Which dynamic queries can be maintained efficiently? For constant-size changes, it is known that constant-depth circuits or, equivalently, first-order updates suffice for maintaining many important queries, among them reachability, tree isomorphism, and the word problem for context-free languages. In other words, these queries are in the dynamic complexity class DynFO. We show that most of the existing results for constant-size changes can be recovered for batch changes of polylogarithmic size if one allows circuits of depth O(log log n) or, equivalently, first-order updates that are iterated O(log log n) times.

翻译：哪些动态查询能够被高效维护？对于恒定大小的变化，已知恒定深度电路或等价的一阶更新足以维护许多重要查询，包括可达性、树同构以及上下文无关语言的字问题。换言之，这些查询属于动态复杂性类DynFO。我们证明，若允许深度为O(log log n)的电路或等价地迭代O(log log n)次的一阶更新，则对于多对数规模的批量变化，大多数现有关于恒定大小变化的结果仍可成立。