Dynamic Complexity under Definable Changes

TL;DR

This paper explores the limits of dynamic complexity in the DynFO framework, showing that many queries like reachability and context-free language membership can be maintained under definable changes, while some cannot.

Contribution

It extends the understanding of what can be maintained dynamically under definable changes, including new results for reachability and context-free languages.

Findings

All (uniform) AC^1 queries can be maintained by first-order dynamic programs under parameter-free definable changes.

Reachability in undirected graphs and DAGs is first-order maintainable under specific definable changes.

Context-free languages are maintainable under Sigma_1-defined changes.

Abstract

This paper studies dynamic complexity under definable change operations in the DynFO framework by Patnaik and Immerman. It is shown that for changes definable by parameter-free first-order formulas, all (uniform) $AC^1$ queries can be maintained by first-order dynamic programs. Furthermore, many maintenance results for single-tuple changes are extended to more powerful change operations: (1) The reachability query for undirected graphs is first-order maintainable under single tuple changes and first-order defined insertions, likewise the reachability query for directed acyclic graphs under quantifier-free insertions. (2) Context-free languages are first-order maintainable under $\Sigma_1$-defined changes. These results are complemented by several inexpressibility results, for example, that the reachability query cannot be maintained by quantifier-free programs under definable, quantifier-free deletions.