# What is Counting problem: Definition and 58 Discussions

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

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{\displaystyle c_{R}(x)=\vert \{y\mid R(x,y)\}\vert \,}
is the corresponding counting function and

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{\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).

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