  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.
