# AC0

AC0 is a complexity class used in circuit complexity. It is the smallest class in the AC hierarchy, and consists of all families of circuits of depth O(1) and polynomial size, with unlimited-fanin AND gates and OR gates. (We allow NOT gates only at the inputs).[1] It thus contains NC0, which has only bounded-fanin AND and OR gates.[2]

From a descriptive complexity viewpoint, DLOGTIME-uniform AC0 is equal to the descriptive class FO+BIT of all languages describable in first-order logic with the addition of the BIT operator, or alternatively by FO(+, $\times$), or by Turing machine in the logarithmic hierarchy.[3]

In 1984 Furst, Saxe, and Sipser showed that calculating the parity of an input cannot be decided by any AC0 circuits, even with non-uniformity.[4][5] It follows that AC0 is not equal to NC1, because a family of circuits in the latter class can compute parity.[1] More precise bounds follow from switching lemma. Using them, it has been shown that there is an oracle separation between PH and PSPACE.

Integer addition and subtraction are computable in AC0, but multiplication is not (at least, not under the usual binary or base-10 representations of integers).

## References

1. ^ a b Arora & Barak (2009) p.118
2. ^ Arora & Barak (2009) p.117
3. ^ *N. Immerman Descriptive complexity (1999 Springer), page 85.
4. ^ Furst, Merrick; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Math. Syst. Theory 17: 13–27. ISSN 0025-5661. Zbl 0534.94008.
5. ^ Arora & Barak (2009) p.287