Parameter-specialization functions

Fix an acceptable numbering $({\phi }_e{)}_{e\in \mathbb{N}}$.

For $n\in \mathbb{N}$, define the recursively coded input tuple by ${\mathrm{Inp}}_0:=<mi\; mathvariant="normal">⌜</mi>(){<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}},{\mathrm{Inp}}_{n+1}(b_1,\dots ,b_{n+1}):=<mi\; mathvariant="normal">⌜</mi>b_1,{\mathrm{Inp}}_n(b_2,\dots ,b_{n+1}){<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}}\mathrm{.}$

For $m,n\in \mathbb{N}$, a total computable function $s_m^n:{\mathbb{N}}^{m+1}\to \mathbb{N}$ is a parameter-specialization function when for all indices $e$, all parameters $a_1,\dots ,a_m$, all inputs $b_1,\dots ,b_n$, and the sequence coding of A natural-number coding of finite sequences, one has ${\phi }_{s_m^n(e,a_1,\dots ,a_m)}({\mathrm{Inp}}_n(b_1,\dots ,b_n))\simeq {\phi }_e({\mathrm{Inp}}_{m+n}(a_1,\dots ,a_m,b_1,\dots ,b_n))\mathrm{.}$

• The function $s_m^n$ hard-wires the first $m$ inputs and leaves the last $n$ inputs free.

• The case $n=1$ gives a residual unary program on the stipulated coded-input convention: it is evaluated at ${\mathrm{Inp}}_1(b)=<mi\; mathvariant="normal">⌜</mi>b,{\mathrm{Inp}}_0{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}}$, not at the bare number $b$.