Difference between revisions of "Primitive recursion"
From Encyclopedia of Mathematics
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| − | A means of defining functions with natural number arguments and values. One says that an | + | {{TEX|done}} |
| + | 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 | 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 | + | 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 | + | 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|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|Primitive recursive function]]; [[Partial recursive function|Partial recursive function]]). |
====References==== | ====References==== | ||
Revision as of 13:43, 27 September 2014
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=33422
Primitive recursion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_recursion&oldid=33422
This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article