×

zbMATH — the first resource for mathematics

Extended triangular norms. (English) Zbl 1165.03010
A triangular norm \(T\) (a non-decreasing commutative associative binary operation on \([0,1]\) with neutral element \(e= 1\)) is extended to act on fuzzy truth values (special fuzzy subsets of \([0,1]\)) based on a (possibly different) t-norm \(T_*\). Analytical forms of extended t-norms \(T\) based on \(T_*\) are discussed for several special cases. First of all, the case \(T=\min\) is discussed. Next, the extended continuous \(T\) based on \(T_*= \min\) is considered. For \(T= T_L\) (Łukasiewicz t-norm given by \(T_L(x,y)= \max(0,x+ y-1)\)), the case of \(T_*\) being a continuous Archimedean t-norm is studied. Also, the case \(T= T_*= \pi\) (product t-norm) is involved. Finally, extensions of continuous triangular norms based on the drastic product t-norm \(T_D\) (\(T_D(x,y)= 0\) whenever \((x,y)\in [0,1[^2\)) is considered. The introduced results may serve to develop new families of type-2 fuzzy logic systems.

MSC:
03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Doctor, F.; Hagras, H.; Callaghan, V., A type-2 fuzzy embedded agent to realize ambient intelligence in ubiquitous computing environments, Information sciences, 171, 309-334, (2005)
[2] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press, Inc. New York · Zbl 0444.94049
[3] Fuller, R.; Keresztfalvi, T., T-norm based addition of fuzzy intervals, Fuzzy sets and systems, 51, 155-159, (1992)
[4] Hagras, H.A., A hierarchical type-2 fuzzy logic control architecture for autonomous robots, IEEE transactions on fuzzy systems, 12, 4, 524-539, (2004)
[5] Karnik, N.N.; Mendel, J.M., Operations on type-2 fuzzy sets, Fuzzy sets and systems, 122, 327-348, (2000) · Zbl 1010.03047
[6] Karnik, N.N.; Mendel, J.M.; Liang, Q., Type-2 fuzzy logic systems, IEEE transactions on fuzzy systems, 7, 6, 643-658, (1999)
[7] Kawaguchi, M.F.; Miyakoshi, M., Extended t-norms as logical connectives of fuzzy truth values, Multiple valued logic, 8, 1, 53-69, (2002) · Zbl 1024.03026
[8] Klement, E.P.; Mesiar, R.; Pap, E., Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms, Fuzzy sets and systems, 104, 3-13, (2000) · Zbl 0953.26008
[9] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002
[10] Mendel, J.M., Advances in type-2 fuzzy sets and systems, Information sciences, 177, 84-110, (2007) · Zbl 1117.03060
[11] Mendel, J.M., Uncertain rule-based fuzzy logic systems: introduction and new directions, (2001), Prentice-Hall PTR Upper Saddle River, NJ · Zbl 0978.03019
[12] Mesiar, R., Triangular-norm-based addition of fuzzy intervals, Fuzzy sets and systems, 91, 231-237, (1997) · Zbl 0919.04011
[13] Mizumoto, M.; Tanaka, K., Some properties of fuzzy sets of type-2, Information and control, 31, 4, 312-340, (1976) · Zbl 0331.02042
[14] Nguyen, H.T., A note on the extension principle for fuzzy sets, Journal of mathematical analysis and applications, 64, 369-380, (1978) · Zbl 0377.04004
[15] Sepúlveda, R.; Castillo, O.; Melin, P.; Rodrı´ guez-Dı´az, A., Experimental study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic, Information sciences, 177, 2023-2048, (2007)
[16] Starczewski, J.T., A triangular type-2 fuzzy logic system, (), 7231-7238
[17] J.T. Starczewski, L. Rutkowski, Neuro-fuzzy systems of type-2, in: 1st International Conference on Fuzzy Systems and Knowledge Discovery, vol. 2, Singapore, 2002, pp. 458-462. · Zbl 1057.68669
[18] J.T. Starczewski, Extended triangular norms on Gaussian fuzzy sets, in: EUSFLAT-LFA Conference, Barcelona, Spain, 2005, pp. 872-877.
[19] Walker, C.; Walker, E., The algebra of fuzzy truth values, Fuzzy sets and systems, 2, 309-347, (2005) · Zbl 1064.03020
[20] D. Wu, W. Tan, Type-2 fuzzy logic controller for the liquid-level process, in: Proceedings of the IEEE-FUZZ 2004, Budapest, Hungary, 2004, pp. 248-253.
[21] Uncu, O.; Turksen, I.B., Discrete interval type 2 fuzzy system models using uncertainty in learning parameters, IEEE transactions fuzzy systems, 15, 1, 90-106, (2007)
[22] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning—I, Information sciences, 8, 199-249, (1975) · Zbl 0397.68071
[23] Zadeh, L.A., Toward a generalized theory of uncertainty (GTU)—an outline, Information sciences, 172, 1-40, (2005) · Zbl 1074.94021
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.