Holographic proofs and derandomization

We derive a stronger consequence of EXP having polynomial-size circuits than was known previously, namely that there is a simulation of P in MAPOLYLOG that fools all deterministic polynomial-time adversaries. Using the connection between circuit lower bounds and derandomization, we obtain uniform assumptions for derandomizing BPP. Our results strengthen the space-randomness tradeoffs of Sipser, Nisan and Wigderson, and Lu. We show a partial converse: oracle circuit lower bounds for EXP imply that there are efficient simulations of P that fool deterministic polynomial-time adversaries. We also consider a more quantitative notion of simulation, where the measure of success of the simulation is the fraction of inputs of a given length on which the simulation works. Among other results, we show that if there is no polynomial time bound t such that P can be simulated well by MATIME(t), then for any ε>0 there is a simulation of BPP in P that works for all but 2^nε inputs of length n. This is a uniform strengthening of a recent result of Goldreich and Wigderson. Finally, we give an unconditional simulation of multitape Turing machines operating in probabilistic time t by Turing machines operating in deterministic time O(2^t). We show similar results for randomized NC¹ circuits. Our proofs are based on a combination of techniques in the theory of derandomization with results on holographic proofs.