  The article investigates classes of queries maintainable by conjunctive
queries (CQs) and their extensions and restrictions in the dynamic complexity
framework of Patnaik and Immerman. Starting from the basic language of
quantifier-free conjunctions of positive atoms, it studies the impact of
additional operators and features - such as union, atomic negation and
quantification - on the dynamic expressiveness, for the standard semantics as
well as for Delta-semantics.
  Although many different combinations of these features are possible, they
basically yield five important fragments for the standard semantics,
characterized by the addition of (1) arbitrary quantification and atomic
negation, (2) existential quantification and atomic negation, (3) existential
quantification, (4) atomic negation (but no quantification), and by (5) no
addition to the basic language at all. While fragments (3), (4) and (5) can be
separated, it remains open whether fragments (1), (2) and (3) are actually
different. The fragments arising from Delta-semantics are also subsumed by the
standard fragments (1), (2) and (4). The main fragments of DynFO that had been
studied in previous work, DynQF and DynProp, characterized by quantifier-free
update programs with or without auxiliary functions, respectively, also fit
into this hierarchy: DynProp coincides with fragment (4) and DynQF is strictly
above fragment (4) and within fragment (3).
  As a further result, all (statically) FO-definable queries are captured by
fragment (2) and a complete characterization of these queries in terms of
non-recursive dynamic programs with existential update formulas with one
existential quantifier is given.
