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| In [[computability theory]] a '''numbering''' is the assignment of [[natural number]]s to a [[Set (mathematics)|set]] of objects such as functions, [[rational number]]s, [[Graph (mathematics)|graph]]s, or words in some [[language]]. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using [[computable function]]s, to these different types of objects.
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| Common examples of numberings include [[Gödel numbering]]s in [[first-order logic]] and [[admissible numbering]]s of the set of partial computable functions.
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| == Definition and examples ==
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| A '''numbering''' of a set <math>S \!</math> is a [[partial function|partial]] [[surjective function]] from <math>\mathbb{N}</math> to ''S'' (Ershov 1999:477). The value of a numbering ν at a number ''i'' (if defined) is often written ν<sub>'i''</sub> instead of the usual <math>\nu(i) \!</math>.
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| For example, the set of all finite subsets of <math>\mathbb{N}</math> has a numbering γ in which <math>\gamma(\emptyset) = 0</math> and <math>\gamma(\{a_0, \ldots, a_k\}) = \sum_{i \leq k} 2^{a_i}</math> (Ershov 1999:477).
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| As a second example, a fixed Gödel numbering <math>\varphi_i</math> of the computable partial functions can be used to define a numbering ''W'' of the [[recursively enumerable set]]s, by letting by ''W''(''i'') be the domain of φ<sub>''i''</sub>.
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| == Types of numberings ==
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| A numbering is '''total''' if it is a total function. If the [[domain (function)|domain]] of a partial numbering is [[recursively enumerable]] then there always exists an equivalent total numbering (equivalence of numberings is defined below).
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| A numbering η is '''decidable''' if the set <math>\{ (x,y) : \eta(x) = \eta(y)\}</math> is a decidable set.
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| A numbering η is '''single-valued''' if η(''x'') = η(''y'') if and only if ''x''=''y''; in other words if η is an injective function. A single-valued numbering of the set of partial computable functions is called a [[Friedberg numbering]].
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| == Comparison of numberings ==
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| There is a [[partial ordering]] on the set of all numberings. Let
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| :<math>\nu_1: \subseteq \mathbb{N} \to S_1</math>
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| and
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| :<math>\nu_2: \subseteq \mathbb{N} \to S_2</math>
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| be two numbering. Then <math>\nu_1</math> is '''reducible''' to <math>\nu_2</math>, written <math>\nu_1 \le \nu_2</math>,
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| if
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| :<math>\exists f \in \mathbf{P}^{(1)} \, \forall i \in \mathrm{Domain}(\nu_1) : \nu_1(i) = \nu_2 \circ f(i).</math>
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| If <math>\nu_1 \le \nu_2</math> and <math>\nu_1 \ge \nu_2</math> then <math>\nu_1</math> is '''equivalent''' to <math>\nu_2</math>; this is written <math>\nu_1 \equiv \nu_2</math>.
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| == Computable numberings ==
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| When the objects of the set ''S'' are sufficiently "constructive", it is common to look at numberings that can be effectively decoded (Ershov 1999:486). For example, if ''S'' consists of recursively enumerable sets, the numbering η is '''computable''' if the set of pairs (''x'',''y'') where ''y''∈η(''x'') is recursively enumerable. Similarly, a numbering ''g'' of partial functions is computable if the relation ''R''(''x'',''y'',''z'') = "[''g''(''x'')](''y'') = ''z''" is partial recursive (Ershov 1999:487).
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| A computable numbering is called '''principal''' if every computable numbering of the same set is reducible to it. Both the set of all r.e. subsets of <math>\mathbb{N}</math> and the set of all partial computable functions have principle numberings (Ershov 1999:487). A principle numbering of the set of partial recursive functions is known as an [[admissible numbering]] in the literature.
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| == See also ==
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| * [[Complete numbering]]
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| * [[Cylindrification]]
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| * [[Gödel numbering]]
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| ==References==
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| * Y.L. Ershov (1999), "Theory of numberings", ''Handbook of Computability Theory'', Elsevier, pp. 473–506.
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| * V.A. Uspenskiĭ, A.L. Semenov (1993), ''Algorithms: Main Ideas and Applications'', Springer.
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| [[Category:Theory of computation]]
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| [[Category:Computability theory]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Since he was eighteen he's been operating as an info officer but he ideas on changing it. To play lacross is the thing I love most of all. Alaska is exactly where I've always been living.
Also visit my blog post online reader