AC0

File:Diagram of an AC0 Circuit.svg

Diagram of an AC⁰ circuit: The n input bits are on the bottom and the top gate produces the output; the circuit consists of AND- and OR-gates of polynomial fan-in each, and the alternation depth is bounded by a constant.

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). It thus contains NC⁰, which has only bounded-fanin AND and OR gates.

Example problems

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).

Descriptive complexity

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 predicate, or alternatively by FO(+, $\times$), or by Turing machine in the logarithmic hierarchy.

Separations

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. It follows that AC⁰ is not equal to NC¹, because a family of circuits in the latter class can compute parity. More precise bounds follow from switching lemma. Using them, it has been shown that there is an oracle separation between the polynomial hierarchy and PSPACE.