# 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 (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 (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.

PR does not contain "PR-complete" problems (assuming, e.g., reductions that belong to ELEMENTARY). In practice, many problems that are not in PR but just beyond are ${\displaystyle {\text{𝐅}}_{\omega }}$-complete (Schmitz 2016).

## References

• S. Barry Cooper (2004), Computability Theory, Chapman & Hall. ISBN 1-58488-237-9
• Herbert Enderton (2011), Computability Theory, Academic Press. ISBN 978-0-12-384-958-8
• Sylvain Schmitz (2016), "Complexity Hierarchies Beyond Elementary", ACM Transactions on Computation Theory 8. doi:10.1145/2858784