A Downward Collapse within the Polynomial Hierarchy
<|reference_start|>A Downward Collapse within 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 $\psigkone = \psigktwo$ then $\sigmak = \pik = \ph$. We extend this to obtain a more general downward collapse result.<|reference_end|>
@article{hemaspaandra1999a,
    title={A Downward Collapse within the Polynomial Hierarchy},
    author={Edith Hemaspaandra, Lane A. Hemaspaandra, and Harald Hempel},
    journal={SIAM Journal on Computing, 28, 383-393, 1999},
    year={1999},
    number={earlier version appears as UR-CS-TR-96-630},
    archivePrefix={arXiv},
    eprint={cs/9910007},
    primaryClass={cs.CC}
}