- The paper surveys 25 years of research on decidability problems within the core Petri net framework, excluding extensions.
- While some properties like boundedness and reachability are decidable but complex, most behavioral equivalences prove to be undecidable for general Petri nets.
- Decidability for temporal logic model checking in Petri nets varies significantly based on the specific logic and included predicates.
Decidability Issues for Petri Nets: A Survey
The paper "Decidability Issues for Petri Nets" by Javier Esparza and Mogens Nielsen offers a comprehensive survey of research centered around the decidability problems for Petri nets over a span of 25 years. The survey meticulously gathers essential results associated with fundamental properties, equivalence notions, and temporal logics. Notably, this survey excludes extensions of the Petri net model, thereby focusing solely on the core framework.
Overview of Petri Nets
Petri nets, established by C.A. Petri in 1962 for modeling parallel processes, have been closely studied alongside vector addition systems introduced by Karp and Miller. Despite the philosophical differences between these approaches, they are mathematically equivalent, fueling research into various decidability questions. The paper groups its highlights into three main sections: specific properties, behavioral equivalences, and model checking for temporal logics.
Decidability of Specific Properties
Despite the expressive power of Petri nets, several significant properties related to their verification are decidable, albeit with high complexities. The boundedness problem, for instance, was shown to be decidable by Karp and Miller, who identified "token generators" through their construction of the coverability tree; however, the process is non-primitive recursive in complexity. Rackoff improved upon this with an algorithm requiring double-exponential space, closely matching Lipton's lower bounds.
The reachability problem was first proven decidable by Mayr and later simplified by Kosaraju, yet remains a significant issue given its computational complexity. By contrast, certain subclasses, such as persistent or conflict-free nets, offer more tractable solutions.
Behavioral Equivalences
Equivalence problems for Petri nets generally yield undecidable outcomes across various notions of behavior. Marking equivalence, trace equivalence, and language equivalence are notably undecidable, with trace and language equivalence proving undecidable even for BPP-nets—highlighting the limited expressive power of these nets. However, for subclasses with semilinear reachability sets or certain structural properties, equivalence problems can be resolved, leading to decidable outcomes.
Temporal Logics
The paper discusses the decidability of model checking problems for both linear and branching time temporal logics. Unsurprisingly, many branching time logics demonstrate undecidability due to the expressive complexity inherent in Petri nets. Conversely, linear time temporal logics showcase decidability when lacking place predicates, while inclusion typically leads to undecidability. Notably, even fragmented logics with restrictions on Boolean operators yield complex decision problems tightly linked to the reachability problem.
Implications and Future Directions
The survey underscores the Petri nets' role as a source of naturally non-primitive recursive problems, presenting challenges yet offering rich avenues for computational theory and practical applications in concurrent system analysis. The complexity results emerging from this study not only highlight intrinsic difficulties but also guide the continued exploration of decidable subclasses and conditions under which certain behaviors and properties can be computed efficiently.
Moving forward, further exploration into complexity reductions, subclass decidability, and the impact of temporal logic extensions on Petri nets holds promise for advancing both theoretical and practical applications, potentially bridging the gap between expressiveness and computational feasibility.