Definition:Minimization/Relation

Definition

Let $\map \RR {n_1, n_2, \ldots, n_k, y} $ be a $k + 1$-ary relation on $\N^{k + 1}$.

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

Then the minimization operation on $\RR$ is written as:

$\mu y \map \RR {n, y}$

and is specified as follows:

$\mu y \map \RR {n, y} = \begin{cases}

\text{the smallest } y \in \N \text{ for which } \map \RR {n, y} \text{ holds} & : \text{if there exists such a } y \\ \text{undefined} & : \text{otherwise} \end{cases}$

We can consider the definition for a function to be a special case of this.

Compare the bounded minimization operation which is always a total function.