The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in $\P^\numP$ relies on the fact that, under mild technical conditions on the complexity class $\mathcal{C}$, we have $\exists \,\mathcal{C} \subset \BP \cdot \oplus \,\mathcal{C}$. More concretely, there is a randomized reduction which transforms nonempty sets and the ... more >>>