AI 中文总结
本文在乘法表模型中研究群同构的低深度电路复杂度，证明其深度2电路下界，给出深度2½均匀电路的拟多项式规模上界，采用不同于前期拟群通用策略的群特定结构方法。
AI 中文摘要
本文研究乘法（Cayley）表模型下群同构的低深度电路复杂度，证明了群同构的首个电路下界，即所有判定群同构的深度2布尔电路族均需拟多项式规模。同时给出了深度2½均匀电路的拟多项式规模上界。1970至2025年的一系列前期结果逐步将电路深度从多项式降至3½，均依赖生成元-枚举器策略且更通用地适用于拟群。与之不同，本文的深度2½构造采用完全不同的策略，利用群更特定的结构：为每个群猜测合成列、其项的生成元及合成因子的同构类型，随后归纳验证两个合成列各层级对应的扩张是否相容。该方法的核心部分结合了短表示猜想的大量研究成果，以及群扩张与上同调的算法理论。
英文摘要
In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. We complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size. A sequence of previous results from 1970-2025 progressively reduced the circuit depth from polynomial to $3\frac{1}{2}$; all of these results relied on the generator-enumerator strategy and, in fact, applied more generally to quasigroups. In contrast, our depth-$2\frac{1}{2}$ construction follows a fundamentally different strategy that exploits structure more specific to groups. We guess a composition series for each group, together with generators for its terms and the isomorphism types of its composition factors. We then inductively verify that the corresponding extensions at each level of the two composition series are compatible. A central part in this approach brings to bear the extensive work on the Short Presentation Conjecture, in tandem with the algorithmic theory of group extensions and cohomology.
Comments62 pages
