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 is a Turing machine, that halt.

PR strictly contains ELEMENTARY.

PR does not contain "PR-complete" problems (assuming, e.g., reductionss that belong to ELEMENTARY).

Hierarchy

The PR class can be divided into an infinite hierarchy of increasingly large complexity levels, according to the fast-growing hierarchy.

steps.

time.

class is ELEMENTARY.

class is TOWER, which can be equivalently written as the class of problems that can be solved in tetration-time.

is PR.

-complete (Schmitz 2016).

References

External links

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