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(2^{2^{poly(n)}},O(n)). Providing a matching lower bound, we show that there is some fixed ground tree rewrite graph whose first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists 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.

💡 Research Summary

The paper “The First‑Order Theory of Ground Tree Rewrite Graphs” investigates the algorithmic complexity of the first‑order (FO) theory of graphs generated by ground tree rewrite systems (GTRS). A GTRS is a term‑rewriting system where all rules are variable‑free; each rule replaces a subtree by another subtree. The transition graph of a GTRS, called a ground tree rewrite graph (GTRG), can model hierarchical multithreaded programs and generalises push‑down systems.

The authors establish three main results.

1. Upper bound – For any GTRG, the uniform FO theory (the set of pairs ⟨GTRS, φ⟩ such that φ holds in the graph of the system) lies in the alternating‑time class
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