Defying Upward and Downward Separation

Upward and downward separation results link the collapse of small and large classes, and are a standard tool in complexity theory. We study the limitations of upward and downward separation. .pp We show that the exponential-time limited nondeterminism hierarchy does not robustly possess downward separation. We show that probabilistic classes do not robustly possess Hartmanis-Immerman-Sewelson upward separation. Though NP is known to robustly possess Hartmanis-Immerman-Sewelson upward separation, we show that NP does not robustly possess Hartmanis-Immerman-Sewelson upward separation with respect to strong (immunity) separation. On the other hand, we provide a structural sufficient condition for upward separation.

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