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). In practice, many problems that are not in PR but just beyond are Failed to parse (syntax error): {\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.
* Schmitz, Sylvain (2016). "Complexity Hierarchies beyond Elementary". ACM Transactions on Computation Theory 8: 1–36. doi:10.1145/2858784.

## External links

* Complexity Zoo: PR.

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Important complexity classes (more)

Considered feasible|

* DLOGTIME
* AC0
* ACC0
* TC0
* L
* SL
* RL
* NL
* NC
* SC
* CC
* P
* P-complete
* ZPP
* RP
* BPP
* BQP
* APX

Suspected infeasible|

* UP
* NP
* NP-complete
* NP-hard
* co-NP
* co-NP-complete
* AM
* QMA
* PH
* ⊕P
* PP
* #P
* #P-complete
* IP
* PSPACE
* PSPACE-complete

Considered infeasible|

* EXPTIME
* NEXPTIME
* EXPSPACE
* 2-EXPTIME
* ELEMENTARY
* PR
* R
* RE
* ALL

Class hierarchies|

* Polynomial hierarchy
* Exponential hierarchy
* Grzegorczyk hierarchy
* Arithmetical hierarchy
* Boolean hierarchy

Families of classes|

* DTIME
* NTIME
* DSPACE
* NSPACE
* Probabilistically checkable proof
* Interactive proof system

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