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 ) = | { y ∣ R ( x , y ) } | {\displaystyle c_{R}(x)=\vert \{y\mid R(x,y)\}\vert \,}

is the corresponding counting function and

# R = { ( x , y ) ∣ y ≤ c R ( x ) } {\displaystyle \#R=\{(x,y)\mid y\leq c_{R}(x)\}}

denotes the corresponding decision problem.

Note that cR is a search problem while #R is a decision problem, however cR 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 cR, is to make this binary search possible).