A Downward Collapse within the Polynomial Hierarchy
Abstract
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 = then σk = πk = . We extend this to obtain a more general downward collapse result.
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