CC-circuits and the expressive power of nilpotent algebras

TL;DR

This paper explores the expressive power of CC-circuits in relation to finite nilpotent algebras, proposing an algebraic perspective on a longstanding conjecture and analyzing the complexity of algebraic problems.

Contribution

It establishes an equivalence between CC-circuits and circuits over nilpotent algebras, and formulates a new algebraic version of a known conjecture about circuit size.

Findings

Bounded depth CC-circuits have the same expressive power as circuits over nilpotent algebras.

Under the conjecture, quasipolynomial algorithms exist for identity and equation problems.

If the conjecture fails, certain algebraic problems become NP- or coNP-complete.

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.