Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 12F / iii / Solution

Let $g(n,k)=r(n)$, where $r$ is the total primitive-recursive parity function from part (ii), and apply unbounded minimization in $k$:If $n$ is even, then $g(n,0)=0$ and $p(n)=0$. If $n$ is odd, then $g(n,k)=1$ for every $k$, so the search never terminates and $p(n)$ is undefined. Hence $p$ is partial recursive. Because every function built without minimization is total, $p$ cannot be defined without minimization; the functions $s$ and $r$ can.

It remains to characterize the functions built using only the initial functions and composition. They are exactly the functionsfor some $c\in \mathbb{N}$ and some input coordinate $i$.

Indeed, the zero function is the first form, the successor function and each projection function have the second form, and composing functions of these forms preserves the classification: a constant outer function remains constant, while an outer function $y_j+c$ selects one inner function and adds $c$. This proves necessity by structural induction. Conversely, applying the successor function $c$ times to zero produces the constant $c$, and applying it $c$ times to ${\pi }_i^k$ produces $x_i+c$.

Every function in this class is directly computable: an algorithm either writes the fixed constant $c$, or copies the $i$th input and performs $c$ successor steps.

Solved by gpt-5.6-sol high.