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 ${\displaystyle (M,k)}${\displaystyle (M,k)}, where ${\displaystyle M}${\displaystyle M} is a Turing machine and ${\displaystyle k}${\displaystyle k} is an integer, if ${\displaystyle M}${\displaystyle M} halts within ${\displaystyle k}${\displaystyle k} steps then output ${\displaystyle M}${\displaystyle M}; otherwise output nothing. Then the union of the outputs, over all possible inputs (${\displaystyle M}${\displaystyle M}, ${\displaystyle k}${\displaystyle k}), is exactly the set of ${\displaystyle M}${\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 ${\displaystyle {\text{F}}_{0}}${\displaystyle {\text{F}}_{0}} class is the class of problems that can be solved in ${\displaystyle n+O(1)}${\displaystyle n+O(1)} time. That is, there exists a Turing machine and a constant ${\displaystyle C}${\displaystyle C}, such that given an input of length ${\displaystyle n}${\displaystyle n}, the machine solves it and halts within ${\displaystyle n+C}${\displaystyle n+C} steps.

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

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

The ${\displaystyle {\text{F}}_{3}}${\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 ${\displaystyle \bigcup _{n\in \mathbb {N} }{\text{F}}_{n}}${\displaystyle \bigcup _{n\in \mathbb {N} }{\text{F}}_{n}} is PR.

In practice, many problems that are not in PR but just beyond it are ${\displaystyle {\text{F}}_{\omega }}${\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.

• Complexity classes