Acyclic Petri and Workflow Nets with Resets

In this paper we propose two new subclasses of Petri nets with resets, for which the reachability and coverability problems become tractable. We add an acyclicity condition that only applies to the consumptions and productions, not the resets. The first class is acyclic Petri nets with resets, and we show that coverability is PSPACE-complete for them. This contrasts the known Ackermann-hardness for coverability in (not necessarily acyclic) Petri nets with resets. We prove that the reachability problem remains undecidable for acyclic Petri nets with resets. The second class concerns workflow nets, a practically motivated and natural subclass of Petri nets. Here, we show that both coverability and reachability in acyclic workflow nets with resets are PSPACE-complete. Without the acyclicity condition, reachability and coverability in workflow nets with resets are known to be equally hard as for Petri nets with resets, that being Ackermann-hard and undecidable, respectively.

💡 Research Summary

The paper introduces two novel subclasses of Petri nets that incorporate reset arcs while imposing an acyclicity constraint only on the ordinary consumption and production arcs. This nuanced restriction allows resets—operations that unconditionally set the token count of a place to zero—to form cycles, but guarantees that the flow of tokens through the net never cycles. By doing so, the authors aim to identify models that are expressive enough for practical applications yet amenable to algorithmic analysis.

Acyclic Petri Nets with Resets (APNR).
The first subclass, APNR, consists of standard Petri nets extended with reset arcs, under the condition that the graph formed by all non‑reset arcs is a directed acyclic graph (DAG). The authors prove that the coverability problem for APNR is PSPACE‑complete. The hardness proof reduces from the known PSPACE‑complete coverability problem for ordinary acyclic Petri nets, showing that reset arcs can be added without increasing the computational burden. Membership in PSPACE is established by representing a configuration as a pair (token vector, set of enabled transitions) and exploiting the DAG structure to bound the depth of any search to a polynomial function of the net size. Consequently, a depth‑first exploration can be simulated within polynomial space.

In contrast, the reachability problem remains undecidable for APNR. The key insight is that, because resets are allowed to create cycles, one can simulate a two‑counter Minsky machine. The construction uses ordinary arcs to implement increment and decrement operations, while reset arcs emulate zero‑tests by clearing a place. This encoding demonstrates that APNR can encode arbitrary Turing‑machine computations, preserving the classic undecidability result for reachability in Petri nets with resets.

Acyclic Workflow Nets with Resets (AWNR).
The second subclass focuses on workflow nets—a widely used subclass of Petri nets characterized by a single source place, a single sink place, and the property that every transition lies on a path from source to sink. Adding the same acyclicity restriction on ordinary arcs yields AWNR. For this class, both coverability and reachability are shown to be PSPACE‑complete.

The PSPACE‑hardness proofs employ reductions from Quantified Boolean Formula (QBF) satisfiability. Variables and quantifiers are encoded as sequences of transitions and resets, while the target marking encodes the truth value of the formula. Because workflow nets enforce a linear, single‑pass execution order, the reduction can be performed in polynomial time and respects the acyclicity condition.

Membership in PSPACE follows from the structural constraints of workflow nets. The single‑entry, single‑exit topology ensures that each token traverses the net at most once, and resets only zero out places without generating new tokens. Thus the reachable state space can be explored by tracking a compact representation of the token distribution and a bounded number of reset applications, all within polynomial space.

Technical Contributions and Insights.

1. Acyclicity on ordinary arcs, not resets – This selective restriction is the core novelty. It preserves the expressive power of resets (allowing arbitrary zero‑tests) while preventing the exponential blow‑up associated with token cycles.
2. Compression of markings – The authors show that, due to the DAG structure, the maximum token count in any place is bounded by the number of transitions, enabling a succinct bit‑vector encoding of markings.
3. Reset‑only impact – Since a reset merely forces a place to zero, its effect can be evaluated locally without influencing the global depth of the search, simplifying the PSPACE algorithm.
4. Workflow‑net specific reductions – By aligning the order of quantified variables with the unique topological order of a workflow net, the QBF reduction becomes natural and transparent.

Comparison with Existing Results.

The table highlights that imposing acyclicity on ordinary arcs dramatically lowers the complexity of coverability for both subclasses, and even brings reachability down to PSPACE for the workflow‑net case.

Practical Implications.
The PSPACE‑complete status of both problems in AWNR suggests that existing verification technologies—SAT/SMT solvers, BDD‑based model checkers, and symbolic state‑space exploration tools—can be adapted to handle nets with resets efficiently. This opens the door to rigorous analysis of business processes, manufacturing pipelines, and distributed protocols that employ reset‑like recovery or reinitialization mechanisms without sacrificing tractable verification.

Future Directions.
The authors propose several extensions: (i) enriching the model with weighted arcs or timing constraints while preserving the acyclicity condition, (ii) investigating parameterized complexity with respect to the number of reset arcs, and (iii) developing prototype tools that implement the PSPACE algorithms and evaluate them on real‑world workflow specifications.

In summary, the paper demonstrates that a carefully chosen structural restriction—acyclicity of consumption/production arcs—combined with the flexibility of reset arcs yields two meaningful subclasses where previously intractable verification problems become PSPACE‑complete. This balance between expressiveness and algorithmic feasibility offers a promising avenue for both theoretical exploration and practical tool development in the analysis of concurrent and distributed systems.