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  "body": "\n## Definition\n\nAn *oracle machine* is a [[Turing machine|758EYCN27B]] with these extra parts:\n\n1. a second tape, the *oracle tape*, on which the machine can read and write;\n2. three special states: a *query state* $q_?$, a *yes state* $q_{\\mathrm{yes}}$ and a *no state* $q_{\\mathrm{no}}$.\n\nLet $O \\subseteq \\N$ be a set, the *oracle*.\nThe machine $M$ *with oracle* $O$, written $M^O$, runs like a Turing machine, except at the query state.\nSuppose $M^O$ enters $q_?$ while the oracle tape holds the binary code of a number $n$.\nThen its next state is $q_{\\mathrm{yes}}$ if $n \\in O$ and $q_{\\mathrm{no}}$ if $n \\notin O$.\nA partial function $f \\colon \\N \\to \\N$ is *computable relative to* $O$ if there is an oracle machine $M$ with this property.\nFor every $x$, the run of $M^O$ on input $x$ halts with output $f(x)$ when $f(x)$ is defined, and does not halt otherwise.\n\n## Remarks\n\nExample: let $M$ copy its input $x$ to the oracle tape, enter $q_?$, and then output $1$ from $q_{\\mathrm{no}}$ and $0$ from $q_{\\mathrm{yes}}$.\nFor each oracle $O$, the machine $M^O$ halts on every input and computes the characteristic function of $\\N \\setminus O$.\n\nNon-example: a device that may query two sets $O_1$ and $O_2$ is not an oracle machine, because an oracle machine queries one fixed set.\nIt can be simulated by an oracle machine with the single oracle $\\set{2n : n \\in O_1} \\cup \\set{2n + 1 : n \\in O_2}$.\n\nA machine that never enters $q_?$ is an ordinary Turing machine, so every computable function is computable relative to every oracle.\nOracle machines can be listed effectively as $M_0, M_1, \\ldots$, and $\\Phi_e^O$ often denotes the partial function computed by $M_e^O$.\n",
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