First-order logic with the reachability predicate (FO(Reach)) is an important means of specification in system analysis. Its decidability status is known for some individual types of infinite-state systems such as pushdown (decidable) and vector addition systems (undecidable).
This work aims at a general understanding of which types of systems admit decidability. As a unifying model, we employ valence systems over graph monoids, which feature a finite-state control and are parameterized by a monoid to represent their storage mechanism. As special cases, this includes pushdown systems, various types of counter systems (such as vector addition systems) and combinations thereof. Our main result is a complete characterization of those graph monoids where FO(Reach) is decidable for the resulting transition systems.