Decision Problems for Petri Nets with Names

We prove several decidability and undecidability results for nu-PN, an extension of P/T nets with pure name creation and name management. We give a simple proof of undecidability of reachability, by reducing reachability in nets with inhibitor arcs to it. Thus, the expressive power of nu-PN strictly surpasses that of P/T nets. We prove that nu-PN are Well Structured Transition Systems. In particular, we obtain decidability of coverability and termination, so that the expressive power of Turing machines is not reached. Moreover, they are strictly Well Structured, so that the boundedness problem is also decidable. We consider two properties, width-boundedness and depth-boundedness, that factorize boundedness. Width-boundedness has already been proven to be decidable. We prove here undecidability of depth-boundedness. Finally, we obtain Ackermann-hardness results for all our decidable decision problems.

💡 Research Summary

The paper investigates a novel extension of classical Place/Transition (P/T) Petri nets, called nu‑Petri Nets (nu‑PN), which incorporates pure name creation (the ν‑operator) and explicit name management (comparison, duplication, and deletion). By enriching the net model with dynamically generated identifiers, nu‑PN can express behaviours that are impossible in ordinary P/T nets, notably the ability to simulate inhibitor arcs. The authors systematically explore the decidability landscape of several fundamental verification problems for nu‑PN, establishing both positive and negative results, and they precisely locate the computational hardness of the decidable cases.

Undecidability of Reachability
The first major contribution is a concise proof that the reachability problem for nu‑PN is undecidable. The authors construct a polynomial‑time reduction from reachability in Petri nets equipped with inhibitor arcs—a known undecidable problem—to reachability in nu‑PN. The reduction exploits the fact that a freshly generated name can serve as a unique token that blocks a transition precisely when a certain place is empty, thereby mimicking the semantics of an inhibitor arc. Consequently, nu‑PN strictly subsumes the expressive power of inhibitor‑arc nets and, by extension, exceeds that of ordinary P/T nets.

Well‑Structured Transition System (WSTS) Properties
Despite the undecidability of reachability, the paper shows that nu‑PN belong to the class of Well‑Structured Transition Systems. The authors define a quasi‑order on markings based on multiset inclusion of name‑annotated tokens and prove that this order is a well‑quasi‑order (wqo). They further demonstrate monotonicity of the transition relation with respect to this order, even in the presence of the ν‑operator, by carefully handling the introduction of fresh names as minimal elements. These two ingredients (wqo and monotonicity) satisfy the classic definition of a WSTS, allowing the authors to invoke generic WSTS algorithms.

From the WSTS framework, they derive decidability of coverability (whether a marking that dominates a given target exists) and termination (whether all executions are finite). The coverability algorithm proceeds via backward reachability, constructing a finite basis of upward‑closed sets, while termination is decided by checking for the existence of infinite strictly increasing sequences under the quasi‑order.

Strict WSTS and Boundedness
A deeper analysis reveals that nu‑PN are not only WSTS but strict WSTS. This stricter notion ensures that every infinite sequence of markings must contain a pair that is strictly comparable, which in turn guarantees the existence of a finite bound on the size of minimal elements. Leveraging this property, the authors prove that the boundedness problem—whether the net can generate arbitrarily many tokens—is decidable for nu‑PN. Their approach decomposes boundedness into two orthogonal dimensions:

1. Width‑boundedness: limits the cardinality of the set of distinct names that may appear in any reachable marking. This property had already been shown decidable in prior work, and the paper re‑affirms it using the strict WSTS framework.

2. Depth‑boundedness: limits the multiplicity of each individual name (i.e., how many tokens may carry the same identifier). The authors present a reduction from the halting problem of a two‑counter machine to depth‑boundedness, establishing its undecidability. The construction encodes counter values as the depth of a particular name, and the ν‑operator enables unbounded duplication, thereby simulating an unbounded counter.

Thus, while overall boundedness remains decidable due to the strict WSTS structure, its two components exhibit contrasting decidability statuses, highlighting a nuanced boundary between tractable and intractable aspects of name‑based dynamics.

Complexity Upper and Lower Bounds
For each of the decidable problems (coverability, termination, boundedness, width‑boundedness), the authors provide an upper bound by adapting standard WSTS algorithms, which are known to be non‑primitive recursive in the worst case. To complement these upper bounds, they prove Ackermann‑hardness: each problem can encode the computation of the Ackermann function, implying that any algorithm solving them must have at least Ackermann‑level time or space complexity. The hardness proofs involve encoding a sequence of increasingly large counters into the net’s name structure, thereby forcing any decision procedure to simulate an Ackermann‑type growth.

Implications and Future Work
The results paint a comprehensive picture of the verification landscape for systems that manipulate dynamically generated identifiers—a scenario common in mobile processes, security protocols, and dynamic network configurations. On the one hand, the undecidability of reachability warns that full state‑space exploration is infeasible. On the other hand, the WSTS and strict WSTS properties guarantee that many practically relevant safety and liveness questions (coverability, termination, boundedness) remain algorithmically tractable, albeit with extremely high worst‑case complexity. The dichotomy between decidable width‑boundedness and undecidable depth‑boundedness suggests that future research could focus on identifying useful subclasses of nu‑PN where depth‑boundedness becomes decidable, perhaps by restricting the form of name duplication or by imposing syntactic constraints.

In summary, the paper makes three core contributions: (1) a clean undecidability proof for reachability via inhibitor‑arc reduction, (2) the establishment of nu‑PN as (strict) WSTS leading to decidability of several key verification problems, and (3) precise complexity characterisation showing Ackermann‑hardness for all decidable cases while pinpointing depth‑boundedness as the sole undecidable boundedness variant. These insights significantly advance the theoretical understanding of name‑aware concurrent models and lay groundwork for future tool development targeting dynamic identifier management.