Formalizing Lattice-Theoretical Aspects of Rough and Fuzzy Sets

  • Adam Grabowski
  • Takashi Mitsuishi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)


Fuzzy sets and rough sets are well-known approaches to incomplete or imprecise data. In the paper we briefly report how these frameworks were successfully encoded with the help of one of the leading computer proof assistants in the world. Even though fuzzy sets are much closer to the set theory implemented within the Mizar library than rough sets, lattices as a basic viewpoint appeared a very feasible one. We focus on the lattice-theoretical aspects of rough and fuzzy sets to enable the application of external theorem provers like EQP or Prover9 as well as to translate them into TPTP format widely recognized in the world of automated proof search. The paper is illustrated with the examples taken just from one of the largest repositories of computer-checked mathematical knowledge – the Mizar Mathematical Library. Our formal development allows both for further generalizations, building on top of the existing knowledge, and even merging of these approaches.


  1. 1.
    Bryniarski, E.: Formal conception of rough sets. Fundamenta Informaticae 27(2/3), 109–136 (1996)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17(2–3), 191–209 (1990)zbMATHCrossRefGoogle Scholar
  4. 4.
    Estaji, A.A., Hooshmandasl, M.R., Davvaz, B.: Rough set theory applied to lattice theory. Inf. Sci. 200, 108–122 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18(1), 145–174 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Grabowski, A.: Mechanizing complemented lattices within Mizar type system. J. Autom. Reasoning 55(3), 211–221 (2015). doi: 10.1007/s10817-015-9333-5 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grabowski, A.: The formal construction of fuzzy numbers. Formalized Math. 22(4), 321–327 (2014)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Grabowski, A.: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1–2), 65–79 (2013)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Grabowski, A.: On the computer-assisted reasoning about rough sets. In: Dunin-Kȩplicz, B., Jankowski, A., Szczuka, M. (eds.) Monitoring, Security and Rescue Techniques in Multiagent Systems. Advances in Soft Computing, vol. 28, pp. 215–226. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Grabowski, A., Jastrzȩbska, 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
  12. 12.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formalized Reasoning 3(2), 153–245 (2010)zbMATHGoogle Scholar
  13. 13.
    Grätzer, G.: General Lattice Theory. Birkhäuser, Boston (1998)zbMATHGoogle Scholar
  14. 14.
    Järvinen, J.: Lattice theory for rough sets. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 400–498. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  15. 15.
    Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Fuzzy Sets Syst. 101, 125–135 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets Syst. 91, 165–175 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Korniłowicz, A.: On rewriting rules in Mizar. J. Autom. Reasoning 50(2), 203–210 (2013)zbMATHCrossRefGoogle Scholar
  18. 18.
    Mitsuishi, T., Endou, N., Shidama, Y.: The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Math. 9(2), 351–356 (2001)Google Scholar
  19. 19.
    Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets Syst. 135(1), 5–9 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991)zbMATHCrossRefGoogle Scholar
  22. 22.
    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reasoning 50(2), 217–228 (2013)zbMATHCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Wiedijk, F.: Formal proof - getting started. Not. AMS 55(11), 1408–1414 (2008)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Yao, Y.Y.: Two views of the rough set theory in finite universes. Int. J. Approximate Reasoning 15(4), 291–317 (1996)zbMATHCrossRefGoogle Scholar
  26. 26.
    Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)zbMATHCrossRefGoogle Scholar
  27. 27.
    Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhu, W.: Generalized rough sets based on relations. Inf. Sci. 177(22), 4997–5011 (2007)zbMATHCrossRefGoogle Scholar
  29. 29.
    Zhu, W.: Topological approaches to covering rough sets. Inf. Sci. 177(6), 1499–1508 (2007)zbMATHCrossRefGoogle 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 SciencesKobeJapan

Personalised recommendations