AC0[p] Lower Bounds against MCSP via the Coin Problem

Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p ≥ 2 (the circuit class AC0[p]). Namely, we show that MCSP requires d-depth AC0[p] circuits of size at least exp(N0.49/d), where N = 2 is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N -bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2 + N−0.49, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC0 circuit with MCSP-oracle gates. 2012 ACM Subject Classification Theory of computation → Circuit complexity; Theory of computation → Problems, reductions and completeness