We investigate the constant-depth circuit complexity of the Isomorphism Problem, Minimum Generating Set Problem (MGS), and Sub(quasi)group Membership Problem (Membership) for groups and quasigroups (=Latin squares), given as input in terms of their multiplication (Cayley) tables. Despite decades of research on these problems, lower bounds for these problems even against depth-$2$ AC circuits remain unknown. Perhaps surprisingly, Chattopadhyay, Tor\'an, and Wagner (FSTTCS 2010; ACM Trans. Comput. Theory, 2013) showed that Quasigroup Isomorphism could be solved by AC circuits of depth $O(\log \log n)$ using $O(\log^2 n)$ nondeterministic bits, a class we denote $\exists^{\log^2(n)}FOLL$. We narrow this gap by improving the upper bound for many of these problems to $quasiAC^0$, thus decreasing the depth to constant. In particular, we show: - MGS for quasigroups is in $\exists^{\log^2(n)}\forall^{\log n}NTIME(\mathrm{polylog}(n))\subseteq quasiAC^0$. Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1996) conjectured that this problem was $\exists^{\log^2(n)}P$-complete; our results refute a version of that conjecture for completeness under $quasiAC^0$ reductions unconditionally, and under polylog-space reductions assuming EXP $\neq$ PSPACE. - MGS for groups is in $AC^{1}(L)$, improving on the previous upper bound of P (Lucchini & Thakkar, J. Algebra, 2024). - Quasigroup Isomorphism belongs to $\exists^{\log^2(n)}AC^0(DTISP(\mathrm{polylog},\log)\subseteq quasiAC^0$, improving on the previous bound of $\exists^{\log^2(n)}L\cap\exists^{\log^2(n)}FOLL\subseteq quasiFOLL$ (Chattopadhyay, Tor\'an, & Wagner, ibid.; Levet, Australas. J. Combin., 2023). Our results suggest that understanding the constant-depth circuit complexity may be key to resolving the complexity of problems concerning (quasi)groups in the multiplication table model.

翻译：我们研究了群和拟群（=拉丁方）的乘法（Cayley）表输入模型下，同构问题、最小生成集问题（MGS）以及子（拟）群成员问题（Membership）的常数深度电路复杂度。尽管对这些问题的研究已持续数十年，但即便针对深度为2的AC电路，其下界仍然未知。令人意外的是，Chattopadhyay、Torán和Wagner（FSTTCS 2010；ACM Trans. Comput. Theory, 2013）证明了拟群同构问题可通过深度为$O(\log \log n)$、使用$O(\log^2 n)$非确定比特的AC电路解决——该类记为$\exists^{\log^2(n)}FOLL$。我们通过将许多此类问题的上界改进为$quasiAC^0$来缩小这一差距，从而将深度降低至常数。具体而言，我们证明：- 拟群的MGS问题属于$\exists^{\log^2(n)}\forall^{\log n}NTIME(\mathrm{polylog}(n))\subseteq quasiAC^0$。Papadimitriou和Yannakakis（J. Comput. Syst. Sci., 1996）曾猜想该问题是$\exists^{\log^2(n)}P$完全的；我们的结果无条件反驳了该猜想在$quasiAC^0$归约下的完全性版本，并在假设EXP ≠ PSPACE的条件下反驳了多项式对数空间归约下的版本。- 群的MGS问题属于$AC^{1}(L)$，改进了此前的P上界（Lucchini & Thakkar, J. Algebra, 2024）。- 拟群同构问题属于$\exists^{\log^2(n)}AC^0(DTISP(\mathrm{polylog},\log)\subseteq quasiAC^0$，改进了此前的$\exists^{\log^2(n)}L\cap\exists^{\log^2(n)}FOLL\subseteq quasiFOLL$上界（Chattopadhyay, Torán, & Wagner, 同上；Levet, Australas. J. Combin., 2023）。我们的结果表明，理解常数深度电路复杂度可能是解决乘法表模型下（拟）群相关问题复杂性的关键。