Nonuniform ACC Circuit Lower Bounds

Abstract

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MOD m gates, where m > 1 is an arbitrary constant. We prove the following. ---NEXP, the class of languages accepted in nondeterministic exponential time, does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. ---E NP , the class of languages recognized in 2 O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2 n o(1) size. The lower bound gives an exponential size-depth tradeoff: for every d, m there is a δ > 0 such that E NP doesn’t have depth- d ACC circuits of size 2 n δ with MOD m gates. Previously, it was not known whether EXP NP had depth-3 polynomial-size circuits made out of only MOD 6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail these lower bounds. The algorithms combine known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a strengthening of the author’s prior work.