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 >>>

An equation over a finite group G is an expression of form
w_1 w_2...w_k = 1_G, where each w_i is a variable, an inverted
variable, or a constant from G; such an equation is satisfiable
if there is a setting of the variables to values in G ... more >>>