Definition:Minimization/Function

Definition

Let $f: \N^{k + 1} \to \N$ be a total function.

Let $n = \tuple {n_1, n_2, \ldots, n_k} \in \N^k$ be fixed.

Then the minimization operation on $f$ is written as:

$\map {\mu y} {\map f {n, y} = 0}$

and is specified as follows:

$\map {\mu y} {\map f {n, y} = 0} = \begin{cases} \text{the smallest } y \in \N \text{ such that } \map f {n, y} = 0 & : \text{if there exists such a } y \\ \text{undefined} & : \text{otherwise} \end{cases}$

Note that if $f: \N^{k + 1} \to \N$ is a total function, and $g: \N^k \to \N$ is given by:

$\map g n = \map {\mu y} {\map f {n, y} = 0}$

then, in general, $g$ will be a partial function.