A Downward Translation in the Polynomial Hierarchy

Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k ? 2, if P \Sigma p k [1] = P \Sigma p k [2] then \Sigma p k = \Pi p k = PH. We extend this to obtain a more general downward collapse result. 1 Introduction The theory of NP-completeness does not resolve the issue of whether P and NP are equal. However, it does unify the issues of whether thousands of natural problems---the NPcomplete problems---have deterministic polynomial-time algorithms. The study of downward collapse is similar in spirit. By proving downward collapses, we seek to tie together central open issues regarding the computing power of complexity classes. For example, the main result of this paper shows that (for k ? 2) the issue of whether the kth level of the polynomial hierarchy is close..