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'''PR''' is the complexity class of all [[primitive recursive function]]s—or, equivalently, the set of all [[formal language]]s that can be decided by such a function. This includes addition, multiplication, exponentiation, [[tetration]], etc. |
'''PR''' is the complexity class of all [[primitive recursive function]]s—or, equivalently, the set of all [[formal language]]s that can be decided in [[time complexity|time]] bounded by such a function. This includes [[addition]], [[multiplication]], [[exponentiation]], [[tetration]], etc. |
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The [[Ackermann function]] is an example of a function that is ''not'' primitive recursive, showing that PR is strictly contained in [[R (complexity)|R]] (Cooper 2004:88). |
The [[Ackermann function]] is an example of a function that is ''not'' primitive recursive, showing that PR is strictly contained in [[R (complexity)|R]] (Cooper 2004:88). |
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On the other hand, we can "enumerate" any [[recursively enumerable set]] (see also its complexity class [[RE (complexity)|RE]]) by a primitive-recursive function in the following sense: given an input (''M'',& |
On the other hand, we can "enumerate" any [[recursively enumerable set]] (see also its complexity class [[RE (complexity)|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. |
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PR strictly contains [[ELEMENTARY]]. |
PR strictly contains [[ELEMENTARY]]. |
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PR does not contain "PR-complete" problems (assuming, e.g., |
PR does not contain "PR-complete" problems (assuming, e.g., [[many-one reduction|reduction]]s that belong to ELEMENTARY). In practice, many problems that are not in PR but just beyond are <math>\text{𝐅}_\omega</math>-complete (Schmitz 2016). |
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== References == |
== References == |
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* {{cite book|author=S. Barry Cooper |year=2004|title=Computability Theory|publisher= Chapman & Hall|isbn=1-58488-237-9}} |
* {{cite book|author=S. Barry Cooper |year=2004|title=Computability Theory|publisher= Chapman & Hall|isbn=1-58488-237-9}} |
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* {{cite book|author=Herbert Enderton |year=2011|title=Computability Theory|publisher= Academic Press|isbn=978-0-12-384-958-8}} |
* {{cite book|author=Herbert Enderton |year=2011|title=Computability Theory|publisher= Academic Press|isbn=978-0-12-384-958-8}} |
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* {{cite journal|arxiv=1312.5686|doi=10.1145/2858784|title=Complexity Hierarchies beyond Elementary|year=2016|last1=Schmitz|first1=Sylvain|journal=ACM Transactions on Computation Theory|volume=8|pages=1–36}} |
* {{cite journal|arxiv=1312.5686|doi=10.1145/2858784|title=Complexity Hierarchies beyond Elementary|year=2016|last1=Schmitz|first1=Sylvain|journal=ACM Transactions on Computation Theory|volume=8|pages=1–36|s2cid=15155865 }} |
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==External links == |
==External links == |
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Latest revision as of 07:50, 26 July 2022
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 -complete (Schmitz 2016).
References
[edit]- 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.