PR (complexity)
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PR is the complexity class of all primitive recursive functionsβor, 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 , where is a Turing machine and is an integer, if halts within steps then output ; otherwise output nothing. Then the union of the outputs, over all possible inputs (, ), is exactly the set of that halt.
PR strictly contains ELEMENTARY.
PR does not contain "PR-complete" problems (assuming, e.g., reductions that belong to ELEMENTARY).
Hierarchy
Summarize
Perspective
The PR class can be divided into an infinite hierarchy of increasingly large complexity levels, according to the fast-growing hierarchy.
The class is the class of problems that can be solved in time. That is, there exists a Turing machine and a constant , such that given an input of length , the machine solves it and halts within steps.
The class is the class of problems that can be solved in time.
The class is ELEMENTARY.
The class is TOWER, which can be equivalently written as the class of problems that can be solved in tetration-time.
The union is PR.
In practice, many problems that are not in PR but just beyond it are -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: 1β36. arXiv:1312.5686. doi:10.1145/2858784. S2CID 15155865.
External links
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