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The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas.It is shown that, with this restriction, the reachability query cannot be dynamically maintained, neither with binary auxiliary relations nor with unary auxiliary functions, and that ternary auxiliary relations are more powerful with respect to graph queries than binary auxiliary relations.Further inexpressibility results are given for the reachability query in a different setting as well as for a syntactical restriction of quantifier-free update formulas. Moreover inexpressibility results for some other queries are presented.

Abstract. It is shown that the finite satisfiability problem for two-variable logic over structures with one total preorder relation, its induced successor relation, one linear order relation and some further unary relations is EXPSPACE-complete. Actually, EXPSPACEcompleteness already holds for structures that do not include the induced successor relation. As a special case, the EXPSPACE upper bound applies to two-variable logic over structures with two linear orders. A further consequence is that satisfiability of twovariable logic over data words with a linear order on positions and a linear order and successor relation on the data is decidable in EXPSPACE.As a complementing result, it is shown that over structures with two total preorder relations as well as over structures with one total preorder and two linear order relations, the finite satisfiability problem for two-variable logic is undecidable.

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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 ∆-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 ∆-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 ∃ 1 FO-programs is given.