= Universal partial computable function
{title2=$U(e,\mathbf x)\simeq\varphi_e(\mathbf x)$}
A universal partial computable function simulates the program with index $e$ on input $\mathbf x$, reproducing both its output and its divergence. Decode the finite program and repeatedly apply its transition rule. This gives a <partial computable function> of the code and inputs. There can be no total computable evaluator whose rows are precisely all <total computable functions>, because $n\mapsto U(n,n)+1$ would be a missing diagonal row.
Back to article page
