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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 (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 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「PR (complexity)」の詳細全文を読む スポンサード リンク
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