, building on top of the existing knowledge, and even merging of these approaches. The paper is illustrated with selected examples of definitions, theorems, and proofs taken from rough and fuzzy set theory formulated in the Mizar language.

Keywords

Membership Function Fuzzy Number Proof Assistant Initial Comparison Mizar Mathematical Library 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bryniarski, E.: Formal conception of rough sets. Fundamenta Informaticae 27(2/3), 109–136 (1996)MathSciNetGoogle Scholar
  2. 2.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. International Journal of System Sciences 9(6), 613–626 (1978)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17(2–3), 191–209 (1990)CrossRefGoogle Scholar
  4. 4.
    Grabowski, A.: The formal construction of fuzzy numbers. Formalized Mathematics 22(4), 313–319 (2014)CrossRefGoogle Scholar
  5. 5.
    Grabowski, A.: On the computer certification of fuzzy numbers. In: Ganzha, M., Maciaszek, L., Paprzycki, M. (eds.) Proceedings of Federated Conference on Computer Science and Information Systems, FedCSIS 2013, pp. 51–54 (2013)Google Scholar
  6. 6.
    Grabowski, A.: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1-2), 65–79 (2013)MathSciNetGoogle Scholar
  7. 7.
    Grabowski, A.: On the computer-assisted reasoning about rough sets. In: Dunin-Kęplicz, B., Jankowski, A., Szczuka, M. (eds.) Monitoring, Security and Rescue Techniques in Multiagent Systems. Advances in Soft Computing, vol. 28, pp. 215–226 (2005)Google Scholar
  8. 8.
    Grabowski, A., Jastrzębska, M.: Rough set theory from a math-assistant perspective. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 152–161. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)MathSciNetGoogle Scholar
  10. 10.
    Kacprzak, M., Kosiński, W.: On lattice structure and implications on ordered fuzzy numbers. In: Proc. of EUSFLAT 2011, pp. 267–274 (2011)Google Scholar
  11. 11.
    Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Fuzzy Sets and Systems 101, 125–135 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets and Systems 91, 165–175 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Koleśnik, R., Prokopowicz, P., Kosiński, W.: Fuzzy calculator – useful tool for programming with fuzzy algebra. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 320–325. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Korniłowicz, A.: On rewriting rules in Mizar. Journal of Automated Reasoning 50(2), 203–210 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kosiński, W., Prokopowicz, P., Ślęzak, D.: Fuzzy numbers with algebraic operations: algorithmic approach. In: Kłopotek, M., Wierzchoń, S.T., Michalewicz, M. (eds.) Intelligent Information Systems, IIS 2002, Poland, pp. 311–320 (2002)Google Scholar
  16. 16.
    Kosiński, W., Prokopowicz, P., Ślęzak, D.: Ordered fuzzy numbers. Bulletin of the Polish Academy of Sciences, Sér. Sci. Math. 51(3), 327–338 (2003)Google Scholar
  17. 17.
    Mitsuishi, T., Endou, N., Shidama, Y.: The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics 9(2), 351–356 (2001)Google Scholar
  18. 18.
    Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135(1), 5–9 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Naumowicz, A., Korniłowicz, A.: A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 67–72. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991)CrossRefGoogle Scholar
  21. 21.
    Pąk, K.: Methods of lemma extraction in natural deduction proofs. Journal of Automated Reasoning 50(2), 217–228 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Urban, J., Sutcliffe, G.: Automated reasoning and presentation support for formalizing mathematics in Mizar. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 132–146. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Wiedijk, F.: Formal proof – getting started. Notices of the AMS 55(11), 1408–1414 (2008)MathSciNetGoogle Scholar
  24. 24.
    Zadeh, L.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BiałystokBiałystokPoland
  2. 2.University of Marketing and Distribution SciencesNishi-ku, KobeJapan

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