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

Abstract

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.

An optimal construction of Hanf sentences

“…Finally, we should also mention the normal form by Schwentick and Barthelmann [24] that rejoins the two formalizations of locality by Gaifman and by Hanf. Gaifman's and Hanf's theorems have found applications in finite model theory and in particular in parametrized complexity. Namely, they lead to efficient parametrized algorithms deciding whether a formula holds in a (finite) structure [25,19,8,9,16,3,21,17,18] and even to more general algorithms that list all the satisfying assignments [5,15]. Hanf's theorem was also used in the transformation of logical formulas into different automata models [27,13,24,2,11,1,12].…”

The First-Order Theory of Ground Tree Rewrite Graphs

“…A complexity gap in this context exists for the first-order theory of the one-step rewrite graph of a semi-Thue system (word rewrite system): It is known to be 2EXPSPACE-hard and decidable but it is not known to be elementary [26]. One may try to tackle this problem using techniques similar to those used in this paper.…”