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  "body": "\n## Definition\n\nLet $k \\ge 1$ and let $f \\colon \\N^k \\rightharpoonup \\N$ be a partial function, where $\\N$ is the set of [[natural numbers|0MWKJ44V5A]].\nFix an encoding of tuples of natural numbers as words on the input alphabet, for example in unary separated by a marker.\nThe function $f$ is *computable*, or *partial computable*, if there is a [[Turing machine|758EYCN27B]] $M$ with the following property for every $\\bar x \\in \\N^k$.\n\n1. If $f(\\bar x)$ is defined, then $M$ halts on input $\\bar x$ with the code of $f(\\bar x)$ on its tape.\n2. If $f(\\bar x)$ is undefined, then $M$ does not halt on input $\\bar x$.\n\nA *total computable function* is a computable function defined at every point of $\\N^k$.\n\n## Remarks\n\nExample: addition $(x, y) \\mapsto x + y$ is computable.\nWith unary codes $1^x 0 1^y$, a machine that replaces the $0$ by $1$ and erases the last $1$ halts with $1^{x+y}$ on its tape.\n\nNon-example: the function $h(e) = 1$ if the machine with index $e$ halts on input $e$, and $h(e) = 0$ otherwise, is total but not computable.\nNo machine satisfies clause 1 for $h$; see the [[halting problem|Z7TN2B35B0]].\n\nA partial function is computable if and only if it is partial [[general recursive|HBK5QHC1GC]].\nThe definition does not depend on the choice of a reasonable encoding or a variant of Turing machines.\n",
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