Class Description

CC0: Constant-Depth MODm Circuits

The set of problems solvable by by constant-depth circuits having only MODm gates for constant .

It was shown in [CW22] that any symmetric function can be computed by depth-3 CC0 circuits with size 2O(na) for any constant a. Such a circuit uses a modulus m with size at most (1/a)2/a.

This is in contrast with AC0 which requires size 2O(n1/(d-1)) for depth d circuits computing arbitrary symmetric functions [Has87]. Even AC0[m] for prime power m requires size 2O(n1/(2d)) for depth d [Raz87] [Smo87].

Linked From

ACC0: AC0 With Arbitrary MOD Gates

Same as AC0[m], but now the constant-depth circuit can contain MOD m gates for any m.

Contained in TC0.

Indeed, can be simulated by depth-3 threshold circuits of quasipolynomial size [Yao90].

According to [All96], there is no good evidence for the existence of cryptographically secure functions in ACC0.

There is no non-uniform ACC0 circuits of polynomial size for NTIMES[2n] and no ACC0 circuit of size 2nO(1) for ENP (The class E with an NP oracle). These are the only two known nontrivial lower bounds against ACC0 and were recently discovered by [Wil11].

Contains 4-PBP [BT88].

See also: QACC0 and CC0.

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