The purpose of this exercise is to prove the following fact:...

The purpose of this exercise is to prove the following fact: (*) There exists a (total) recursive function $\psi(x, y)$ such that if we set $\psi_{x}=$ $\lambda y . \psi(x, y)$, then the set $\left\{\psi_{x}: x \in \mathbb{N}\right\}$ is precisely the set of all primitive recursive functions of one variable. (a) Show that if $f \in \mathcal{F}_{p}$, then the following two conditions are equivalent: (i) $f$ is primitive recursive; (ii) there exists an index $i$ and a primitive recursive function $g \in \mathcal{F}_{p}$ such that the machine whose index is $i$ computes $f$ and the computation time $T\left(i, x_{1}, x_{2}, \ldots, x_{p}\right)$ is less than or equal to $g\left(x_{1}, x_{2}, \ldots, x_{p}\right)$. (b) We will make use of Ackerman's function $\zeta$ and of the functions $\zeta_{n}=$ $\lambda x . \zeta(n, x)$. Show that if $f$ is a primitive recursive function of one variable, then there exist two integers $n$ and $A$ such that, for all $x$, we have $$ f(x) \leq \sup \left(A, \zeta_{n}(x)\right) . $$ (c) Let $g$ be the function of four variables defined by $$ \begin{gathered} g(i, A, n, x)=\mu y \leq \sup (A, \zeta(n, x)) \\ {\left[\exists t \leq \sup (A, \zeta(n, x))(i, t, x, y) \in C^{1}\right] ;} \end{gathered} $$ [recall that $(i, t, x, y) \in C^{1}$ means that when the machine whose index is $i$ is set in operation with $x$ on its first band, it will halt at the instant $t$ with output $y$ ]. Show that, for all $i, A$, and $n$, the function $\lambda x \cdot g(i, A, n, x)$ is primitive recursive and that, conversely, if $f$ is any primitive recursive function of one variable, then there exist integers $i, A$, and $n$ such that $f=\lambda x . g(i, A, n, x)$. (d) Use these results to prove $(*)$. (e) Show that there exists a recursive set that is not primitive recursive.

The purpose of this exercise is to prove the following fact:
(*) There exists a (total) recursive function $\psi(x, y)$ such that if we set $\psi_{x}=$ $\lambda y . \psi(x, y)$, then the set $\left\{\psi_{x}: x \in \mathbb{N}\right\}$ is precisely the set of all primitive recursive functions of one variable.
(a) Show that if $f \in \mathcal{F}_{p}$, then the following two conditions are equivalent:
(i) $f$ is primitive recursive;
(ii) there exists an index $i$ and a primitive recursive function $g \in \mathcal{F}_{p}$ such that the machine whose index is $i$ computes $f$ and the computation time $T\left(i, x_{1}, x_{2}, \ldots, x_{p}\right)$ is less than or equal to $g\left(x_{1}, x_{2}, \ldots, x_{p}\right)$.
(b) We will make use of Ackerman's function $\zeta$ and of the functions $\zeta_{n}=$ $\lambda x . \zeta(n, x)$. Show that if $f$ is a primitive recursive function of one variable, then there exist two integers $n$ and $A$ such that, for all $x$, we have
$$
f(x) \leq \sup \left(A, \zeta_{n}(x)\right) .
$$
(c) Let $g$ be the function of four variables defined by
$$
\begin{gathered}
g(i, A, n, x)=\mu y \leq \sup (A, \zeta(n, x)) \\
{\left[\exists t \leq \sup (A, \zeta(n, x))(i, t, x, y) \in C^{1}\right] ;}
\end{gathered}
$$
[recall that $(i, t, x, y) \in C^{1}$ means that when the machine whose index is $i$ is set in operation with $x$ on its first band, it will halt at the instant $t$ with output $y$ ]. Show that, for all $i, A$, and $n$, the function $\lambda x \cdot g(i, A, n, x)$ is primitive recursive and that, conversely, if $f$ is any primitive recursive function of one variable, then there exist integers $i, A$, and $n$ such that $f=\lambda x . g(i, A, n, x)$.
(d) Use these results to prove $(*)$.
(e) Show that there exists a recursive set that is not primitive recursive.

Key Concepts

Primitive Recursive Functions

Primitive recursive functions form a class of total functions defined from a set of basic functions (such as the zero, successor, and projection functions) using operations like composition and primitive recursion. They are characterized by their guaranteed termination and play a central role in computability theory as a well-behaved subset of all computable functions.

Total Recursive Functions

Total recursive functions, also known as computable functions, include all functions that can be computed by an algorithm (or Turing machine) and always produce an output for every input. They represent the full range of effective computation and encompass primitive recursive functions as well as more complex functions that may require unbounded search or minimisation.

Universal Function for Primitive Recursion

A universal function in the context of primitive recursive functions is a single total recursive function that, when appropriately parameterized, can simulate every primitive recursive function of one variable. This concept is used to effectively enumerate the entire class of primitive recursive functions and highlights the idea of ‘programs as data’ in recursion theory.

Ackermann's Function

Ackermann's function is a well-known example of a computable function that is not primitive recursive due to its extremely fast growth rate. In the context of the problem, variations of Ackermann's function are used to provide natural bounds and to demonstrate that every primitive recursive function can be majorized by a function of Ackermann-type, thus linking the growth rates of primitive recursive functions to a non-primitive recursive benchmark.

Bounded Minimization

Bounded minimization refers to the process of searching for the smallest number that satisfies a given predicate, with the search restricted to a known finite bound. This method is crucial in defining functions in a way that ensures termination by limiting the search space, thereby allowing one to maintain the property of being primitive recursive even when using the minimization operator.

Computation Time in Recursion Theory

The concept of computation time in recursion theory involves analyzing the number of steps a machine takes to compute a function. By establishing that a function’s computation time is bounded by a primitive recursive function, one can argue that the overall function is primitive recursive. This connection between time bounds and function classification is essential in proving the equivalence of different characterizations of primitive recursive functions.

Recursive Sets vs Primitive Recursive Sets

Recursive sets are those for which membership can be decided by an algorithm, meaning there is a total computable (recursive) characteristic function for the set. The existence of a recursive set that is not primitive recursive illustrates the strict inclusion of primitive recursive sets within the broader class of recursive sets, showing that some decidable sets require computational resources or strategies that exceed the limitations of primitive recursion.

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