Derandomizing Arthur—Merlin games under uniform assumptions

Abstract. We study how the nondeterminism versus determinism problem and the time versus space problem are related to the problem of derandomization. In particular, we show two ways of derandomizing the complexity class AM under uniform assumptions, which was only known previously under nonuniform assumptions. First, we prove that either AM = NP or it appears to any nondeterministic polynomial time adversary that NP is contained in deterministic subexponential time infinitely often. This implies that to any nondeterministic polynomial time adversary, the graph nonisomorphism problem appears to have subexponential-size proofs infinitely often, the first nontrivial derandomization of this problem without any assumption. Next, we show that either all of BPP = P, AM = NP, and $ {\bf PH} \subseteq \oplus{\bf P} $ hold, or for any $ t(n) = 2^{\Omega(n)} $, $ {\bf DTIME}(t(n)) \subseteq {\bf DSPACE}(t^\epsilon(n)) $ infinitely often for any constant $ \epsilon > 0 $. Similar tradeoffs also hold for a whole range of parameters. This improves previous results and seems to be the first example of an interesting condition that implies three derandomization results at once.