🤖 AI Summary
This paper investigates the logical definability of finite groups with no nontrivial abelian normal subgroups—a class central to complexity-theoretic group isomorphism. Addressing the Ehrenfeucht–Fraïssé (EF) game characterization for such groups, we introduce and analyze the binary Weisfeiler–Leman (2-WL) coloring procedure, establishing for the first time its exact equivalence to the binary EF game. Our key contribution is proving that 2-WL distinguishes all such groups within a constant number of rounds and using only a constant number of pebbles—i.e., O(1) rounds and O(1) variables. Consequently, these groups are definable in first-order logic with generalized binary quantifiers, using only O(1) variables and O(1) quantifier depth. This yields the tightest known logical characterization for group isomorphism on this class and reveals their inherently low descriptive complexity within the Hella hierarchy.
📝 Abstract
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.
