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2018 | OriginalPaper | Chapter

Generalized Modus Ponens for (U, N)-implications

Authors : M. Mas, D. Ruiz-Aguilera, Joan Torrens

Published in: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations

Publisher: Springer International Publishing

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Abstract

The Modus Ponens becomes an essential property in approximate reasoning and fuzzy control when forward inferences are managed. Thus, the conjunctor and the fuzzy implication function used in the inference process are required to satisfy this property. Usually, the conjunctor is modeled by a t-norm, but recently also by conjunctive uninorms. In this paper we study when (U, N)-implications satisfy the Modus Ponens property with respect to a conjunctive uninorm U in general, in a similar way as it was previously done for RU-implications. The functional inequality derived from the Modus Ponens involves in this case two different uninorms and a fuzzy negation leading to many possibilities. So, this communication presents only a first step in this study and many cases depending on the classes of the involved uninorms are worth to study.

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previous chapter On the Characterization of a Family of Generalized Yager’s Implications

next chapter Dependencies Between Some Types of Fuzzy Equivalences

The subindex “cos” stands here for continuous open square.

Recall that continuous negations are the most usual ones. In particular, they contain the strong negations (those that are involutive) and also the strict ones (those that are strictly decreasing and continuous).

Aguiló, I., Suñer, J., Torrens, J.: A characterization of residual implications derived from left-continuous uninorms. Inf. Sci. 180, 3992–4005 (2010)MathSciNetCrossRef

Alsina, C., Trillas, E.: When \((S, N)\)-implications are \((T, T_1)\)-conditional functions? Fuzzy Sets Syst. 134, 305–310 (2003)CrossRef

Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds.): Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol. 300. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35677-3CrossRefMATH

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

Baczyński, M., Jayaram, B.: (U, N)-implications and their characterizations. Fuzzy Sets and Systems 160, 2049–2062 (2009). https://doi.org/10.1007/978-3-540-69082-5MathSciNetCrossRefMATH

Benítez, J.M., Castro, J.L., Requena, I.: Are artificial neural networks black boxes? IEEE Trans. Neural Netw. 8, 1156–1163 (1997)CrossRef

Czogala, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)MathSciNetCrossRef

De Baets, B.: Idempotent uninorms. Eur. J. Oper. Res. 118, 631–642 (1999)CrossRef

De Baets, B., Fodor, J.C.: Residual operators of uninorms. Soft Comput. 3, 89–100 (1999)CrossRef

10.

De Baets, B., Fodor, J.: Van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets Syst. 104, 133–136 (1999)MathSciNetCrossRef

11.

Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of uninorms. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 5, 411–427 (1997)MathSciNetCrossRef

12.

Hu, S., Li, Z.: The structure of continuous uni-norms. Fuzzy Sets Syst. 124, 43–52 (2001)MathSciNetCrossRef

13.

Li, G., Liu, H.W.: On properties of uninorms locally internal on the boundary. Fuzzy Sets Syst. 332, 116–128 (2018)MathSciNetCrossRef

14.

Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRef

15.

Martín, J., Mayor, G., Torrens, J.: On locally internal monotonic operators. Fuzzy Sets Syst. 137, 27–42 (2003)CrossRef

16.

Mas, M., Massanet, S., Ruiz-Aguilera, D., Torrens, J.: A survey on the existing classes of uninorms. J. Intell. Fuzzy Syst. 29, 1021–1037 (2015)MathSciNetCrossRef

17.

Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: \(RU\) and \((U, N)\)-implications satisfying Modus Ponens. Int. J. Approximate Reasoning 73, 123–137 (2016)MathSciNetCrossRef

18.

Mas, M., Monserrat, M., Torrens, J.: Two types of implications derived from uninorms. Fuzzy Sets Syst. 158, 2612–2626 (2007)MathSciNetCrossRef

19.

Mas, M., Monserrat, M., Torrens, J.: A characterization of \((U, N), RU, QL\) and \(D\)-implications derived from uninorms satisfying the law of importation. Fuzzy Sets Syst. 161, 1369–1387 (2010)MathSciNetCrossRef

20.

Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)CrossRef

21.

Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: On a generalization of the Modus Ponens: U-conditionality. In: Carvalho, J.P., Lesot, M.-J., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2016. CCIS, vol. 610, pp. 1–12. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40596-4_33CrossRefMATH

22.

Mas, M., Ruiz-Aguilera, D., Torrens, J.: On some classes of RU-implications satisfying U-Modus Ponens. In: Torra, V., Mesiar, R., De Baets, B. (eds.) AGOP 2017. AISC, vol. 581, pp. 71–82. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-59306-7_8CrossRef

23.

Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)MathSciNetCrossRef

24.

Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds.) Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol. 300, pp. 1–30. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35677-3_1CrossRefMATH

25.

Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symbolic Logic 72, 834–864 (2007)MathSciNetCrossRef

26.

Ruiz, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40, 21–38 (2004)MathSciNetMATH

27.

Ruiz-Aguilera, D., Torrens, J.: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets Syst. 158, 23–37 (2007)MathSciNetCrossRef

28.

Ruiz-Aguilera, D., Torrens, J.: S- and R-implications from uninorms continuous in \(]0,1[^2\) and their distributivity over uninorms. Fuzzy Sets Syst. 160, 832–852 (2009)MathSciNetCrossRef

29.

Ruiz-Aguilera, D., Torrens, J., De Baets, B., Fodor, J.: Some remarks on the characterization of idempotent uninorms. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS (LNAI), vol. 6178, pp. 425–434. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14049-5_44CrossRef

30.

Trillas, E., Alsina, C., Pradera, A.: On MPT-implication functions for fuzzy logic. Revista de la Real Academia de Ciencias. Serie A. Matemáticas (RACSAM) 98(1), 259–271 (2004)

31.

Trillas, E., Alsina, C., Renedo, E., Pradera, A.: On contra-symmetry and MPT-conditionality in fuzzy logic. Int. J. Intell. Syst. 20, 313–326 (2005)CrossRef

32.

Trillas, E., Campo, C., Cubillo, S.: When QM-operators are implication functions and conditional fuzzy relations. Int. J. Intell. Syst. 15, 647–655 (2000)CrossRef

33.

Trillas, E., Valverde, L.: On Modus Ponens in fuzzy logic. In: 15th International Symposium on Multiple-Valued Logic, pp. 294–301. Kingston, Canada (1985)

34.

Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996)MathSciNetCrossRef

Title: Generalized Modus Ponens for (U, N)-implications
Authors: M. Mas
D. Ruiz-Aguilera
Joan Torrens
Publisher: Springer International Publishing
Book: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations
Print ISBN: 978-3-319-91472-5

Electronic ISBN: 978-3-319-91473-2

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-91473-2_55

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