Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 11H / b / Solution

The S-m-n theorem says that parameters can be compiled effectively into program indices: for a standard effective numbering, there is a total computable function $s_m^n$ with ${\phi }_{s_m^n(e,\mathbf{a})}(\mathbf{x})={\phi }_e(\mathbf{a},\mathbf{x})$ for partial computable functions of $m+n$ arguments.

Write $K=\{e:{\phi }_e(e)\text{ halts}\}$, a recursively enumerable halting problem. If $X$ is recursively enumerable, make a program with parameter $x$ which ignores its input and runs a semidecision procedure for $x\in X$, halting when that procedure halts. By the S-m-n theorem its index $f(x)$ is total computable and $f(x)\in K⟺x\in X$. Conversely, $X{\le }_{m}K$ implies that $X$ is recursively enumerable by the previous construction. Thus $\boxed{X\text{ is r.e.}⟺X{\le }_{m}K}$.