# Primitive recursion

A means of defining functions with natural number arguments and values. One says that an $(n+1)$-place function $f(x_1,\dots,x_n,y)$ is obtained by primitive recursion from an $n$-place function $g(x_1,\dots,x_n)$ and an $(n+2)$-place function $h(x_1,\dots,x_n,y,z)$ if for all natural number values of $x_1,\dots,x_n,y$ one has

$$f(x_1,\dots,x_n,0)=g(x_1,\dots,x_n)$$

and

$$f(x_1,\dots,x_n,y+1)=h(x_1,\dots,x_n,y,f(x_1,\dots,x_n,y)).$$

For given $g$ and $h$ such a function $f$ exists always and is unique. For $n=0$ the defining equations for $f$ can be written as

$$f(0)=a,\quad f(x+1)=h(x,f(x)).$$

A fundamental property of primitive recursion is that for any meaningful specification of the notion of computability, a function $f$ obtained from computable functions $g$ and $h$ by means of primitive recursion is itself computable (cf. Computable function). Primitive recursion is one of the basic ways for generating all primitive recursive and all partial recursive functions from an initial set of basic functions (cf. Primitive recursive function; Partial recursive function).

#### References

[1] | V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian) |

[2] | A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian) |

[3] | H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165 |

#### Comments

#### References

[a1] | C. Calude, "Theories of computational complexity" , North-Holland (1988) |

[a2] | Yu.I. Manin, "A course in mathematical logic" , Springer (1977) (Translated from Russian) |

[a3] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1959) pp. Chapts. IX; XI, §54 |

[a4] | P. Odifreddi, "Classical recursion theory" , North-Holland (1989) pp. Chapt. II; esp. pp. 199ff |

**How to Cite This Entry:**

Primitive recursion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Primitive_recursion&oldid=34257