Count-Free Weisfeiler--Leman and Group Isomorphism
Nathaniel A. Collins, Michael Levet
TL;DR
The paper tackles Group Isomorphism from a descriptive-complexity perspective, focusing on the power of counting in the WL framework. It shows that count-free WL Version I, combined with bounded nondeterminism and limited counting, yields efficient parallel isomorphism tests for direct products of simple groups, coprime extensions with Abelian normals and polyloglog solvable complements, and graphical class-2, exponent-$p$ groups; it also establishes that higher-arity count-free games cannot distinguish Abelian groups, suggesting counting is sometimes necessary to place GpI in P. The work connects WL, pebble games, and first-order logics to derive explicit parallel complexities (e.g., $\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ and related classes) for these families, while extending CFI/Mekler constructions to analyze the limits of count-free approaches. Collectively, the results illuminate which counting resources are essential for efficient GpI, and where count-free techniques suffice. The findings have implications for descriptive complexity, parallel isomorphism testing, and the design of circuit-based group isomorphism tests with restricted counting. The paper also identifies open questions about the exact logical characterizations of the $q$-ary count-free game and the full reach of canonicalization in count-free settings.
Abstract
We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler--Leman Version I algorithm for groups (Brachter & Schweitzer, LICS 2020) in tandem with limited non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: - Direct products of non-Abelian simple groups. - Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an $O(1)$-generated solvable group with solvability class $\text{poly} \log \log n$. This notably includes instances where the complement is an $O(1)$-generated nilpotent group. This problem was previously known to be in $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011), and the complexity was recently improved to $\textsf{L}$ (Grochow & Levet, FCT 2023). - Graphical groups of class $2$ and exponent $p > 2$ (Mekler, J. Symb. Log., 1981) arising from the CFI and twisted CFI graphs (Cai, Fürer, & Immerman, Combinatorica 1992) respectively. In particular, our work improves upon previous results of Brachter & Schweitzer (LICS 2020). We finally show that the $q$-ary count-free pebble game is unable to distinguish even Abelian groups. This extends the result of Grochow & Levet (ibid), who established the result in the case of $q = 1$. The general theme is that some counting appears necessary to place Group Isomorphism into $\textsf{P}$.