Definition:Primitive Recursion

Primitive Recursion on Several Variables
Let $$f: \N^k \to \N$$ and $$g: \N^{k+2} \to \N$$ be functions.

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

Then the function $$h: \N^{k+1} \to \N$$ is obtained from $$f$$ and $$g$$ by primitive recursion if:
 * $$\forall n \in \N: h \left({n_1, n_2, \ldots, n_k, n}\right) = \begin{cases}

f \left({n_1, n_2, \ldots, n_k}\right) & : n = 0 \\ g \left({n_1, n_2, \ldots, n_k, n-1, h \left({n_1, n_2, \ldots, n_k, n-1}\right)}\right) & : n > 0 \end{cases} $$

Primitive Recursion on One Variable
Let $$a \in \N$$ be a natural number.

Let $$g: \N^2 \to \N$$ be a function.

Then the function $$h: \N \to \N$$ is obtained from the constant $$a$$ and $$g$$ by primitive recursion if:
 * $$\forall n \in \N: h \left({n}\right) = \begin{cases}

a & : n = 0 \\ g \left({n-1, h \left({n-1}\right)}\right) & : n > 0 \end{cases} $$

It can be seen that this is a special case of primitive recursion on several variables, with $$k = 0$$ and $$f$$ replaced by the constant function $$f_a$$.

Primitive Recursion on Partial Functions
Let $$f: \N^k \to \N$$ and $$g: \N^{k+2} \to \N$$ be partial functions.

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

Then the partial function $$h: \N^{k+1} \to \N$$ is obtained from $$f$$ and $$g$$ by primitive recursion if:
 * $$\forall n \in \N: h \left({n_1, n_2, \ldots, n_k, n}\right) \approx \begin{cases}

f \left({n_1, n_2, \ldots, n_k}\right) & : n = 0 \\ g \left({n_1, n_2, \ldots, n_k, n-1, h \left({n_1, n_2, \ldots, n_k, n-1}\right)}\right) & : n > 0 \end{cases} $$ where $$\approx$$ is as defined in Partial Function Equality.

Note that $$h \left({n_1, n_2, \ldots, n_k, n}\right)$$ is defined only when:
 * $$h \left({n_1, n_2, \ldots, n_k, n - 1}\right)$$ is defined;
 * $$g \left({n_1, n_2, \ldots, n_k, n-1, h \left({n_1, n_2, \ldots, n_k, n-1}\right)}\right)$$ is defined.