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  "body": "\n## Definition\n\nLet $A$ and $B$ be subsets of $\\N$.\nA *Turing reduction* from $A$ to $B$ is an [[oracle machine|Z5CCVBME2V]] $M$ such that $M$ with oracle $B$ halts on every input $x \\in \\N$.\nMoreover, the output is $1$ if $x \\in A$ and $0$ if $x \\notin A$.\nThe set $A$ is *Turing reducible* to $B$, written $A \\le_T B$, if such an $M$ exists.\nThe sets $A$ and $B$ are *Turing equivalent*, written $A \\equiv_T B$, if $A \\le_T B$ and $B \\le_T A$.\n\n## Remarks\n\nExample: for every $B \\subseteq \\N$, the complement $\\N \\setminus B$ satisfies $\\N \\setminus B \\le_T B$.\nThe machine asks the oracle whether $x \\in B$ and outputs the opposite answer.\n\nNon-example: the [[halting problem|Z7TN2B35B0]] $K$ satisfies $K \\not\\le_T \\emptyset$.\nAn oracle machine with the empty oracle computes only what an ordinary Turing machine computes, and $K$ is not a [[computable set|PPGD5KW8JC]].\n\nThe relation $\\le_T$ is reflexive and transitive.\nA set $A$ satisfies $A \\le_T \\emptyset$ exactly when $A$ is computable.\nEvery [[many-one reduction|JTAH37JTHD]] gives a Turing reduction.\nThe converse fails: $\\N \\setminus K \\le_T K$, but $\\N \\setminus K \\not\\le_m K$, since $K$ is computably enumerable and $\\N \\setminus K$ is not.\n",
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