CC-circuits and the expressive power of nilpotent algebras
Abstract
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 and discuss a new 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 in which checking identities is coNP-complete, and solving equations is NP-complete.
Keywords
Cite
@article{arxiv.1911.01479,
title = {CC-circuits and the expressive power of nilpotent algebras},
author = {Michael Kompatscher},
journal= {arXiv preprint arXiv:1911.01479},
year = {2023}
}