The Boolean hierarchy and the polynomial hierarchy: a closer connection

It is shown that if the Boolean hierarchy collapses to level k , then the polynomial hierarchy collapses to BH₃(k ), where BH₃(k) is the kth level of the Boolean hierarchy over Σ₂^p. 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. This results also points to some previously unexplored connections between the Boolean and query hierarchies of Δ₂^p and Δ₃^p