Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

Abstract

Impagliazzo and Wigderson proved a uniform hardness vs. randomness “gap theorem” for BPP. We show an analogous result for AM: Either Arthur-Merlin protocols are very strong and everything in \( \textrm{E = DTIME}(2^{O(n)}) \) can be proved to a subexponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm such that it is infeasible to come up with inputs on which the algorithm fails. We also show that if Arthur-Merlin protocols are not very strong (in the sense explained above) then \( \textrm{AM} \cap \textrm{coAM} = \textrm{NP} \cap \textrm{coNP} \)

Our technique combines the nonuniform hardness versus randomness tradeoff of Miltersen and Vinodchandran with “instance checking”. A key ingredient in our proof is identifying a novel “resilience” property of hardness vs. randomness tradeoffs.