PR (complexity)

PR is the complexity class of all primitive recursive functions – or, equivalently, the set of all formal languages that can be decided 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.

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

External links

Complexity Zoo: PR.

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 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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