Decidable First-Order Theories of One-Step Rewriting in Trace Monoids

Abstract

We prove that the first-order theory of the one-step rewriting relation associated with a trace rewriting system is decidable but in general not elementary. This extends known results on semi-Thue systems but our proofs use new methods; these new methods yield the decidability of local properties expressed in first-order logic augmented by modulo-counting quantifiers. Using the main decidability result, we define several subclasses of trace rewriting systems for which the confluence problem is decidable.