Fuzzy Relation Equations with Fuzzy Quantifiers

  • Nhung CaoEmail author
  • Martin Štěpnička
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)


In this paper, we follow the previous works on fuzzy relation compositions based on fuzzy quantifiers and we introduce systems of fuzzy relation equations stemming from compositions based on fuzzy quantifiers. We address the question, whether such systems under some specific conditions may become solvable, and we provide a positive answer. Based on the computational forms of the compositions using fuzzy quantifiers, we explain a way of getting solutions of the systems. In addition to showing some new properties and theoretical results, we provide readers with illustrative examples.


Fuzzy relation equations Mamdani-Assilian model Implicative model Fuzzy (generalized) quantifiers 



This research was partially supported by the NPU II project LQ1602 “IT4Innovations excellence in science” provided by the MŠMT.


  1. 1.
    Burda, M.: Linguistic fuzzy logic in R. In: Proceedigns of the IEEE International Conference on Fuzzy Systems, Istanbul, Turkey (2015)Google Scholar
  2. 2.
    Cao, N., Štěpnička, M., Holčapek, M.: An extension of fuzzy relational compositions using generalized quantifiers. In: Proceedings of the 16th World Congress of the International Fuzzy Systems Association (IFSA) and 9th Conference of the European Society for Fuzzy-Logic and Technology (EUSFLAT), Advances in Intelligent Systems Research, vol. 89, pp. 49–58. Atlantis press, Gijón (2015)Google Scholar
  3. 3.
    Cao, N., Štěpnička, M., Holčapek, M.: Extensions of fuzzy relational compositions based on generalized quantifer. Fuzzy Sets Syst. (in press). doi: 10.1016/j.fss.2017.04.009
  4. 4.
    Cao, N., Štěpnička, M., Holčapek, M.: Non-preservation of chosen properties of fuzzy relational compositions based on fuzzy quantifiers. In: 2017 IEEE International Conference on Fuzzy Systems. (in press)Google Scholar
  5. 5.
    De Baets, B.: Analytical solution methods for fuzzy relational equations. In: Dubois, D., Prade, H. (eds.) The Handbook of Fuzzy Set Series, vol. 1, pp. 291–340. Academic Kluwer Publishers, Boston (2000)Google Scholar
  6. 6.
    Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer, Boston (1989)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dvořák, A., Holčapek, M.: L-fuzzy quantifiers of type \(\langle 1\rangle \) determined by fuzzy measures. Fuzzy Sets Syst. 160(23), 3425–3452 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dvořák, A., Holčapek, M.: Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices. Inf. Sci. 185, 205–229 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dvořák, A., Holčapek, M.: Type \(\langle 1,1\rangle \) fuzzy quantifiers determined by fuzzy measures. Part I. Basic definitions and examples. Fuzzy Sets Syst. 242, 31–55 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gottwald, S., Pedrycz, W.: Solvability of fuzzy relational equations and manipulation of fuzzy data. Fuzzy Sets Syst. 18, 45–65 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ignjatović, J., Ćirić, M., Šešelja, B., Tepavčević, A.: Fuzzy relational inequalities and equations, fuzzy quasi-orders, closures and openings of fuzzy sets. Fuzzy Sets Syst. 260, 1–24 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Klawonn, F.: Fuzzy points, fuzzy relations and fuzzy functions. In: Novák, V., Perfilieva, I. (eds.) Discovering the World with Fuzzy Logic, pp. 431–453. Springer, Berlin (2000)Google Scholar
  13. 13.
    Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey (1995)zbMATHGoogle Scholar
  14. 14.
    Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man Mach. Stud. 7, 1–13 (1975)CrossRefzbMATHGoogle Scholar
  15. 15.
    Nosková, L.: Systems of fuzzy relation equation with inf-\(\rightarrow \) composition solvability and solutions. J. Electr. Eng. 12(s), 69–72 (2005)zbMATHGoogle Scholar
  16. 16.
    Perfilieva, I., Lehmke, S.: Correct models of fuzzy if-then rules are continuous. Fuzzy Sets Syst. 157, 3188–3197 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Perfilieva, I., Nosková, L.: System of fuzzy relation equations with inf-\(\rightarrow \) composition: complete set of solutions. Fuzzy Sets Syst. 159, 2256–2271 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Qu, X.B., Sun, F., Wang, T.F.: Matrix elementary transformations in solving systems of fuzzy relation equations. Appl. Soft Comput. 31, 25–29 (2015)CrossRefGoogle Scholar
  19. 19.
    Sanchez, E.: Resolution of composite fuzzy relation equations. Inf. Control 30, 38–48 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Štěpnička, M., De Baets, B.: Interpolativity of at-least and at-most models of monotone single-input single-output fuzzy rule bases. Inf. Sci. 234, 16–28 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Štěpnička, M., De Baets, B., Nosková, L.: Arithmetic fuzzy models. IEEE Trans. Fuzzy Syst. 18, 1058–1069 (2010)CrossRefGoogle Scholar
  22. 22.
    Štěpnička, M., Holčapek, M.: Fuzzy relational compositions based on generalized quantifiers. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems, PT II (IPMU 2014). Communications in Computer and Information Science, vol. 443, pp. 224–233. Springer, Berlin (2014)Google Scholar
  23. 23.
    Štěpnička, M., Jayaram, B.: On the suitability of the Bandler-Kohout subproduct as an inference mechanism. IEEE Trans. Fuzzy Syst. 18(2), 285–298 (2010)CrossRefGoogle Scholar
  24. 24.
    Štěpnička, M., Jayaram, B.: Interpolativity of at-least and at-most models of monotone fuzzy rule bases with multiple antecedent variables. Fuzzy Sets Syst. 297, 26–45 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy Modeling, CE IT4InnovationsUniversity of OstravaOstravaCzech Republic

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