Dynamic Complexity under Definable Changes

Abstract

In the setting of dynamic complexity, the goal of a dynamic program is to maintain the result of a fixed query for an input database that is subject to changes, possibly using additional auxiliary relations. In other words, a dynamic program updates a materialized view whenever a base relation is changed. The update of query result and auxiliary relations is specified using first-order logic or, equivalently, relational algebra. The original framework by Patnaik and Immerman only considers changes to the database that insert or delete single tuples. This article extends the setting to definable changes , also specified by first-order queries on the database, and generalizes previous maintenance results to these more expressive change operations. More specifically, it is shown that the undirected reachability query is first-order maintainable under single-tuple changes and first-order defined insertions, likewise the directed reachability query for directed acyclic graphs is first-order maintainable under insertions defined by quantifier-free first-order queries. These results rely on bounded bridge properties , which basically say that, after an insertion of a defined set of edges, for each connected pair of nodes there is some path with a bounded number of new edges. While this bound can be huge, in general, it is shown to be small for insertion queries defined by unions of conjunctive queries. To illustrate that the results for this restricted setting could be practically relevant, they are complemented by an experimental study that compares the performance of dynamic programs with complex changes, dynamic programs with single changes, and with recomputation from scratch. The positive results are complemented by several inexpressibility results. For example, it is shown that—unlike for single-tuple insertions—dynamic programs that maintain the reachability query under definable, quantifier-free changes strictly need update formulas with quantifiers. Finally, further positive results unrelated to reachability are presented: it is shown that for changes definable by parameter-free first-order formulas, all LOGSPACE-definable (and even AC 1 -definable) queries can be maintained by first-order dynamic programs.