On computation of the compositional rule of inference under triangular norms. (English) Zbl 0782.68110

Summary: The paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under Archimedean \(t\)-norms, when both the observation and the relation parts are given by H. Hellendoorn’s [Fuzzy Sets Syst. 35, No. 2, 163-183 (1990; Zbl 0704.03006)] \(\varphi\)-function.


68T35 Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence


Zbl 0704.03006
Full Text: DOI


[1] Da, R., A critical study of widely used fuzzy implication operators and their influence on the inference rules in fuzzy expert systems, (Ph.D. Thesis (1990), State University of Ghent)
[2] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[3] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262
[4] Dubois, D.; Martin-Clouarie, R.; Prade, H., Practical computing in fuzzy logic, (Gupta, M. M.; Yamakawa, T., Fuzzy Computing: Theory, Hardware, and Applications (1988), North-Holland: North-Holland Amsterdam), 11-34
[5] Fullér, R.; Kereszfalvi, T., t-Norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155-159 (1992)
[6] Hellendoorn, H., Closure properties of the compositional rule of inference, Fuzzy Sets and Systems, 35, 163-183 (1990) · Zbl 0704.03006
[7] Hellendoorn, H., The generalized modus ponens considered as a fuzzy relation, Fuzzy Sets and Systems, 46, 29-48 (1992) · Zbl 0773.03016
[8] Martin-Clouaire, R., Semantics and computation of the generalized modus ponens: The long paper, Internat. J. Approximate Reasoning, 3, 195-217 (1989) · Zbl 0689.94006
[9] Margrez, P.; Smets, P., Fuzzy modus ponens: A new model suitable for applications in knowledge-based systems, Internal. J. Intelligent Systems, 4, 181-200 (1989) · Zbl 0672.03010
[10] Mizumoto, M.; Zimmermann, H.-J., Comparison of fuzzy reasoning methods, Fuzzy Sets and Systems, 8, 253-283 (1982) · Zbl 0501.03013
[11] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001
[12] Yager, R. R., Approximate reasoning as a basis for rule-based expert systems, IEEE Trans. Systems Man Cybernet., 14, 636-643 (1984) · Zbl 0555.68066
[13] Zadeh, L. A., The role of fuzzy logic in the management of uncertainty in expert systems, Fuzzy Sets and Systems, 11, 199-228 (1983) · Zbl 0553.68049
[14] Zadeh, L. A., The concept of linguistic variable and its applications to approximate reasoning, Parts I, II, III, Inform. Sci., 9, 43-80 (1975) · Zbl 0404.68075
[15] Zimmermann, H.-J., Fuzzy Sets, Decision Making and Expert Systems (1987), Reidel: Reidel Dordrecht-Boston
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.