群阶逻辑中的子群可达性

中文摘要 AI 辅助

我们研究不动点逻辑（FP）及其扩展在定义可定义置换群的可达子群的生成集方面的表达能力。该操作可通过Schreier-Sims算法在多项式时间内计算，在图同构和图规范化的群论方法中处于核心地位，尤其为有界颜色类图的多项式时间规范化提供了基础——而目前尚未发现能刻画该类图的自然P类逻辑。我们首先证明，该操作一般无法在任何P类逻辑中表达，这一局限源于可达子群不一定存在多项式大小的对称生成集。不过，我们证明当基础群存在可定义的有序生成集时，该可达子群操作可通过在不动点逻辑中加入群阶算子（FP + ord）来定义，具体通过在FP + ord中部分模拟Schreier-Sims算法实现。作为推论，当基础群为阿贝尔群时，带计数的不动点逻辑（FPC）也可定义该操作；特别地，FPC可定义任何具有阿贝尔颜色的图的自同构群，尽管它无法对这类图进行规范化。

英文摘要

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

发表机构

• University of Cambridge（剑桥大学）

机构由 AI 辅助整理，请以论文原文为准。