PR (complexity)
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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.
PR functions can be explicitly enumerated, whereas functions in R cannot be (since otherwise the halting problem would be decidable). That is, PR is a "syntactic" class whereas R is "semantic."
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.
See also
| Important complexity classes (more) |
| P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NL | NC | P-C | PSPACE | PSPACE-C |
| EXPTIME | EXPSPACE | PR | RE | Co-RE | RE-C | Co-RE-C | R | BQP | BPP | RP | ZPP | PCP | IP | PH |
