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Parameterized R-implications. (English) Zbl 1010.03018

This is a very extensive paper focusing on the wide class of fuzzy implications called R-implications because they appear in residuated lattices on the basis of the residuation property \[ a\otimes b\leq c \qquad\text{ iff }\qquad a\leq b\to c, \] where \(\otimes\) is a binary monoidal operation (product) and \(\to\) is the corresponding residuation. Note that the residuation is an algebraic counterpart to the classical modus ponens rule in many-valued logic (unfortunately, these facts are not mentioned in the paper). The interval \([0,1]\) can be seen as a residuated lattice if the product operation \(\otimes\) is a left-continuous t-norm (a binary operation on \([0,1]\) which is commutative, associative, monotone, with 1 as the unit). The author focuses on a class of R-implications being residuations of the Schweizer-Sklar family of t-norms, which is a parametrized family of continuous t-norms (with the exception of the parameter \(p=\infty\)). The author considers fuzzy IF-THEN rules interpreted as fuzzy relations based on the above family of R-implications provided that the antecedent and succedent form trapezoids. Furthermore, he derives a possible defuzzification operation on the basis of that. He then makes a lot computations with not very clearly stated goal. Moreover, he ignores results of formal fuzzy logic, which can be seen from the citations but also from some formulations and terminology (e.g. the product logic is hidden under the unused term quotient or Goguen logic). A lot of useless tables and unclear formulations could be avoided or simplified when clearly referring to results of formal fuzzy logic in the narrow sense and the theory of t-norms; cf. P. Hájek [Metamathematics of fuzzy logic. Kluwer, Dordrecht (1998; Zbl 0937.03030)], E. P. Klement, R. Mesiar and E. Pap [Triangular norms. Kluwer, Dordrecht (2000; Zbl 0972.03002)], V. Novák, I. Perfilieva and J. Močkoř Mathematical principles of fuzzy logic. Kluwer, Dordrecht (1999; Zbl 0940.03028)].

MSC:

03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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[1] Bandler, W.; Kohout, L., Semantics of implication operators and fuzzy relational products, (Mamdani, E. H.; Gaines, B. R., Fuzzy Reasoning and Its Applications (1981), Academic Press: Academic Press London) · Zbl 0435.68042
[2] Bonissone, P., Summarizing and propagating uncertain information with triangular norms, Internat. J. Approximate Reasoning, 1, 1, 71-102 (1987)
[3] Cao, Z.; Kandel, A., Applicability of some fuzzy implication operators, Fuzzy Sets and Systems, 31, 151-186 (1989)
[4] De Beets, B.; Kerre, E. E., The generalized modus ponens and the triangular fuzzy data model, Fuzzy Sets and Systems, 59, 1, 305-317 (1993) · Zbl 0799.03021
[5] Dubois, D.; Prade, H., Fuzzy sets in approximate reasoning, Part 1Inference with possibility distributions, Fuzzy Sets and Systems, 40, 1, 143-202 (1991) · Zbl 0722.03017
[6] Dubois, D.; Prade, H.; Ughetto, L., Checking the coherence and redundancy of fuzzy knowledge bases, IEEE Trans. Fuzzy Systems, 5, 3, 398-417 (1997)
[7] El Hajjaji, A.; Rachid, A., Explicit formulas for fuzzy controller, Fuzzy Sets and Systems, 62, 135-141 (1994)
[8] Hall, L., The choice of ply operator in fuzzy intelligent systems, Fuzzy Sets and Systems, 34, 2, 135-144 (1990) · Zbl 0689.68108
[9] Kiszka, J.; Kochanski, M.; Sliwinska, D., The influence of some fuzzy implication operators on the accuracy of a fuzzy modelPart I, Fuzzy Sets and Systems, 15, 111-128 (1985) · Zbl 0581.94033
[10] Kiszka, J.; Kochanski, M.; Sliwinska, D., The influence of some fuzzy implication operators on the accuracy of a fuzzy modelPart II, Fuzzy Sets and Systems, 15, 223-240 (1985) · Zbl 0581.94034
[11] Martin-Clouaire, R., Semantics and computation of the generalized modus ponensthe long paper, Internat. J. Approximate Reasoning, 3, 195-217 (1989) · Zbl 0689.94006
[12] Mizumoto, M., Fuzzy inference using max-∧ composition in the compositional rule of inference, (Gupta, M.; Sanchez, E., Approximate Reasoning in Decision Analysis (1982), North-Holland: North-Holland Amsterdam) · Zbl 0503.94032
[13] M. Mizumoto, Comparison of various fuzzy reasoning methods, Proc. Second International Fuzzy Systems Association Conf., 1987.; M. Mizumoto, Comparison of various fuzzy reasoning methods, Proc. Second International Fuzzy Systems Association Conf., 1987.
[14] B. Schott, T. Whalen, Sources of error in fuzzy inference, Proc. 1990 North American Fuzzy Information Processing Society Conference, 1990, pp. 223-226.; B. Schott, T. Whalen, Sources of error in fuzzy inference, Proc. 1990 North American Fuzzy Information Processing Society Conference, 1990, pp. 223-226.
[15] B. Schott, T. Whalen, Analysis of error in fuzzy inference, Proc. 1991 North American Fuzzy Information Processing Society Conference, 1991, pp. 78-81.; B. Schott, T. Whalen, Analysis of error in fuzzy inference, Proc. 1991 North American Fuzzy Information Processing Society Conference, 1991, pp. 78-81.
[16] Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Pub. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203
[17] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Pub. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001
[18] Trillas, E.; Valverde, L., On mode and inference in approximate reasoning, (Gupta, M. M.; etal., Approximate Reasoning in Expert Systems (1985), Elsevier, North-Holland: Elsevier, North-Holland Amsterdam) · Zbl 0546.03015
[19] E. Trillas, L. Valverde, On mode and indistinguishability in the setting of fuzzy logic, in: J. Kacprzyk, R. Yager (Eds.), Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, Verlag TUV Rhineland, 1985.; E. Trillas, L. Valverde, On mode and indistinguishability in the setting of fuzzy logic, in: J. Kacprzyk, R. Yager (Eds.), Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, Verlag TUV Rhineland, 1985.
[20] Trunen, E., On fuzzy intuitionistic logic, (Zadeh, L.; Kacprzyk, J., Fuzzy Logic for the Management of Uncertainty (1992), Wiley: Wiley New York)
[21] Uehara, K.; Fujise, M., Fuzzy inference based on families of a-level sets, IEEE Trans. Fuzzy Systems, 1, 2, 111-124 (1993)
[22] L. Ughetto, D. Dubois, H. Prade, Efficient inference procedures with fuzzy inputs, Proc. Sixth IEEE Internat. Conf. on Fuzzy Systems, Vol. 1, 1997, pp. 567-572.; L. Ughetto, D. Dubois, H. Prade, Efficient inference procedures with fuzzy inputs, Proc. Sixth IEEE Internat. Conf. on Fuzzy Systems, Vol. 1, 1997, pp. 567-572. · Zbl 1038.68115
[23] T. Whalen, Consensus and selectivity in fuzzy rule based interpolation, Proc. N. Amer. Fuzzy Information Processing Society, 1994.; T. Whalen, Consensus and selectivity in fuzzy rule based interpolation, Proc. N. Amer. Fuzzy Information Processing Society, 1994.
[24] Whalen, T., Exact solutions for interacting rules in generalized modus ponens with parameterized implication operators, Inform. Sci.: Intell. Systems, 92, 1, 211-232 (1996) · Zbl 0882.03021
[25] T. Whalen, Mixed-logic interpolation of differentiable functions, Proc. 1998 IEEE World Congr. on Computational Intelligence, 1998, pp. 961-966.; T. Whalen, Mixed-logic interpolation of differentiable functions, Proc. 1998 IEEE World Congr. on Computational Intelligence, 1998, pp. 961-966.
[26] Whalen, T.; Gim, G.; Schott, B., Control of error in fuzzy logic modeling, Fuzzy Sets and Systems, 80, 1, 23-35 (1996)
[27] Whalen, T.; Schott, B., Issues in fuzzy production systems, Internat. J. Man-Machine Stud., 19, 57-71 (1983)
[28] Whalen, T.; Schott, B., Alternative logics for approximate reasoninga comparative study, Internat. J. Man-Machine Stud., 22, 327-346 (1985)
[29] Whalen, T.; Schott, B., Presumption and prejudice in logical inference, Internat. J. Approximate Reasoning, 3, 5, 359-382 (1989)
[30] Whalen, T.; Schott, B., Usuality, regularity, and fuzzy set logic, Internat. J. Approximate Reasoning, 6, 4, 481-504 (1992) · Zbl 0757.03014
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