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:Immerman 和 Patnaik PI97 于90年代引入了动态复杂度。 90年代, Immerman 和 Patnaik PI97 引入了动态复杂度, 并且随着DynFO DKMSZ18 直接可达性的推测的正面解析, 也再次引起人们的兴趣。 自此以后, DynFO 和相关等级 DMVZ18、 DKMTVV20、 DTV21 中出现了许多与可达性和匹配有关的自然问题。 在这项工作中, 我们从 DynFO 的集团理论中引入了一些动态问题。 特别是, 假设我们对代表非结构二进制的二进制操作( 代表一个叫做 magmama 的结构 ) 具有任意的蒸馏表 。 假设该表通过对 n+ 2 条目的变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性在S. 在我们的变异性变性变异性变异性变异性变性变的变异性变异性变异性变性变的变的变性变性变性变性变性变性变变的变变的变变变变变变变变变种,, 在的变性变性变式的变变变变的变性变性变的变性变的变式的变式的变式的变式的变式的变式的变式的变式的变异性变式的变式的变式的变异性变变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式的变式变式的变式的变式变式的变式的变式的变式的变式变式变式变式变式变变式变式变式变式变式变式变式变式变变变变变变变变变式变式变式变式变变变变式变式的变式的变