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).

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

The Failed to parse (syntax error): {\displaystyle \text{𝐅}_0} class is the class of problems that can be solved in ${\displaystyle n+O(1)}$ time. That is, there exists a Turing machine and a constant ${\displaystyle C}$, such that given an input of length ${\displaystyle n}$, the machine solves it and halts within ${\displaystyle n+C}$ steps.

The Failed to parse (syntax error): {\displaystyle \text{𝐅}_1} class is the class of problems that can be solved in ${\displaystyle {\mathsf {poly}}(n)}$ time.

The Failed to parse (syntax error): {\displaystyle \text{𝐅}_2} class is ELEMENTARY.

The Failed to parse (syntax error): {\displaystyle \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 Failed to parse (syntax error): {\displaystyle \bigcup_{n \in \N} \text{𝐅}_n} is PR.

In practice, many problems that are not in PR but just beyond it are Failed to parse (syntax error): {\displaystyle \text{𝐅}_\omega} -complete (Schmitz 2016).

• 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.

• Complexity Zoo: PR.