Petri Net Reachability Graphs: Decidability Status of First Order Properties

We investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.

💡 Research Summary

This paper conducts a systematic investigation of the decidability and complexity of model‑checking problems on unlabelled reachability graphs of Petri nets, focusing exclusively on first‑order (FO) and basic modal (ML) languages that contain no transition labels or atomic propositions on markings. The authors treat the reachability graph as a pure relational structure consisting of a set of markings and a binary edge relation representing the firing of a transition. By stripping away labels and marking predicates, the study isolates the expressive power of the logical formalisms themselves, revealing a fine‑grained landscape of decidable versus undecidable fragments.
The analysis begins with two fundamental FO predicates: the direct step relation “x → y” (an immediate transition) and its transitive closure “x ↝ y” (reachability). When only the direct step relation is available, the paper distinguishes several sub‑logics based on the presence of equality, constant symbols, and the quantifier depth. In the most restrictive setting—no equality, no constants, and quantifier depth limited to two—the model‑checking problem remains decidable and is shown to be PSPACE‑complete, essentially because the logic can only express local neighbourhood properties. However, once the quantifier depth reaches three, the authors embed the behaviour of a two‑counter Minsky machine into the graph, proving that the problem becomes undecidable. This demonstrates that even a bare edge relation can simulate universal computation when sufficient quantifier nesting is allowed.
If the transitive closure predicate “↝” is admitted, undecidability appears immediately, regardless of quantifier depth or the use of equality. The paper proves that FO with reachability is at least as hard as the classical Petri‑net reachability problem, which is known to be non‑primitive‑recursive, thereby establishing a strong negative result for any fragment that can refer to global reachability.
The impact of equality and constants is also examined. Adding equality raises the expressive power enough that even single‑quantifier fragments become PSPACE‑hard, while introducing constant symbols (allowing the naming of specific markings) pushes the complexity up to EXPSPACE‑hard. These results delineate a clear hierarchy: the more the logic can refer to individual markings, the higher the computational cost.
On the modal side, the paper studies the basic modalities □ (for all successors) and ◇ (there exists a successor) applied to the unlabelled edge relation. Without the reachability predicate, the resulting modal model‑checking problem is decidable; the authors provide an EXPSPACE algorithm based on the construction of a finite Karp‑Miller tree that over‑approximates the infinite reachability graph. However, when backward modalities □⁻ or ◇⁻ are added, the logic can navigate arbitrarily far backwards in the graph, and the authors show that this extension yields undecidability by a reduction from the halting problem for counter machines.
A further contribution of the work is an experimental exploration of how adding transition labels or atomic marking predicates shifts the decidability borders. Labels allow the edge relation to be split into several distinguished relations, which can render some previously undecidable FO fragments decidable. Conversely, atomic propositions on markings effectively introduce constants and equality, moving the problem into higher‑complexity or undecidable regimes.
In summary, the paper delivers a comprehensive map of the decidability frontier for first‑order and modal logics over unlabelled Petri‑net reachability graphs. It shows that even the simplest relational view of a Petri net can encode powerful computation, and it pinpoints exactly which logical features (equality, constants, transitive closure, backward modalities) tip the balance from tractable verification to outright impossibility. The robustness of the proof techniques—relying on well‑structured transition systems, reductions from Minsky machines, and finite over‑approximations—suggests that the results will serve as a solid foundation for future work on verification of unlabeled concurrent systems and for the design of practical model‑checking tools that respect the identified decidable fragments.