Abstrakt
We show that CC-circuits of bounded depth have the same expressive power as circuits over finite nilpotent algebras from congruence modular varieties. We use this result to phrase a new algebraic version of Barrington, Straubing and Therien's conjecture, which states that CC-circuits of bounded depth need exponential size to compute AND.
Furthermore we investigate the complexity of deciding identities and solving equations in a fixed nilpotent algebra. Under the assumption that the conjecture is true, we obtain quasipolynomial algorithms for both problems.
On the other hand, if AND is computable by uniform CC-circuits of bounded depth and polynomial size, we can construct a nilpotent algebra in which checking identities is coNP-complete, and solving equations is NP-complete.