Introduction to the concept of recursiveness of fuzzy functions. (English) Zbl 0639.03044
The hypothesis that the class of recursive functions is identical with the one of functions being computable by a Turing machine is known as “Church’s soft hypothesis”. The main purpose of this article is to extend this hypothesis to the fuzzy calculus. Let \(W=([0,1],\oplus,\otimes,\leq)\) be an ordered semiring. The fuzzy functions are considered as associating an output to an input with membership degree belonging to W. For such W-functions the concepts of computability and recursiveness are extended, obtaining the so-called W- computability and W-recursiveness, respectively. Some properties of W- recursive functions are analyzed and the equivalence between W-recursive and W-computable functions is shown.
Reviewer: S.Miura
MSC:
| 03D20 | Recursive functions and relations, subrecursive hierarchies |
| 03B52 | Fuzzy logic; logic of vagueness |
References:
| [1] | Chang, C. L., Interpretation and execution of fuzzy programs, (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura, M., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (1975), Academic Press: Academic Press New York), 191-218 · Zbl 0322.68057 |
| [2] | Chang, S. K., On the execution of fuzzy program using finite-state machines, IEEE Trans. Comput., 21, 241-253 (1972) · Zbl 0243.68009 |
| [3] | Church, A., An unsolvable problem of elementary number theory, Amer. J. Math., 58, 345-363 (1936) · JFM 62.0046.01 |
| [4] | Clares, B., Una introduccion a la \(W\)-calculabilidad: Operaciones basicas, Stochastica, 7, 111-135 (1983) · Zbl 0557.03026 |
| [5] | Clares, B.; Delgado, M., On the \(W\)-computability of fuzzy predicates (1985), IFSA |
| [6] | Kleene, S. C., General recursive functions of natural numbers, Math. Ann., 112, 340-353 (1936) · Zbl 0014.19402 |
| [7] | Ostasiewicz, W., A new approach to fuzzy programming, Fuzzy Sets and Systems, 7, 139-152 (1982) · Zbl 0474.68007 |
| [8] | Post, E., Finite combinatory processes — formulation I, J. Symbolic Logic, 1, 103-105 (1936) · JFM 62.1060.01 |
| [9] | Prade, H., Fuzzy programming: Why and how?, Busefal, 5, 76-89 (1981) |
| [10] | Santos, E. S., Fuzzy and probabilistic programs, (Gupta, M. M.; Saridis, G. N.; Gaines, B. R., Fuzzy Automata and Decision Processes (1977), North-Holland: North-Holland Amsterdam), 133-148 · Zbl 0378.68035 |
| [11] | Tanaka, K.; Mizumoto, M., Fuzzy programs and their execution, (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura, M., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (1975), Academic Press: Academic Press New York) · Zbl 0314.68008 |
| [12] | Turing, A. M., On computable numbers, with an application to the entscheidungproblem, (Proc. London Math. Soc., 42 (1936)), 230-265, (2) · Zbl 0016.09701 |
| [13] | Vila, M. A.; Clares, B., On the \(W\)-computability of fuzzy functions (1984), FISAL |
| [14] | Zadeh, L. A., Fuzzy algorithms, Inform. and Control, 12, 94-102 (1968) · Zbl 0182.33301 |
| [15] | Santos, E. S., Computability by probabilistic Turing Machines, Trans. Amer. Math. Soc., 159, 165-184 (1971) · Zbl 0246.02030 |
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