In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

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 <?TeX $\mathsf {AC}$?> Math 1 circuits remain unknown. Perhaps surprisingly, Chattopadhyay, Torán, and Wagner (FSTTCS 2010; ACM Trans. Comput. Theory, 2013) showed that Quasigroup Isomorphism could be solved by <?TeX $\mathsf {AC}$?> Math 2 circuits of depth O(log log n) using O(log 2n) nondeterministic bits, a class we denote <?TeX $\exists ^{\log ^{2}n}\mathsf {FOLL}$?> Math 3 . We narrow this gap by improving the upper bound for these problems to <?TeX $\mathsf {quasiAC^0}$?> Math 4 , thus decreasing the depth to constant.

In this paper, we consider relational structures arising from Comer's finite field construction, where the cosets need not be sum free. These Comer schemes generalize the notion of a Ramsey scheme and may be of independent interest. As an application, we give the first finite representation of $34_{65}$. This leaves $33_{65}$ as the only remaining relation algebra in the family $N_{65}$ with a flexible atom that is not known to be finitely representable. Motivated by this, we complement our upper bounds with some lower bounds. Using a SAT solver, we show that $33_{65}$ is not finitely representable on fewer than $24$ points, and that $33_{65}$ does not admit a cyclic group representation on fewer than $120$ points. We also employ a SAT solver to show that $34_{65}$ is not representable on fewer than $24$ points. Fundamenta Informaticae final journal version; previous conference version appeared in RAMiCS 2023

We report our experiences implementing standards-based grading at scale in an Algorithms course, which serves as the terminal required CS Theory course in our department's undergraduate curriculum. The course had 200-400 students, taught by two instructors, eight graduate teaching assistants, and supported by two additional graders and several undergraduate course assistants. We highlight the role of standards-based grading (SBG) in supporting our students during the COVID-19 pandemic. We conclude by detailing the successes and adjustments we would make to the course structure.