# Multiple recursion

A form of recursion using several variables simultaneously. The set of values of these variables is ordered lexicographically. This definition subsumes numerous concrete recursive descriptions. If in such a description the unknown function is not substituted into itself, then it results in a primitive recursion. In general, multiple recursion leads outside the limits of the set of primitive recursive functions, since by a double recursion (performed with respect to two variables) it is possible to construct a function which is universal for the primitive recursive functions (similarly, for $ k $-
recursive functions there is a $ ( k+ 1 ) $-
multiple universal function); cf. Universal function. All possible forms of multiple recursion can be reduced to the following normal form:

$$ \phi ( n _ {1} \dots n _ {k} ) = 0 \ \textrm{ if } n _ {1} \dots n _ {k} = 0, $$

$$ \phi ( n _ {1} + 1 \dots n _ {k} + 1 ) = \beta ( n _ {1} \dots n _ {k} , \phi _ {1} \dots \phi _ {k} ) , $$

where

$$ \phi _ {i} = \phi ( n _ {1} + 1 \dots n _ {i-} 1 + 1 , n _ {i} , $$

$$ \gamma _ {1} ^ {(} i) ( n _ {1} \dots n _ {k} , \phi ( n _ {1} + 1 \dots n _ {k-} 1 + 1 , n _ {k} )) \dots $$

$$ \dots {}\gamma _ {k-} 1 ^ {(} i) ( n _ {1} \dots n _ {k} , \phi ( n _ {1} + 1 \dots n _ {k-} 1 + 1 , n _ {k} ) ) ) . $$

#### References

[1] | R. Peter, "Recursive functions" , Acad. Press (1967) (Translated from German) |

#### Comments

It is more common to speak of simultaneous recursion, rather than multiple recursion; cf., e.g., [a1].

#### References

[a1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) |

[a2] | H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165 |

**How to Cite This Entry:**

Multiple recursion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Multiple_recursion&oldid=47932