Continuous Reachability for Unordered Data Petri nets is in PTime

Unordered data Petri nets (UDPN) are an extension of classical Petri nets with tokens that carry data from an infinite domain and where transitions may check equality and disequality of tokens. UDPN are well-structured, so the coverability and termination problems are decidable, but with higher complexity than for Petri nets. On the other hand, the problem of reachability for UDPN is surprisingly complex, and its decidability status remains open. In this paper, we consider the continuous reachability problem for UDPN, which can be seen as an over-approximation of the reachability problem. Our main result is a characterization of continuous reachability for UDPN and polynomial time algorithm for solving it. This is a consequence of a combinatorial argument, which shows that if continuous reachability holds then there exists a run using only polynomially many data values.

💡 Research Summary

The paper investigates the continuous reachability problem for Unordered Data Petri Nets (UDPN), an extension of classical Petri nets where each token carries a datum from an infinite domain and transitions may test equality or disequality of these data values. While coverability and termination are known to be decidable for UDPN (albeit with higher complexity than for ordinary Petri nets), the decidability of the standard reachability problem remains open, and its complexity is expected to be very high. Continuous reachability—allowing tokens to be split into arbitrary positive rational amounts—serves as an over‑approximation of the standard reachability relation and is therefore of practical interest for verification.

The authors’ main contribution is twofold. First, they prove a combinatorial bound (Theorem 1) stating that if a rational (ℚ) or positive‑rational (ℚ⁺) marking f is reachable from an initial marking i via a continuous run, then there exists a continuous run ρ that uses only a polynomially bounded number of distinct data values. Formally, |dval(ρ)| ≤ |dval(i) ∪ dval(f)| + 1 + maxₜ |vars(t)|, where dval(·) denotes the set of data values appearing in a marking or run, and vars(t) is the set of variables actually used by transition t. This result shows that one never needs to explore the infinite data domain; a small “witness” run suffices.

Second, leveraging this bound, the paper presents a polynomial‑time algorithm for deciding continuous reachability (Theorem 2). The algorithm proceeds as follows:

1. Data‑value restriction – Compute the finite set of candidate data values given by the bound above. This set is of size polynomial in the size of the net and the two markings.

2. State‑equation formulation – Encode the effect of a run as a linear equation Δ(ρ) = f – i, where Δ(ρ) is the sum of the displacement vectors of the fired transitions, each multiplied by its rational firing coefficient. The variables of this system are the coefficients of the steps and the assignments of data values to transition variables.

3. Histogram compression – Instead of enumerating each occurrence of a data value, the algorithm aggregates identical patterns into histograms. A histogram records, for a given data value, how many tokens of each type are consumed and produced across all steps. This reduces the number of rows/columns of the linear system from exponential (if each data value were treated separately) to polynomial.

4. Implication constraints – Equality and disequality tests in transitions are expressed as logical implications among histogram entries. These constraints are added to the linear system, yielding a mixed system of linear equations and Boolean implications.

5. Net transformation – To avoid spurious solutions introduced by the compression, the net is transformed into a normal form where each transition’s consumption/production pattern is simplified. The histogram technique is then applied to this transformed net, guaranteeing that any solution of the compressed system corresponds to a genuine continuous run.

6. Solving – The resulting polynomial‑size system can be solved using standard linear‑programming techniques (or Gaussian elimination for the rational part) combined with a simple SAT‑style propagation for the implication constraints. If a solution exists, the algorithm reconstructs a concrete continuous run; otherwise, it reports non‑reachability.

Complexity analysis shows that each step runs in time polynomial in the size of the net, the number of variables, and the size of the two markings. Since continuous reachability for ordinary Petri nets is already known to be PTIME‑complete, the authors conclude that continuous reachability for UDPN is also PTIME‑complete.

The paper’s technical novelty lies in the combinatorial argument that yields the data‑value bound, and in adapting the histogram compression technique—originally developed for data‑free nets—to the data‑rich setting while handling equality/disequality constraints via implications. By doing so, the authors bridge the gap between the high complexity of standard reachability and the tractability of its continuous over‑approximation, providing a practical tool for verification.

From a verification perspective, the result enables efficient over‑approximation of coverability for UDPN, as continuous reachability can be used as a fast pre‑filter before invoking more expensive exact analyses. The authors suggest future work on extending the approach to integer (ℤ) reachability, integrating the algorithm into existing Petri‑net analysis tools, and exploring experimental evaluation on real‑world models that employ data‑rich tokens.