  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.
