×

A theory of approximate reasoning with type-2 fuzzy set. (English) Zbl 1473.03013

Summary: In this paper, an attempt is made to study approximate reasoning based on a Type-2 fuzzy set theory. In the process, we have examined the underlying fuzzy logic structure on which the reasoning is formulated. We have seen that the partial/incomplete/imprecise truth-values of elements of a type-2 fuzzy set under consideration forms a lattice. We propose two new lattice operations which ultimately help us to define a residual and thereby making the structure of truth-values a residuated lattice. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Our proposal is illustrated with typical artificial examples.

MSC:

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Castillo, O., Angulor, L.A., Castro, J.R. and Valdez, M.G., “A comparative study of type-1 fuzzy logic sys- tems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems”, Information Sciences, 354, (2016), 257-274.
[2] Galatos, N., Residuated lattices: An algebraic glimpse at substructural logics, Elsevier, 2007. · Zbl 1171.03001
[3] Hajek, P., Metamathematics of fuzzy logic, Kluwer Academic Press, The Netherlands, 1998. · Zbl 0937.03030
[4] John, R.I., “Type-2 fuzzy sets: an appraisal of theory and applications”, International Journal of Uncer- tainty, Fuzziness and Knowledge-Based Systems, 6(6), (1998), 563-576. · Zbl 1087.68639
[5] Liang, Q., Karnik,N.N. and Mendel, J.M., “Type-2 fuzzy logic systems”, IEEE Transactions on Fuzzy Systems, 7(6), (1999), 643-658.
[6] Mendel, J. and John, R. , “Type-2 fuzzy sets made simple”, IEEE Transactions on Fuzzy Systems, 10(2), (2002), 117-127.
[7] Mizumoto, M. and Kokichi, T., “Fuzzy-Fuzzy Automata”, Kybernetes, 5, (1976), 107-112. · Zbl 0334.94020
[8] Mizumoto, M. and Kokichi, T., “Some properties of fuzzy sets of Type-2”, Information and Control, 31, (1976), 312-340. · Zbl 0331.02042
[9] Mizumoto, M. and Kokichi, T., “Fuzzy Sets of Type-2 under Algebraic Product and Algebraic Sum”, Fuzzy Sets and System, 5, (1981), 277-290. · Zbl 0457.04005
[10] Morgan, W. and Dilworth, R.P., “Residuated lattices”, Transactions on American Mathematical Society, 45, (1939), 335-354. · Zbl 0021.10801
[11] Pavelka, J.P., “On fuzzy logic Part I”, Zeistchr. f. math. Logik and Grundlagun d. Math. Bd, 25, (1979), 45-52. · Zbl 0435.03020
[12] Pavelka, J.P., “On fuzzy logic Part II”, Zeistchr. f. math. Logik and Grundlagun d. Math. Bd, 25, (1979), 119-134. · Zbl 0446.03015
[13] Pavelka, J.P., “On fuzzy logic Part III”, Zeistchr. f. math. Logik and Grundlagun d. Math. Bd, 25, (1979), 447-464. · Zbl 0446.03016
[14] Walker, C.L. and Walker, E.A., “The algebra of fuzzy truth values”, International Journal of Approximate Reasoning, 149, (2005), 309-347. · Zbl 1064.03020
[15] Walker, C.L. and Walker, E.A., “Automorphisms of the algebra of fuzzy truth values”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 14, (2006), 711-732. · Zbl 1114.03053
[16] Walker, C.L. and Walker, E.A., “Sets with type-II operations”, International Journal of Approximate Rea- soning, 50, (2009), 63-71. · Zbl 1193.03079
[17] Zadeh, L.A., “Fuzzy sets”, Information and Control, 8, (1965), 338-353. · Zbl 0139.24606
[18] Zadeh, L.A., “The concept of linguistic variable and its application to approximate reasoning I”, Information Sciences, 8, (1975), 199-249. · Zbl 0397.68071
[19] Zadeh, L.A., “A computational approach to fuzzy quantifiers in natural languages”, Computers and Mathe- matics with applications, 9, (1983), 149-184. · Zbl 0517.94028
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.