The First-Order Theory of Ground Tree Rewrite Graphs

We prove that the complexity of the uniform first-order theory of ground tree rewrite graphs is in ATIME(22 poly(n) , O(n)). Providing a matching lower bound, we show that there is a fixed ground tree rewrite graph whose first-order theory is hard for ATIME(22 poly(n) , poly(n)) with respect to logspace reductions. Finally, we prove that there is a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory. For a long version of this paper with complete proofs see [11].