Universal program

In computability theory, a universal program is a partial computable function $U:\mathbb{N}\rightharpoonup \mathbb{N}$ such that, for every program index $e$ and input $x$,
$U⟨e,x⟩\simeq {\varphi }_e(x)$
where ${\varphi }_e:\mathbb{N}\rightharpoonup \mathbb{N}$ is a computable denumeration of partial recursive functions, and $⟨,⟩:\mathbb{N}\to \mathbb{N}\to \mathbb{N}$ is a (computable) pairwise encoding of naturals.

References

cutland1980-computability