Abstract-Barrington, Straubing and 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 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 • OR • CC 0 = OR • AND • 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.
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