Counting problem (complexity)
# Counting problem (complexity)

In computational complexity theory and computability theory, a counting problem is a type of computational problem. If R is a search problem then

    $c_{R}(x)=\vert \{y\mid R(x,y)\}\vert \,$

is the corresponding counting function and

    $\#R=\{(x,y)\mid y\leq c_{R}(x)\}$

denotes the corresponding decision problem.

Note that $c_R$ is a search problem while #R is a decision problem, however $c_R$ can be C Cook-reduced to #R (for appropriate C) using a binary search (the reason #R is defined the way it is, rather than being the graph of $c_R,$ is to make this binary search possible).

## Counting complexity class

If NX is a complexity class associated with non-deterministic machines then #X = {#R | R ∈ NX} is the set of counting problems associated with each search problem in NX. In particular, #P is the class of counting problems associated with NP search problems. Just as NP has NP-complete problems via many-one reductions, #P has complete problems via parsimonious reductions, problem transformations that preserve the number of solutions.

