Complexity Class $\Pi_3^{\mathsf{P}}$

The decision problems of the form $\forall y_1 \exists y_2 \forall y_3\, W(x, y_1, y_2, y_3)$, where the $y_i$ have size polynomial in $|x|$ and $W$ is decidable in polynomial time. Equivalently, the problems solvable in $\mathsf{coNP}$ with a $\Sigma_2^{\mathsf{P}}$ oracle. These are the complements of the problems in $\Sigma_3^{\mathsf{P}}$.