PR (complexity)

PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided in time bounded 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 (Cooper 2004:88).

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 ${\displaystyle (M,k)}$, where ${\displaystyle M}$ is a Turing machine and ${\displaystyle k}$ is an integer, if ${\displaystyle M}$ halts within ${\displaystyle k}$ steps then output ${\displaystyle M}$; otherwise output nothing. Then the union of the outputs, over all possible inputs (${\displaystyle M}$, ${\displaystyle k}$), is exactly the set of ${\displaystyle M}$ that halt.

PR strictly contains ELEMENTARY.

PR does not contain "PR-complete" problems (assuming, e.g., reductions that belong to ELEMENTARY).