Extending Formal Fuzzy Sets with Triangular Norms and Conorms | SpringerLink
Skip to main content
Springer Nature Link
Log in

Extending Formal Fuzzy Sets with Triangular Norms and Conorms

  • Conference paper
  • First Online:
Advances in Fuzzy Logic and Technology 2017 (EUSFLAT 2017, IWIFSGN 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 22879
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 28599
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    http://vsl2014.at/livestream/index.html#Recordings.

  2. 2.

    Widely accepted numbers in kilobytes of zipped code are now gradually replaced in the field by more human-oriented measure, as it is relatively easy to convert the number of lines into the number of pages.

  3. 3.

    All Mizar articles can be browsed online at http://mizar.org/version/current/html/ via MML identifiers.

References

  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:10.1007/978-3-319-20615-8_17

  2. Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  3. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978). doi:10.1080/00207727808941724

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17(2–3), 191–209 (1990). doi:10.1080/03081079008935107

    Article  MATH  Google Scholar 

  5. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, Orlando (1997)

    MATH  Google Scholar 

  6. Grabowski, A.: Basic formal properties of triangular norms and conorms. Formaliz. Math. 25(2), 93–100 (2017). doi:10.1515/forma-2017-0009

    Google Scholar 

  7. Grabowski, A.: Binary relations-based rough sets - an automated approach. Formaliz. Math. 24(2), 143–155 (2016). doi:10.1515/forma-2016-0011

    Article  MATH  Google Scholar 

  8. Grabowski, A.: Lattice theory for rough sets - a case study with Mizar. Fundam. Inform. 147(2–3), 223–240 (2016). doi:10.3233/FI-2016-1406

    Article  MathSciNet  Google Scholar 

  9. Grabowski, A.: Mechanizing complemented lattices within Mizar type system. J. Autom. Reason. 55(3), 211–221 (2015). doi:10.1007/s10817-015-9333-5

    Article  MathSciNet  MATH  Google Scholar 

  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. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015). doi:10.1007/s10817-015-9345-1

    Article  MathSciNet  MATH  Google Scholar 

  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:10.1007/978-3-319-25754-9_31

  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:10.1007/978-3-319-19324-3_15

  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:10.1007/978-3-540-27818-4_10

  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. Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998). doi:10.1007/978-94-011-5300-3

    Book  MATH  Google Scholar 

  17. Klement, E., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000). doi:10.1007/978-94-015-9540-7

    Book  MATH  Google Scholar 

  18. Korniłowicz, A.: Flexary connectives in Mizar. Comput. Lang. Syst. Struct. 44, 238–250 (2015). doi:10.1016/j.cl.2015.07.002

    Google Scholar 

  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. 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:10.1007/s10817-015-9332-6

    Article  MathSciNet  MATH  Google Scholar 

  21. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991). doi:10.1007/978-94-011-3534-4

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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:10.1007/978-3-642-14128-7_12

  24. Wiedijk, F.: Formal proof - getting started. Not. Am. Math. Soc. 55(11), 1408–1414 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Yao, Y.Y.: A comparative study of fuzzy sets and rough sets. Inf. Sci. 109(1–4), 227–242 (1998). doi:10.1016/S0020-0255(98)10023-3

    Article  MathSciNet  MATH  Google Scholar 

  26. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965). doi:10.1016/S0019-9958(65)90241-X

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics, University of Białystok, Konstantego Ciołkowskiego 1M, 15-245, Białystok, Poland

    Adam Grabowski

  2. University of Marketing and Distribution Sciences, 3-1 Gakuen Nishimachi, Nishi-ku, Kobe, 655-2188, Japan

    Takashi Mitsuishi

Authors
  1. Adam Grabowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takashi Mitsuishi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adam Grabowski .

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Eulalia Szmidt

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Slawomir Zadrożny

  4. Department of Telecommunications and Information Processing, Bulgarian Academy of Sciences, Sofia, Bulgaria

    K. T. Atanassov

  5. WIT - Warsaw School of Information Technology, Warsaw, Poland

    Maciej Krawczak

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Cite this paper

Grabowski, A., Mitsuishi, T. (2018). Extending Formal Fuzzy Sets with Triangular Norms and Conorms. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 642. Springer, Cham. https://doi.org/10.1007/978-3-319-66824-6_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66824-6_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66823-9

  • Online ISBN: 978-3-319-66824-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation