Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
fuzzy connectives; fuzzy implication; distributivity; functional equations
fuzzy connectives; fuzzy implication; distributivity; functional equations
Summary:
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) $$ f(\min(x+y,a))=\min(f(x)+f(y),b), $$ where $a,b>0$ and $f\colon[0,a]\to[0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation $$ f(m_1(x+y))=m_2(f(x)+f(y)), $$ where $m_1,m_2$ are functions defined on some intervals of ${\mathbb R}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.
References:
[1] Baczyński, M.: On a class of distributive fuzzy implications. Int. J. Uncertain. Fuzziness Knowledge-Based Systems 9 (2001), 229-238. DOI 10.1142/S0218488501000764 | MR 1821991 | Zbl 1113.03315
[2] Baczyński, M.: On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms. Fuzzy Sets and Systems 161 (2010), 1406-1419. MR 2606422 | Zbl 1204.03029
[3] Baczyński, M.: On the distributivity of fuzzy implications over representable uninorms. Fuzzy Sets and Systems 161 (2010), 2256-2275. MR 2658032 | Zbl 1252.03046
[4] Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing 231, Springer, Berlin Heidelberg 2008. Zbl 1293.03012
[5] Baczyński, M., Jayaram, B.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. IEEE Trans. Fuzzy Syst. 17 (2009), 590-603. DOI 10.1109/TFUZZ.2008.924201
[6] Baczyński, M., Qin, F.: Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms. Int. J. Approx. Reason. 54 (2013), 290-296. DOI 10.1016/j.ijar.2012.10.001 | MR 3021572 | Zbl 1280.03029
[7] Baczyński, M., Szostok, T., Niemyska, W.: On a functional equation related to distributivity of fuzzy implications. In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ IEEE 2013) Hyderabad 2013, pp. 1-5.
[8] Balasubramaniam, J., Rao, C. J. M.: On the distributivity of implication operators over T and S norms. IEEE Trans. Fuzzy Syst. 12 (2004), 194-198. DOI 10.1109/TFUZZ.2004.825075
[9] Combs, W. E., Andrews, J. E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 6 (1998), 1-11. DOI 10.1109/91.660804
[10] Combs, W. E.: Author's reply. IEEE Trans. Fuzzy Syst. 7 (1999), 371-373. DOI 10.1109/TFUZZ.1999.771094
[11] Combs, W. E.: Author's reply. IEEE Trans. Fuzzy Syst. 7 (1999), 477-478. DOI 10.1109/TFUZZ.1999.784215
[12] Baets, B. De: Fuzzy morphology: A logical approach. In: Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, and Neural Network Approach (B. M. Ayyub and M. M. Gupta, eds.), Kluwer Academic Publishers, Norwell 1997, pp. 53-68. Zbl 1053.03516
[13] Dick, S., Kandel, A.: Comments on ``Combinatorial rule explosion eliminated by a fuzzy rule configuration". IEEE Trans. Fuzzy Syst. 7 (1999), 475-477. DOI 10.1109/91.784213
[14] González-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D.: Fuzzy hit-or-miss transform using the fuzzy mathematical morphology based on T-norms. In: Aggregation Functions in Theory and in Practise (H. Bustince et al., eds.), Advances in Intelligent Systems and Computing 228, Springer, Berlin - Heidelberg 2013, pp. 391-403. Zbl 1277.68283
[15] Jayaram, B.: Rule reduction for efficient inferencing in similarity based reasoning. Int. J. Approx. Reason. 48 (2008), 156-173. DOI 10.1016/j.ijar.2007.07.009 | MR 2420665 | Zbl 1184.68511
[16] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. MR 1790096 | Zbl 1087.20041
[17] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality. Państwowe Wydawnictwo Naukowe (Polish Scientific Publishers) and Uniwersytet Śląski, Warszawa-Kraków-Katowice 1985. MR 0788497 | Zbl 1221.39041
[18] Ling, C. H.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189-212. MR 0190575 | Zbl 0137.26401
[19] Mendel, J. M., Liang, Q.: Comments on ``Combinatorial rule explosion eliminated by a fuzzy rule configuration". IEEE Trans. Fuzzy Syst. 7 (1999), 369-371. DOI 10.1109/91.771093
[20] Qin, F., Baczyński, M., Xie, A.: Distributive equations of implications based on continuous triangular norms (I). IEEE Trans. Fuzzy Syst. 20 (2012), 153-167. DOI 10.1109/TFUZZ.2011.2171188
[21] Qin, F., Yang, L.: Distributive equations of implications based on nilpotent triangular norms. Int. J. Approx. Reason. 51 (2010), 984-992. DOI 10.1016/j.ijar.2010.07.005 | MR 2719614 | Zbl 1226.03036
[22] Ruiz-Aguilera, D., Torrens, J.: Distributivity of strong implications over conjunctive and disjunctive uninorms. Kybernetika 42 (2006), 319-336. MR 2253392 | Zbl 1249.03030
[23] Ruiz-Aguilera, D., Torrens, J.: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets and Systems 158 (2007), 23-37. MR 2287424 | Zbl 1114.03022
[24] Trillas, E., Alsina, C.: On the law $[(p\wedge q)\to r]=[(p\to r)\vee(q\to r)]$ in fuzzy logic. IEEE Trans. Fuzzy Syst. 10 (2002), 84-88.
Search
Partner of
