In this paper, fuzzy Turing machines with membership degrees in distributive lattices, which are called lattice-valued fuzzy Turing machines, are studied. First several formulations of lattice-valued fuzzy Turing machines, including in particular deterministic and nondeterministic lattice-valued fuzzy Turing machines (l-DTMcs and l-NTMs), are given. It is shown that l-DTMcs and l-NTMs are not equivalent as the acceptors of fuzzy languages. This contrasts sharply with classical Turing machines. Second, it is shown that lattice-valued fuzzy Turing machines can recognize n-r.e. sets in the sense of Bedregal and Figueira, the super-computing power of fuzzy Turing machines is established in the lattice-setting. Third, it is demonstrated that the truth-valued lattice being finite is a necessary and sufficient condition for the existence of a universal lattice-valued fuzzy Turing machine. For an infinite distributive lattice with a compact metric, it is declared that a universal fuzzy Turing machine exists in an approximate sense. This means, for any prescribed accuracy, there is a universal machine that can simulate any lattice-valued fuzzy Turing machine on it with the given accuracy.
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