The Polynomial Time Hierarchy Collapses If the Boolean Hierarchy Collapses

It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that co-NP⊆NPS, and therefore the polynomial-time hierarchy (PH) collapses to a subclass of ΔP/3. Since the BH is contained in PNP, these results relate the internal structure of PNP to the structure of the PH as a whole. Other conditions that imply the collapse of the BH (and the collapse of the PH in turn) are examined