Definition:Minimization/Relation

Definition
Let $\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) $ be a $k+1$-ary relation on $\N^{k+1}$.

Let $n = \left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ be fixed.

Then the minimization operation on $\mathcal R$ is written as:
 * $\mu y \mathcal R\left({n, y}\right)$

and is specified as follows:
 * $\mu y \mathcal R \left({n, y}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ for which } \mathcal R \left({n, y}\right) \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.