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.
翻译:我们用这一结果来表述和讨论Barrington、Straubing和Th\'erien的代数,其中指出,受约束深度的复方电路需要指数大小来计算和计算。此外,我们还调查在固定的零能力代数中决定身份和解方程的复杂性。根据预测是真实的假设,我们为这两个问题都获得了准极价算法。另一方面,如果并且能够由受约束深度和多面体大小的统一的CC-电路进行计算,我们就可以建造一个以无源代数完成的CONP(cNP-NP-完整问题)的无源代数代数。