// Workers AI · dad joke modeWhat did PR complexity say to its date? "You're reducible to my heart.

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.