Definition:Minimization

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

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

Then the minimization operation on $$f$$ is written as:
 * $$\mu y \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$$

and is specified as follows:
 * $$\mu y \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ such that } f \left({n_1, n_2, \ldots, n_k, y}\right) = 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:
 * $$g \left({n_1, n_2, \ldots, n_k}\right) = \mu y \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$$

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

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

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

Then the minimization operation on $$\mathcal{R}$$ is written as:
 * $$\mu y \mathcal{R}\left({n_1, n_2, \ldots, n_k, y}\right)$$

and is specified as follows:
 * $$\mu y \mathcal{R}\left({n_1, n_2, \ldots, n_k, y}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ for which } \mathcal{R}\left({n_1, n_2, \ldots, n_k, 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.