AC (complexity)

In circuit complexity, AC is a complexity class hierarchy. Each class, ACⁱ, consists of the languages recognized by Boolean circuits with depth ${\displaystyle O(\log ^{i}n)}$ and a polynomial number of unlimited fan-in AND and OR gates.

The name "AC" was chosen by analogy to NC, with the "A" in the name standing for "alternating" and referring both to the alternation between the AND and OR gates in the circuits and to alternating Turing machines.^[1]

The smallest AC class is AC⁰, consisting of constant-depth unlimited fan-in circuits.

The total hierarchy of AC classes is defined as

${\displaystyle {\mbox{AC}}=\bigcup _{i\geq 0}{\mbox{AC}}^{i}}$