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 $(M,k)$, 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).

Hierarchy

The PR class can be divided into an infinite hierarchy of increasingly large complexity levels, according to the fast-growing hierarchy.

The ${\text{𝐅}}_{0}$ class is the class of problems that can be solved in $n+O(1)$ time. That is, there exists a Turing machine and a constant $C$, such that given an input of length $n$, the machine solves it and halts within $n+C$ steps.

The ${\text{𝐅}}_{1}$ class is the class of problems that can be solved in ${\mathsf {poly}}(n)$ time.

The ${\text{𝐅}}_{2}$ class is ELEMENTARY.

The ${\text{𝐅}}_{3}$ class is TOWER, which can be equivalently written as the class of problems that can be solved in tetration-time.

The union $\bigcup _{n\in \mathbb {N} }{\text{𝐅}}_{n}$ is PR.

In practice, many problems that are not in PR but just beyond it are ${\text{𝐅}}_{\omega }$-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.
• Schmitz, Sylvain (2016). "Complexity Hierarchies beyond Elementary". ACM Transactions on Computation Theory. 8: 1–36. arXiv:1312.5686. doi:10.1145/2858784. S2CID 15155865.

External links

• Complexity Zoo: PR.