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    Many-one reducibility implies Turing reducibility — Carmelics
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    Supports→K is Turing complete among the computably enumerable sets

    Many-one reducibility implies Turing reducibility

    Modality & PossibilityTruth & Knowledge
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    K is Turing complete among the computably enumerable setsK is many-one complete among the c.e. setsTherefore any c.e. set is Turing reducible to K

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    Many-one reducibility implies Turing reducibility (A ≤_m B implies A ≤...93%The structure of reducibility among these problems yields at least one...80%The axiom of reducibility is required in Principia Mathematica to allo...78%Mathematical analysis would collapse if the axiom of reducibility is a...78%

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    It is easy to see that \(A \leq_m B\) implies \(A \leq_T B\). (For if \(f(x)\) is a \(m\)-reduction of \(A\) to \(B\), then consider the program which first computes \(f(n)\) and then, using \(B\) an as oracle, checks if \(f(n) \in B\), outputting 1 if so and 0 if not.) It thus follows that \(K\) is also Turing complete—i.e., it embodies the maximum “degree of unsolvability” among the the c.e. sets when this notion is understood in terms of Turing reducibility as well as many-one reducibility.

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