In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fraisse 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 the 2-ary WL is equivalent to the second Ehrenfeucht-Fraisse 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 P; Babai, Codenotti, & Qiao, ICALP 2012). 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 O(1) variables and O(1) quantifier depth.

翻译：本文通过考察Hella（Ann. Pure Appl. Log., 1989）层级中第二类Ehrenfeucht-Fraïssé双射卵石博弈的能力，探索有限群的描述复杂性理论。这是一种Spoiler-Duplicator博弈，其中Spoiler每轮最多可放置两个卵石。尽管该博弈能平凡地解决图同构问题，但对于有限群及其他三元关系结构而言可能非平凡。我们首先提出Weisfeiler-Leman（WL）着色的一种新颖推广，称为2-ary WL。随后证明2-ary WL等价于Hella层级中的第二类Ehrenfeucht-Fraïssé双射卵石博弈。我们的主要结果是：在卵石博弈刻画中，仅需O(1)个卵石和O(1)轮博弈即可识别所有无阿贝尔正规子群的群（该类群的同构测试已知属于P；Babai, Codenotti, & Qiao, ICALP 2012）。特别地，我们证明在最初几轮博弈后，Spoiler可迫使Duplicator在后续每轮博弈中选取这两个群之间的一个同构映射。根据Hella的结论（同上），这等价于这些群可被带广义2-ary量词的一阶逻辑公式（仅使用O(1)个变量和O(1)量词深度）所识别。