Universal and acceptable numberings

Let $({\phi }_e{)}_{e\in \mathbb{N}}$ be a numbering of the partial computable functions.

The numbering is universal when the binary partial function $U(e,x)\simeq {\phi }_e(x)$ is itself partial computable.

The numbering is acceptable when it is universal and there exists a total computable function $s:{\mathbb{N}}^2\to \mathbb{N}$ such that for all $e,a,x\in \mathbb{N}$ one has ${\phi }_{s(e,a)}(x)\simeq {\phi }_e(<mi\; mathvariant="normal">⌜</mi>a,x{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}}),$ where $<mi\; mathvariant="normal">⌜</mi>a,x{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}}$ is the chosen code of the two-term sequence $(a,x)$.

• Universality says one indexed program can simulate the whole numbering.

• Acceptability adds one effective hard-wiring operation: the first input can be compiled into the index while the residual program still receives one natural input.