On the quantifier-free dynamic complexity of Reachability

TL;DR

This paper investigates the limitations of quantifier-free dynamic complexity in maintaining reachability queries, showing that certain restrictions prevent dynamic maintenance and highlighting the relative power of auxiliary relation arities.

Contribution

It proves inexpressibility results for reachability in quantifier-free dynamic complexity, demonstrating the limitations of binary and unary auxiliary relations and the increased power of ternary relations.

Findings

Reachability cannot be dynamically maintained with binary auxiliary relations.

Unary auxiliary functions are insufficient for maintaining reachability.

Ternary auxiliary relations are more powerful than binary ones for graph queries.

Abstract

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.