δ-Uniformly decidable sets and turing machines
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We give a characterization of the archimedean fields in which nontrivial δ-uniform decidable sets exist. More exactly, after we introduce a notion of Turing closure of an archimedean field we prove that such a field posseses nontrivial δ-uniformly decidable sets if and only if it is not Turing closed. Moreover, if a function is δ-uniformly computable on a Turing closed field then it is rational over each of the connected components induced on the halting set by the reals. Finally, given a field which is not Turing closed, we obtain as a consequence that there exists a δ-uniform machine computing a total function which is not rational.
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