Dynamic Graph Queries

Graph databases in many applications---semantic web, transport or biological networks among others---are not only large, but also frequently modified. Evaluating graph queries in this dynamic context is a challenging task, as those queries often combine first-order and navigational features. Motivated by recent results on maintaining dynamic reachability, we study the dynamic evaluation of traditional query languages for graphs in the descriptive complexity framework. Our focus is on maintaining regular path queries, and extensions thereof, by first-order formulas. In particular we are interested in path queries defined by non-regular languages and in extended conjunctive regular path queries (which allow to compare labels of paths based on word relations). Further we study the closely related problems of maintaining distances in graphs and reachability in product graphs. In this preliminary study we obtain upper bounds for those problems in restricted settings, such as undirected and acyclic graphs, or under insertions only, and negative results regarding quantifier-free update formulas. In addition we point out interesting directions for further research

Dynamic Conjunctive Queries

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