The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.

We study the problem of incrementally maintaining an XPath query on an XML database under updates. The updates we consider are node insertion, node deletion, and node relabeling. Our main results are that downward XPath queries can be incrementally maintained in time O(depth(D) · poly(Q)) and conjunctive forward XPath queries in time O(depth(D)· log(width(D))·poly(Q)), where D is the size of the database, Q the size of the query, and depth(D) and width(D) are the nesting depth and maximum number of siblings in the database, respectively. The auxiliary data structures for maintenance are linear in D and polynomial in Q in all these cases.

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