Theorem 4.2 generalizes to provide a characterization of the classes which comprise the Polynomial Hierarchy. For instance, the logics \(\Sigma^1_i\) and \(\Pi^1_i\) uniformly capture the complexity classes \(\Sigma^P_i\) and \(\Pi^P_i\) (where \(\mathsf{SO}\exists = \Sigma^1_1\) ). Moreover \(\mathsf{SO}\) (i.e. full second-order logic) captures \(\textbf{PH}\) itself. On the other hand, it can also be shown that first-order logic (\(\mathsf{FO}\)) captures only a very weak complexity class known as \(\textbf{AC}^0\) consisting of languages decidable by polynomial size circuits of constant depth.