PR (complexity)

PR (complexity)

PR is the complexity class of all primitive recursive functions – or, equivalently, the set of all formal languages that can be decided by such a function. This includes addition, multiplication, exponentiation, tetration, etc.

The Ackermann function is an example of a function that is "not" primitive recursive, showing that PR is strictly contained in R.

PR functions can be explicitly enumerated, whereas functions in R cannot be (since otherwise the halting problem would be decidable). That is, PR is a "syntactic" class whereas R is "semantic."

On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input ("M", "k"), where "M" is a Turing machine and "k" is an integer, if "M" halts within "k" steps then output "M"; otherwise output nothing. Then the union of the outputs, over all possible inputs ("M", "k"), is exactly the set of "M" that halt.

PR strictly contains ELEMENTARY.

ee also

* Primitive recursive function


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