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.^[1]

Example problems

Integer addition and subtraction are computable in AC⁰,^[2] but multiplication is not (specifically, when the inputs are two integers under the usual binary^[3] or base-10 representations of integers).

Since it is a circuit class, like P/poly, AC⁰ also contains every unary language.

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(+, ×), or by Turing machine in the logarithmic hierarchy.^[4]

Separations

In 1984 Furst, Saxe, and Sipser showed that calculating the parity of the input bits (unlike the aforementioned addition/subtraction problems above which had two inputs) cannot be decided by any AC⁰ circuits, even with non-uniformity.^[5][1] 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 the polynomial hierarchy and PSPACE.

References