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Approximation and universality of fuzzy Turing machines

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Abstract

Fuzzy Turing machines are the formal models of fuzzy algorithms or fuzzy computations. In this paper we give several different formulations of fuzzy Turing machine, which correspond to nondeterministic fuzzy Turing machine using max-⋆ composition for some t-norm ⋆ (or NFTM⋆, for short), nondeterministic fuzzy Turing machine (or NFTM), deterministic fuzzy Turing machine (or DFTM), and multi-tape versions of fuzzy Turing machines. Some distinct results compared to those of ordinary Turing machines are obtained. First, it is shown that NFTM⋆, NFTM, and DFTM are not necessarily equivalent in the power of recognizing fuzzy languages if the t-norm ⋆ does not satisfy finite generated condition, but are equivalent in the approximation sense. That is to say, we can approximate an NFTM⋆ by some NFTM with any given accuracy; the related constructions are also presented. The level characterization of fuzzy recursively enumerable languages and fuzzy recursive languages are exploited by ordinary r.e. languages and recursive languages. Second, we show that universal fuzzy Turing machine exists in the approximated sense. There is a universal fuzzy Turing machine that can simulate any NFTM⋆ on it with a given accuracy.

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Authors and Affiliations

  1. College of Computer Science, Shaanxi Normal University, Xi’an, 710062, China

    YongMing Li

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  1. YongMing Li
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Correspondence to YongMing Li.

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Supported by the National Natural Science Foundation of China (Grant No. 10571112), and “TRAPOYT” of China and the National 973 Foundation Research Program (Grant No. 2002CB312200)

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Li, Y. Approximation and universality of fuzzy Turing machines. Sci. China Ser. F-Inf. Sci. 51, 1445–1465 (2008). https://doi.org/10.1007/s11432-008-0089-y

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  • DOI: https://doi.org/10.1007/s11432-008-0089-y

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