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In [[computability theory]] '''complete numberings''' are generalizations of [[Numbering (computability theory)|Gödel numbering]] first introduced by [[A.I. Mal'tsev]] in 1963. They are studied because several important results like the [[Kleene's recursion theorem]] and [[Rice's theorem]], which were originally proven for the Gödel-numbered set of [[computable function]]s, still hold for arbitrary sets with complete numberings.   
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== Definition ==
 
A [[numbering (computability theory)|numbering]] <math>\nu</math> of a set <math>A</math> is called '''complete''' (with respect to an element <math>a \in A</math>) if for every [[partial computable function]] <math>f</math> there exists a [[total computable function]] <math>h</math> so that
:<math> \nu \circ h(i) =
\left\{
\begin{matrix}
\nu \circ f(i) &\mbox{if}\ i \in \mathrm{dom}(f), \\
a &\mbox{otherwise}.
\end{matrix}
\right.
</math>
 
The numbering <math>\nu</math> is called '''precomplete''' if
 
:<math> \nu \circ f(i) = \nu \circ h(i) \qquad i \in \mathrm{dom}(f).\,</math>
 
== Examples ==
 
* any numbering of a [[singleton set]] is complete
* the [[identity function]] on the natural numbers is ''not'' complete
* a [[Numbering (computability theory)|Gödel numbering]] is precomplete
 
== References ==
* A.I. Mal'tsev, ''Sets with complete numberings''. [[Algebra i Logika]], 1963, vol. 2, no. 2, 4-29 (Russian)
 
[[Category:Computability theory]]

Revision as of 02:44, 9 February 2014

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