Defying upward and downward separation

Abstract

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. We show that the exponential-time limited nondeterminism hierarchy does not robustly possess downward separation. We show that probabilistic classes do not robustly possess upward separation. Though NP is known [19] to robustly possess upward separation, we show that NP does not robustly possess upward separation with respect to strong (immunity) separation. On the other hand, we provide a structural sufficient condition for upward separation.