Dynamic Complexity was introduced by Immerman and Patnaik PI97 in the nineties and has seen a resurgence of interest with the positive resolution of their conjecture on directed reachability in DynFO DKMSZ18. Since then many natural problems related to reachability and matching have been placed in DynFO and related classes DMVZ18,DKMTVZ20,DTV21. In this work, we place some dynamic problems from group theory in DynFO. In particular, suppose we are given an arbitrary multiplication table over n elements representing an unstructured binary operation (representing a structure called a magma). Suppose the table evolves through a change in one of its n^2 entries in one step. For a set S of magma elements which also changes one element at a time, we can maintain enough auxiliary information so that when the magma is a group, we are able to answer the Cayley Group Membership (CGM) problem for S and a target t (i.e. "Is t a product of elements from S? ") using an FO query at every step. This places the dynamic CGM problem (for groups) when the ambient magma is specified via a table in DynFO. In contrast, for the table setting, statically CGM was known to be in the class Logspace BarringtonM06. Building on the dynamic CGM result, we can maintain the isomorphism of of two magmas, whenever both are Abelian groups, in DynFO. Our techniques include a way to maintain the powers of the elements of a magma in DynFO using left associative parenthesisation, the notion of cube independence to cube generate a subgroup generated by a set, a way to maintain maximal cube independent sequences in a magma along with some group theoretic machinery available from McKenzieCook. The notion of cube independent sequences is new as far as we know and may be of independent interest. These techniques are very different from the ones employed in Dynamic Complexity so far.

翻译：动态复杂性由Immerman与Patnaik于90年代提出（PI97），并随着DynFO中有向可达性猜想的正面解决（DKMSZ18）而重新引起研究兴趣。此后，许多与可达性和匹配相关的自然问题被归入DynFO及其衍生类（DMVZ18, DKMTVZ20, DTV21）。本研究将群论中的若干动态问题归入DynFO。具体而言，假设我们有一个基于n个元素的任意乘法表，表示一种无结构的二元运算（即幺拟结构）。假设该表通过单步改变其n²个条目中的一个进行演化。对于同样逐次改变一个元素的幺拟元素集合S，我们能够维护足够的辅助信息，使得当该幺拟为群时，我们可以在每一步通过FO查询回答S与目标元素t的凯莱群成员问题（CGM），即"t是否可由S中元素相乘得到？"。这便将动态CGM问题（针对群）在乘法表指定的背景幺拟中归入DynFO。相比之下，在静态乘法表场景下，CGM已知属于对数空间类（BarringtonM06）。基于动态CGM结果，我们还能在DynFO中维护两个幺拟（当两者均为阿贝尔群时）的同构性。我们的技术包括：利用左结合括号化在DynFO中维护幺拟元素幂次的方法；基于立方独立概念生成集合生成子群的立方生成子；维护幺拟中最大立方独立序列的方法，以及结合McKenzieCook的群论工具。据我们所知，立方独立序列的概念是全新的，可能具有独立研究价值。这些方法与当前动态复杂性领域采用的技术截然不同。