- PR (complexity)
**PR**is the complexity class of allprimitive recursive function s – or, equivalently, the set of allformal language s 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 thehalting 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 containsELEMENTARY .**ee also***

Primitive recursive function

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