The goal of this exercise is to show that the precautions...

The goal of this exercise is to show that the precautions which we took in defining the unbounded $\mu$-operator (see Definition $5.19$ ) are necessary. Show that the partial function $\psi(x, y)$ defined by $$ \psi(x, y)= \begin{cases}\phi^{1}(x, y)-\phi^{1}(x, y) & \text { if } y=0 \\ 0 & \text { otherwise }\end{cases} $$ is partial recursive. Define the function $g$ by $g(x)=$ the least integer $y$ such that $\psi(x, y)=0 .$ Show that $g$ is a total function that is not recursive.

The goal of this exercise is to show that the precautions which we took in defining the unbounded $\mu$-operator (see Definition $5.19$ ) are necessary.
Show that the partial function $\psi(x, y)$ defined by
$$
\psi(x, y)= \begin{cases}\phi^{1}(x, y)-\phi^{1}(x, y) & \text { if } y=0 \\ 0 & \text { otherwise }\end{cases}
$$
is partial recursive.
Define the function $g$ by $g(x)=$ the least integer $y$ such that $\psi(x, y)=0 .$
Show that $g$ is a total function that is not recursive.

Key Concepts

Non-Recursive Functions

A function is non-recursive (or non-computable) if there is no algorithm or Turing machine that can compute its value for every input. This is a fundamental notion in computability theory. The function g in the problem, although total, is constructed in such a way that it escapes the bounds of recursive computability. This example highlights how totality does not necessarily imply computability, emphasizing the necessity of careful definitions in recursion theory.

Total Functions

Total functions are those functions that are defined on every input from their domain, meaning they always produce an output. In the context of recursion theory, while many functions defined via the unbounded ?-operator are only partial, the exercise demonstrates a situation where the resulting function g is total — it yields an integer for every input, even though it is built from a partially-defined process.

Partial Recursive Functions

Partial recursive functions are functions that can be computed by a Turing machine or equivalent formalism but might not provide an output for every input; that is, they are defined only on a subset of the natural numbers. They are constructed using initial functions along with operations like composition, primitive recursion, and the unbounded minimization operator. The function ? in the problem is an example and underscores how partial recursion can be used to construct more complex functions.

Unbounded ?-operator

The unbounded ?-operator (minimization operator) is an operation that, given a predicate or function, returns the least number satisfying that predicate if one exists. It is a critical tool in recursion theory to define functions that cannot be generated through just primitive recursion or composition. However, because the search may not terminate when no such number exists, caution is necessary. This operator is central to understanding the boundary between partial and total computable functions.

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