×

A fuzzy logic system based on Schweizer-Sklar t-norm. (English) Zbl 1122.03025

Summary: Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL* is established, and its soundness theorem and completeness theorem are proved. The following facts are pointed out: the well-known formal system SBL\(_\sim\) is a semantic extension of UL*; the fuzzy logic system IMTL\(_\Delta\) is a special case of UL* when two negations in UL* coincide. Moreover, the connections between the system UL* and some fuzzy logic formal systems are investigated. Finally, starting from the concepts of “the strength of an ‘AND’ operator” by R. R. Yager and “the strength of fuzzy rule interaction” by T. Whalen, the essential meaning of a parameter \(p\) in UL* is explained and the use of the fuzzy logic system UL* in approximate reasoning is presented.

MSC:

03B52 Fuzzy logic; logic of vagueness
68T37 Reasoning under uncertainty in the context of artificial intelligence
68T27 Logic in artificial intelligence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hajek, P., Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, 1998.
[2] Esteva, F., Godo, L., Monoidal t-norm based logic: Towards a logic for left-continous t-norms, Fuzzy Sets and Systems, 2001, 124: 271–288. · Zbl 0994.03017 · doi:10.1016/S0165-0114(01)00098-7
[3] Ying, M. S., Perturbation of fuzzy reasoning, IEEE Trans. Fuzzy Sys., 1999, 7: 625–629. · doi:10.1109/91.797985
[4] Ying, M. S., Implications operators in fuzzy logic, IEEE Trans. Fuzzy Sys., 2002, 10: 88–91. · doi:10.1109/91.983282
[5] Wang, G. J., Non-Classical Mathematical Logic and Approximate Reasoning (in Chinese), Beijing: Science Press, 2000.
[6] Wang, G. J., The full implicational Triple I method for fuzzy reasoning, Science in China (in Chinese), Series E, 1999, 29(1): 43–53.
[7] Wang, G. J., Song, Q.Y., A new kind of Triple I method and its logic foundation, Progress in Natural Science, 2003, 13(6): 575–581.
[8] Klement, E. P., Navara, M., Propositional Fuzzy Logics Based on Frank t-norms: A Comparison in Fuzzy Sets, Logics and Reasoning About Knowledge (eds. Dubois, D. et al.), Kluwer Academic Publishers, 1999. · Zbl 0947.03035
[9] Wu, W. M., Generalized tautologies in parametric Kleene’s systems, Fuzzy Systems and Mathematics, 2000, 14(1): 1–7. · Zbl 0941.26014 · doi:10.1016/S0165-0114(97)00353-9
[10] Wang, G. J., Lan, R., Generalized tautologies of the systems H {\(\alpha\)}, J. of Shaanxi Normal University (Natural Science Edition), 2003, 31(2): 1–11. · Zbl 1037.03509
[11] Whalen, T., Parameterized R-implications, Fuzzy Sets and Systems, 2003, 134: 231–281. · Zbl 1010.03018 · doi:10.1016/S0165-0114(02)00215-4
[12] Wang, G. J., Formalized theory of general fuzzy reasoning, Information Sciences, 2004, 160: 251–266. · Zbl 1076.68091 · doi:10.1016/j.ins.2003.09.004
[13] Batyrshin, I., Kaynak, O., Rudas, I., Fuzzy modeling based on generalized conjunction operations, IEEE Transactions on Fuzzy Systems, 2002, 10(5): 678–683. · doi:10.1109/TFUZZ.2002.803500
[14] Song, Q., Kandel, A., Schneider, M., Parameterized fuzzy operators in fuzzy decision making, International Journal of Intelligent Systems, 2003, 18: 971–987. · Zbl 1069.68603 · doi:10.1002/int.10124
[15] Rutkowski, L., Cpalka, K., Flexible neuro-fuzzy systems, IEEE Transactions on Neural Networks, 2003, 14(3): 554–574. · doi:10.1109/TNN.2003.811698
[16] Bubnicki, Z., Uncertain Logics, Variables and Systems, Berlin: Springer, 2002. · Zbl 1005.93001
[17] Schweizar, B., Sklar, A., Associative functions and abstract semigroups, Pub. Math. Debrecen, 1963, 10: 69–81.
[18] Schweizar, B., Sklar, A., Associative functions and statistical triangle inequalities, Pub. Math. Debrecen, 18 1961, 8: 169–186. · Zbl 0107.12203
[19] Klement, E. P., Mesiar, R., Pap, E., Triangular Norms, Volume 8 of Trends in Logic, Dordrecht: Kluwer Academic Publishers, 2000. · Zbl 0972.03002
[20] Esteva, F., Godo, L., Hajek, P., Navara, M., Residuated fuzzy logic with an involutive negation, Arch. Math. Logic, 2000, 39: 103–124. · Zbl 0965.03035 · doi:10.1007/s001530050006
[21] Yager, R. R., Generalized ”AND/OR” operators for multivalued and fuzzy logic, Int. Sym. on Multiple-valued logic (IEEE), 1980, 214–218. · Zbl 0539.03010
[22] Hajek, P., Observations on non-commutative fuzzy logic, Soft Computing, 2003, 8: 38–43.
[23] Jenei, S., Montagna, F., A proof of standard completeness for non-commutative monoidal t-norm logic, Neural Network World, 2003, 5: 481–489.
[24] Cheng, Y. Y., An approach to fuzzy operators (Part I), Fuzzy Mathematics, 1982, 2: 1–10.
[25] Ying, M. S., On fuzzy degrees of fuzzy operators, Fuzzy Mathematics, 1984, 4: 1–6. · Zbl 0597.94032
[26] Pei, D. W., Wang, G. J., The completeness and applications of the formal system L *, Science in China (in Chinese), Series E, 2002, 32(1): 56–64.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.