Topology inside NC$^1$

2026-09-10

AI Summary

本文探讨了ACC^0和NC^1电路复杂性类，通过研究不同拓扑结构如多对数属、交叉数和厚度，证明了特定条件下这些属性对计算能力的影响。

Abstract

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

Problem

• Topology
• ACC^0
• NC^1
• circuit
• genus

Innovation

• constant-width circuits
• polylogarithmic genus
• thickness two