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Fuzzy Bi-implications Generated by t-norms and Fuzzy Negations

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Fuzzy Techniques: Theory and Applications (IFSA/NAFIPS 2019 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1000))

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Abstract

In the literature, there are several forms of extensions of the classical bi-implication for the fuzzy logic, as for example, the axiomatization proposed by Fodor and Roubens [1]. Another way to obtain a generalization is to provide a definition based on the classical equivalence \(\phi \iff \psi \equiv (\phi \Rightarrow \psi )\wedge (\psi \Rightarrow \phi )\), in which the classical operators of conjunction and implication are replaced, respectively, by a t-norm (T) and a fuzzy implication (I). In this paper, we investigate a particular class of fuzzy bi-implications \(B(x,y)=T(I(x,y),I(y,x))\), in which I is a fuzzy (T, N)-implication introduced by Bedregal [2]. We study several properties satisfied by (T, N)-bi-implications, such as the sufficient conditions that they must satisfy in order to be a f-bi-implication.

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Notes

  1. 1.

    If \(T:[0,1]^2\rightarrow [0,1]\) is a t-norm and \(N:[0,1]\rightarrow [0,1]\) is a fuzzy negation, then we say that the pair (T, N) satisfies the law of non-contradiction if \(T(x,N(x))=0\), for all \(x\in [0,1]\) (this law is equivalently stated in [11, p. 55]).

References

  1. Fodor, J.C., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support, vol. 14. Springer, Heidelberg (1994)

    Book  Google Scholar 

  2. Bedregal, B.C.: A normal form which preserves tautologies and contradictions in a class of fuzzy logics. J. Algorithms 62(3), 135–147 (2007)

    Article  MathSciNet  Google Scholar 

  3. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  Google Scholar 

  4. Hájek, P.: Metamathematics of Fuzzy Logic, vol. 4. Springer, Heidelberg (2013)

    Google Scholar 

  5. Behounek, L., Cintula, P., Hájek, P.: Introduction to Mathematical Fuzzy Logic. College Publications (2011). Ch. 1

    Google Scholar 

  6. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, vol. 8. Springer, Heidelberg (2000)

    Google Scholar 

  7. Klement, E., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets Syst. 143(1), 5–26 (2004)

    Article  MathSciNet  Google Scholar 

  8. Klement, E., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families. Fuzzy Sets Syst. 145(3), 411–438 (2004)

    Article  MathSciNet  Google Scholar 

  9. Klement, E., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms. Fuzzy Sets Syst. 145(3), 439–454 (2004)

    Article  MathSciNet  Google Scholar 

  10. Bertei, A., Zanotelli, R., Cardoso, W., Reiser, R., Foss, L., Bedregal, B.: Correlation coefficient analysis based on fuzzy negations and representable automorphisms. In: 2016 IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2016, 24–29 July 2016, Vancouver, BC, Canada, pp. 127–132 (2016)

    Google Scholar 

  11. Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)

    Google Scholar 

  12. Mizumoto, M.: Fuzzy controls under various fuzzy reasoning methods. Inf. Sci. 45(2), 129–151 (1988)

    Article  MathSciNet  Google Scholar 

  13. Dalen, D.: Logic and Structure, 5th edn. Springer, Heidelberg (2013). Universitext

    Book  Google Scholar 

  14. Kleene, S.: Mathematical Logic. Dover Books on Mathematics. Dover Publications, Mineola (2002)

    MATH  Google Scholar 

  15. Recasens, J.: Indistinguishability Operators: Modelling Fuzzy Equalities and Fuzzy Equivalence Relations, vol. 260. Springer, Heidelberg (2010)

    Google Scholar 

  16. Bodenhofer, U.: A compendium of fuzzy weak orders: representations and constructions. Fuzzy Sets Syst. 158(8), 811–829 (2007)

    Article  MathSciNet  Google Scholar 

  17. Bedregal, B.C., Cruz, A.P.: A characterization of classic-like fuzzy semantics. Logic J. IGPL 16(4), 357–370 (2008)

    Article  MathSciNet  Google Scholar 

  18. Novák, V., De Baets, B.: Eq-algebras. Fuzzy Sets Syst. 160(20), 2956–2978 (2009)

    Article  MathSciNet  Google Scholar 

  19. Mesiar, R., Novák, V.: Operations fitting triangular-norm-based biresiduation. Fuzzy Sets Syst. 104(1), 77–84 (1999)

    Article  MathSciNet  Google Scholar 

  20. Ćirić, M., Ignjatović, J., Bogdanović, S.: Fuzzy equivalence relations and their equivalence classes. Fuzzy Sets Syst. 158(12), 1295–1313 (2007)

    Article  MathSciNet  Google Scholar 

  21. Moser, B.: On the t-transitivity of kernels. Fuzzy Sets Syst. 157(13), 1787–1796 (2006)

    Article  MathSciNet  Google Scholar 

  22. Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157(17), 2333–2346 (2006)

    Article  MathSciNet  Google Scholar 

  23. Callejas, C.: What is a fuzzy bi-implication? Master’s thesis, Universidade Federal do Rio Grande do Norte (2012)

    Google Scholar 

  24. Callejas, C., Marcos, J., Bedregal, B.R.C.: On some subclasses of the Fodor-Roubens fuzzy bi-implication. In: Proceedings of Logic, Language, Information and Computation - 19th International Workshop, WoLLIC 2012, 3–6 September 2012, Buenos Aires, Argentina, pp. 206–215 (2012)

    Chapter  Google Scholar 

  25. Callejas, C., Marcos, J., Bedregal, B.: Actions of automorphisms on some classes of fuzzy bi-implications. Mathware Soft Comput. Mag. 20, 94–97 (2013)

    Google Scholar 

  26. Pinheiro, J., Bedregal, B., Santiago, R., Santos, H., Dimuro, G.P.: (T,N)-implications and some functional equations. In: Barreto, G.A., Coelho, R. (eds.) Fuzzy Information Processing, pp. 302–313. Springer, Cham (2018)

    Chapter  Google Scholar 

  27. Pinheiro, J., Bedregal, B., Santiago, R.H., Santos, H.: A study of (T,N)-implications and its use to construct a new class of fuzzy subsethood measure. Int. J. Approximate Reason. 97, 1–16 (2018)

    Article  MathSciNet  Google Scholar 

  28. Pinheiro, J., Bedregal, B., Santiago, R.H.N., Santos, H.: (T, N)-implications. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–6 (2017)

    Google Scholar 

  29. Bustince, H., Barrenechea, E., Pagola, M.: Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158(5), 496–516 (2007)

    Article  MathSciNet  Google Scholar 

  30. Dimuro, G.P., Bedregal, B., Bustince, H., Jurio, A., Baczynski, M., Mis, K.: QL-operations and QL-implication functions constructed from tuples (O, G, N) and the generation of fuzzy subsethood and entropy measures. Int. J. Approximate Reason. 82, 170–192 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

This work is partially supported by Universidade Federal Rural do Semi-Árido - UFERSA (Project PIH10002-2018).

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Authors and Affiliations

  1. Centro Multidisciplinar de Pau dos Ferros, Universidade Federal Rural do Semi-Árido - UFERSA, Pau dos Ferros, RN, Brazil

    Antonio Diego S. Farias & Claudio Callejas

  2. Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte - UFRN, Natal, RN, Brazil

    João Marcos, Benjamín Bedregal & Regivan Santiago

Authors
  1. Antonio Diego S. Farias
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  2. Claudio Callejas
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  3. João Marcos
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  4. Benjamín Bedregal
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  5. Regivan Santiago
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Corresponding author

Correspondence to Antonio Diego S. Farias .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA

    Ralph Baker Kearfott

  2. Center for Computing Research (CIC), Instituto Politecnico Nacional, Ciudad de Mexico, Mexico

    Ildar Batyrshin

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

    Marek Reformat

  4. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Martine Ceberio

  5. Department of Computer Science, University of Texas at El Paso , El Paso, TX, USA

    Vladik Kreinovich

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Farias, A.D.S., Callejas, C., Marcos, J., Bedregal, B., Santiago, R. (2019). Fuzzy Bi-implications Generated by t-norms and Fuzzy Negations. In: Kearfott, R., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds) Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS 2019 2019. Advances in Intelligent Systems and Computing, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-030-21920-8_53

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