## On the compositional rule of inference under triangular norms.(English)Zbl 1018.03511

Summary: The aim of this paper is to provide a close upper bound of the membership function for the compositional rule of inference under an Archimedean t-norm, where both the observation and the relation parts are given by Hellendoorn’s $$\phi$$-function [H. Hellendoorn, Fuzzy Sets Syst. 35, No. 2, 163-183 (1990; Zbl 0704.03006)]. In particular, if the left and right spreads of the observation part are the same as those of the relation part, then this upper bound is the exact membership function, which generalizes the earlier result of R. Fullér and H.-J. Zimmermann [Fuzzy Sets Syst. 51, No. 3, 267-275 (1992; Zbl 0782.68110)] in that the assumption of twice differentiability is deleted.

### MSC:

 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc.

### Citations:

Zbl 0704.03006; Zbl 0782.68110
Full Text:

### References:

 [1] Chung, K. L., A Course in Probability Theory (1974), Academic Press: Academic Press New York · Zbl 0159.45701 [2] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262 [4] Fullér, R.; Keresztfalvi, T., t-Norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155-159 (1992) [5] Fullér, R.; Zimmermann, H. J., On computation of the compositional rule of inference under triangular norms, Fuzzy Sets and Systems, 51, 267-275 (1992) · Zbl 0782.68110 [6] Hellendoorn, H., Closure properties of the compositional rule of inference, Fuzzy Sets and Systems, 35, 163-183 (1990) · Zbl 0704.03006 [7] Zadeh, L. A., The concept of linguistic variable and its applications to approximate reasoning, Part III, Inform. Sci., 9, 43-80 (1975) · Zbl 0404.68075
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