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arXiv · 1911.01479

CC-circuits and the expressive power of nilpotent algebras

Michael Kompatscher
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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.

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Michael Kompatscher. 2019-11-04. CC-circuits and the expressive power of nilpotent algebras. https://doi.org/10.46298/lmcs-18(2%3A12)2022

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