The Boolean Hierarchy and the Polynomial Hierarchy: a Closer Connection

We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH3(k), where BH3(k) is the k th level of the Boolean hierarchy over Σ2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to P [O(log . This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of ∆2 and ∆ P 3 . Namely, BH(k) = co-BH(k) =⇒ BH3(k) = co-BH3(k) P = P =⇒ P ‖[k+1] = P NP‖[k+2].