A New Characterization of ACC(0) and Probabilistic CC0
Barrington, Straubing & Th,rien (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 CC0 circuits. In this work we show that the And function can be computed by uniform probabilistic CC0 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 CC0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, one would obtain a collapse of circuit classes giving ACC(0) = CC0. We present a derandomization of probabilistic CC0 circuits using And and Or gates to obtain ACC(0) = And omicron Or omicron CC0 = Or omicron And omicron CC0. (And and Or gates of sublinear fan-in suffice in non-uniform setting.) 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 - CC0 = rand(log n)- CC0, i.e., probabilistic ACC(0) circuits can be simulated by probabilistic CC0 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 quasi-polynomial size setting.
and function modular counting planarity branching programs boolean circuits probabilistic circuits