A New Characterization of ACC0 and Probabilistic CC0

Abstract

Barrington, Straubing and Therien (1990) conjectured that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC^0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC^0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture. As a consequence of our construction we get that all of ACC^0 can be computed by probabilistic CC^0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC^0=CC^0. We present a derandomization of probabilistic CC^0 circuits using AND and OR gates to obtain ACC^0 = AND o OR o CC^0 = OR o AND o CC^0. AND and OR gates of sublinear fan-in suffice. Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC^0 = rand-ACC^0 = rand-CC^0 = rand(log n)-CC^0, i.e., probabilistic ACC^0 circuits can be simulated by probabilistic CC^0 circuits using only O(log n) random bits. As an application of our results we obtain a characterization of ACC^0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting.