Hardness hypotheses, derandomization, and circuit complexity

Abstract

We consider three complexity-theoretic hypotheses that have been studied in different contexts and shown to have many plausible consequences. - The Measure Hypothesis: NP does not have p-measure 0. - The pseudo-NP Hypothesis: there is an NP Language L such that any DTIME(2ne) Language L′ can be distinguished from L by an NP refuter. - The NP-Machine Hypothesis: there is an NP machine accepting 0* for which no 2ne-time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomization and circuit-size lower bounds that are known to follow from the first two hypotheses also follow from the NP-machine hypothesis. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. © Springer-Verlag Berlin Heidelberg 2004.