Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 4G / a

Fix an effective enumeration $f_{e,n}$ of the $n$-ary partial recursive functions.

The S-m-n theorem says that program inputs can be compiled into an index: for every $m,n\ge 1$ there is a total recursive $s_n^m$ such thatHere $\simeq$ means that either both sides are undefined or both are defined and equal.

The recursion theorem says that every total recursive transformation $h$ of program indices has a semantic fixed point: some $e$ satisfies

The Rice theorem says that every nontrivial extensional property of programs computed by partial recursive functions is undecidable.