A New Characterization of ACC⁰ and Probabilistic CC⁰

Abstract.

Barrington, Straubing & Thérien (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 ACC⁰ 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 ACC⁰ = CC⁰. We present a derandomization of probabilistic CC⁰ circuits using And and Or gates to obtain ACC⁰ = And ο Or ο CC⁰ = Or ο And ο 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 ACC⁰ = rand – ACC⁰ = rand – CC⁰ = rand(log n)– CC⁰, i.e., probabilistic ACC⁰ 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 ACC⁰ by constant width planar nondeterministic branching programs, improving a previous characterization for the quasi-polynomial size setting.