On the Quantifier-Free Dynamic Complexity of Reachability

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 results are obtained including more inexpressibility results for reachability in a different setting, inexpressibility results for some other queries and normal forms for quantifier-free update programs.