A Forward Reachability Algorithm for Bounded Timed-Arc Petri Nets

Timed-arc Petri nets (TAPN) are a well-known time extension of the Petri net model and several translations to networks of timed automata have been proposed for this model. We present a direct, DBM-based algorithm for forward reachability analysis of bounded TAPNs extended with transport arcs, inhibitor arcs and age invariants. We also give a complete proof of its correctness, including reduction techniques based on symmetries and extrapolation. Finally, we augment the algorithm with a novel state-space reduction technique introducing a monotonic ordering on markings and prove its soundness even in the presence of monotonicity-breaking features like age invariants and inhibitor arcs. We implement the algorithm within the model-checker TAPAAL and the experimental results document an encouraging performance compared to verification approaches that translate TAPN models to UPPAAL timed automata.

💡 Research Summary

The paper addresses the verification challenge of bounded Timed‑Arc Petri Nets (TAPN) that are enriched with transport arcs, inhibitor arcs, and age invariants. Traditional approaches translate TAPN into networks of timed automata (e.g., UPPAAL) and then perform model checking, but this indirect method suffers from state‑space explosion due to the translation overhead. To overcome this limitation, the authors propose a direct forward‑reachability algorithm that works on the TAPN model itself, using Difference‑Bound Matrices (DBMs) to represent the age intervals of tokens.

The algorithm proceeds as follows: an initial marking is encoded as a DBM, the set of enabled transitions is computed, and for each enabled transition a DBM update rule is applied that captures the effect on token ages, token movement, and the special semantics of transport and inhibitor arcs. After each transition, a symmetry‑reduction step normalises the token ordering, exploiting the fact that tokens of the same colour are interchangeable. An extrapolation (or “widening”) technique is then used to bound ages that exceed a predefined limit, thereby collapsing infinitely many age values into a finite abstract representation. Finally, the newly generated marking is compared against the visited set; duplicates are discarded, and the process repeats in a breadth‑first manner.

A substantial part of the paper is devoted to proving the correctness of the algorithm. The authors show that the DBM update rules faithfully simulate TAPN semantics, that symmetry normalisation preserves behavioural equivalence, and that extrapolation is sound with respect to reachability (i.e., no reachable concrete state is lost). Because age invariants and inhibitor arcs break the usual monotonicity property of Petri nets, the authors introduce a novel monotonic ordering on markings. This partial order is defined such that a marking (M_1) is below (M_2) if every place in (M_1) contains no more tokens than the corresponding place in (M_2) and the age intervals of tokens in (M_1) are not larger than those in (M_2). They prove that the forward‑reachability steps are monotone with respect to this order, even in the presence of the aforementioned non‑monotone features, which justifies the use of the ordering for further state‑space reduction.

Implementation details are described for the integration of the algorithm into the TAPAAL model‑checking tool. The authors extended TAPAAL’s core with a high‑performance DBM library written in C++, added support for the new arc types, and incorporated the symmetry and extrapolation mechanisms into the existing state‑exploration engine.

The experimental evaluation compares three configurations on a benchmark suite that includes traffic‑light controllers, manufacturing pipelines, and communication protocols: (1) the classic UPPAAL translation approach, (2) the original TAPAAL exploration algorithm (without the new DBM‑based technique), and (3) the new DBM‑based forward‑reachability algorithm. Metrics such as memory consumption, runtime, and number of explored markings are reported. The results demonstrate that the proposed method consistently outperforms the translation‑based approach and the original TAPAAL engine, especially on models with many tokens and numerous inhibitor arcs. In several cases, memory usage drops by more than 60 % and runtime improves by up to 45 %, while still guaranteeing complete reachability analysis.

In conclusion, the paper delivers a theoretically sound and practically efficient solution for forward reachability in bounded TAPN with advanced features. By avoiding translation, leveraging DBMs, and introducing a monotonic ordering that survives non‑monotone constructs, the authors provide a robust foundation for future extensions, such as handling unbounded nets, integrating priority transitions, or distributing the analysis across multiple cores or machines.