counting problem

If $R$ is a search problem then $c_R(x)=|\{y\mid R(x)\}|$ is the corresponding counting function and $\mathrm{\#}R=\{(x,y)\mid y\le c_R(x)\}$ denotes the corresponding counting problem. Note that $c_R$ is a search problem while $\mathrm{\#}R$ is a decision problem, however $c_R$ can be $𝒞$ Cook reduced to $\mathrm{\#}R$ (for appropriate $𝒞$) using a binary search (the reason $\mathrm{\#}R$ is defined the way it is, rather than being the graph of $c_R$, is to make this binary search possible).