primitive recursive function
To define what a primitive recursive function is, the following notations are used:
, where for each .
Definition. The set of primitive recursive functions is the smallest subset of where:
(zero function) , given by ;
(successor function) , given by ;
(projection functions) , where , given by ;
is closed under primitive recursion: If and , then , where
Many of the arithmetic functions that we encounter in basic math are primitive recursive, including addition, multiplication, and exponentiation. More examples can be found in this entry (http://planetmath.org/ExamplesOfPrimitiveRecursiveFunctions).
Primitive recursive functions are computable by Turing machines. In fact, it can be shown that is precisely the set of functions computable by programs using FOR NEXT loops. However, not all Turing-computable functions are primitive recursive: the Ackermann function is one such example.
Every primitive recursive function is total, since it is built from , , and , each of which is total, and that functional composition, and primitive recursion preserve totalness. By including in above, and close it by functional composition and primitive recursion, one gets the set of partial primitive recursive functions.
|Title||primitive recursive function|
|Date of creation||2013-03-22 12:33:19|
|Last modified on||2013-03-22 12:33:19|
|Last modified by||CWoo (3771)|
|Defines||primitive recursive set|
|Defines||primitive recursive predicate|
|Defines||partial primitive recursive function|