CC-circuits and the expressive power of nilpotent algebras

We show that CC-circuits of bounded depth have the same expressive power as polynomials over finite nilpotent algebras from congruence modular varieties. We use this result to phrase and discuss an algebraic version of Barrington, Straubing and Th\'erien'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 with coNP-complete, respectively NP-complete problem.