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 kth level of the Boolean hierarchy over P 2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to PNPNP [O(log n)] . 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 P 2 and P 3 . Namely, BH(k) = co-BH(k) = BH3(k) = co-BH3(k) PNP [k] = PNP [k+1] = PNPNP [k+1] = PNPNP [k+2] . Key words: polynomial time hierarchy, Boolean hierarchy, polynomial time Turing reductions, oracle access, nonuniform algorithms, sparse sets AMS (MOS) subject classifications: 68Q15, 03D15, 03D20

• Type: Conference
• Year: 1990
• Where: COCO
• Authors: Richard Chang, Jim Kadin
• Keywords: Algorithms, Boolean Hierarchy, COCO 1990, Hierarchy Collapses, Polynomial Hierarchy