Second level of the polynomial hierarchy (${\Sigma }_2^{\mathbf{P}}$ and ${\Pi }_2^{\mathbf{P}}$)

A language is in ${\Sigma }_2^{\mathbf{P}}$ when membership has the form $\exists u\forall vR(x,u,v)$ for a polynomial-time predicate $R$ and polynomially bounded strings. Reversing the quantifiers defines ${\Pi }_2^{\mathbf{P}}$.