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Proceedings of the Conference of the European Society for Fuzzy Logic and Technology

International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets

EUSFLAT 2017, IWIFSGN 2017: Advances in Fuzzy Logic and Technology 2017 pp 354–367Cite as

Fuzzy Relation Equations with Fuzzy Quantifiers

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 641))

Abstract

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.

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Notes

  1. 1.

    The operations \(\wedge , \vee , *, \rightarrow \) stand for meet (infimum), join (supremum), multiplication (left-continuous t-norm) and its residual implications, respectively.

  2. 2.

    The closeness is here given by the similarity measure using the fuzzy equivalence from the underlying algebraic structure. In the case of a continuous Archimedean t-norm, the similarity is a dual notion to the metric function induced by the additive generator, which justifies the continuity point of view as well as the terminology [16].

References

  1. Burda, M.: Linguistic fuzzy logic in R. In: Proceedigns of the IEEE International Conference on Fuzzy Systems, Istanbul, Turkey (2015)

    Google Scholar 

  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. 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. 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. 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. Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer, Boston (1989)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gottwald, S., Pedrycz, W.: Solvability of fuzzy relational equations and manipulation of fuzzy data. Fuzzy Sets Syst. 18, 45–65 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey (1995)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  15. Nosková, L.: Systems of fuzzy relation equation with inf-\(\rightarrow \) composition solvability and solutions. J. Electr. Eng. 12(s), 69–72 (2005)

    MATH  Google Scholar 

  16. Perfilieva, I., Lehmke, S.: Correct models of fuzzy if-then rules are continuous. Fuzzy Sets Syst. 157, 3188–3197 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  19. Sanchez, E.: Resolution of composite fuzzy relation equations. Inf. Control 30, 38–48 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Štěpnička, M., De Baets, B., Nosková, L.: Arithmetic fuzzy models. IEEE Trans. Fuzzy Syst. 18, 1058–1069 (2010)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

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

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, CE IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Nhung Cao & Martin Štěpnička

Authors
  1. Nhung Cao
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  2. Martin Štěpnička
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Corresponding author

Correspondence to Nhung Cao .

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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

    Sławomir Zadrożny

  4. Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir T. Atanassov

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

    Maciej Krawczak

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Cao, N., Štěpnička, M. (2018). Fuzzy Relation Equations with Fuzzy Quantifiers. 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 641. Springer, Cham. https://doi.org/10.1007/978-3-319-66830-7_32

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  • DOI: https://doi.org/10.1007/978-3-319-66830-7_32

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66829-1

  • Online ISBN: 978-3-319-66830-7

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