EUSFLAT 2017, IWIFSGN 2017: Advances in Fuzzy Logic and Technology 2017 pp 176-187

Extending Formal Fuzzy Sets with Triangular Norms and Conorms

• Takashi Mitsuishi
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 642)

Abstract

Fuzzy sets is a well-known approach to incomplete or imprecise data. Contrary to the rough sets however, the notion of fuzziness allows for quite natural description in terms of ordinary set theory used by mathematicians and computer scientists. As contemporary mathematics uses more and more methods of computer verification of theorems and discovering their proofs, it is not very strange that also in this area we could observe growing usage of automated proof-assistants. We report on the progress of the development of already well-established framework of fuzzy set theory within one of popular repositories of computerized mathematical knowledge – the Mizar Mathematical Library. Even if the original formal background was created some ten years ago, and during that time it was thoroughly redesigned in order to increase its expressive power and to follow the evolution of underlying proof language, we see the need for further modifications. In this paper, we describe the process of the parametrization of classical operations on fuzzy sets via triangular norms and conorms because as of now, classical union and intersection of corresponding membership functions were defined only based on operations of maximum, and minimum, respectively. We illustrate our development by examples taken from correct and fully verified Mizar code.

References

1. 1.
Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pa̧k, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M. et al. (eds.): Conference on Intelligent Computer Mathematics, CICM 2015. Lecture Notes in Computer Science, vol. 9150, pp. 261–279. Springer (2015). doi:
2. 2.
Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)
3. 3.
Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978). doi:
4. 4.
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17(2–3), 191–209 (1990). doi:
5. 5.
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, Orlando (1997)
6. 6.
Grabowski, A.: Basic formal properties of triangular norms and conorms. Formaliz. Math. 25(2), 93–100 (2017). doi: Google Scholar
7. 7.
Grabowski, A.: Binary relations-based rough sets - an automated approach. Formaliz. Math. 24(2), 143–155 (2016). doi:
8. 8.
Grabowski, A.: Lattice theory for rough sets - a case study with Mizar. Fundam. Inform. 147(2–3), 223–240 (2016). doi:
9. 9.
Grabowski, A.: Mechanizing complemented lattices within Mizar type system. J. Autom. Reason. 55(3), 211–221 (2015). doi:
10. 10.
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
11. 11.
Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015). doi:
12. 12.
Grabowski, A., Mitsuishi, T.: Formalizing lattice-theoretical aspects of rough and fuzzy sets. In: Rough Sets and Knowledge Technology, RSKT 2015. LNAI, vol. 9436, pp. 347–356 (2015). doi:
13. 13.
Grabowski, A., Mitsuishi, T.: Initial comparison of formal approaches to fuzzy and rough sets. In: Artificial Intelligence and Soft Computing, ICAISC 2015. LNAI, vol. 9119, pp. 160–171 (2015). doi:
14. 14.
Grabowski, A., Schwarzweller, C.: Rough concept analysis - theory development in the Mizar system. In: Mathematical Knowledge Management. Lecture Notes in Computer Science, vol. 3119, pp. 130–144 (2004). doi:
15. 15.
Grabowski, A., Schwarzweller, C.: Towards automatically categorizing mathematical knowledge. In: Ganzha, M., Maciaszek, L., Paprzycki, M. (eds.) Proceedings of FedCSIS 2012, pp. 63–68 (2012)Google Scholar
16. 16.
Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998). doi:
17. 17.
Klement, E., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000). doi:
18. 18.
Korniłowicz, A.: Flexary connectives in Mizar. Comput. Lang. Syst. Struct. 44, 238–250 (2015). doi: Google Scholar
19. 19.
Mitsuishi, T., Endou, N., Shidama, Y.: The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formaliz. Math. 9(2), 351–356 (2001)Google Scholar
20. 20.
Naumowicz, A.: Automating boolean set operations in Mizar proof checking with the aid of an external SAT solver. J. Autom. Reason. 55(3), 285–294 (2015). doi:
21. 21.
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991). doi:
22. 22.
Pa̧k, K.: Improving legibility of formal proofs based on the close reference principle is NP-hard. J. Autom. Reason. 55(3), 295–306 (2015)
23. 23.
Urban, J., Sutcliffe, G.: Automated reasoning and presentation support for formalizing mathematics in Mizar. Lecture Notes in Computer Science, vol. 6167, pp. 132–146 (2010). doi:
24. 24.
Wiedijk, F.: Formal proof - getting started. Not. Am. Math. Soc. 55(11), 1408–1414 (2008)
25. 25.
Yao, Y.Y.: A comparative study of fuzzy sets and rough sets. Inf. Sci. 109(1–4), 227–242 (1998). doi:
26. 26.
Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965). doi: