Complexity Class $\mathsf{coNP}$

The decision problems of the form $\forall y\, W(x, y)$, where $y$ has size polynomial in $|x|$ and $W$ is decidable in polynomial time: $x$ is a yes-instance if and only if every candidate $y$ passes the check, so no-instances have short counterexamples. These are the complements of the problems in $\mathsf{NP}$.