Deciding Reachability and the Covering Problem with Diagnostics for Sound Acyclic Free-Choice Workflow Nets

A central decision problem in Petri net theory is reachability asking whether a given marking can be reached from the initial marking. Related is the covering problem (or sub-marking reachbility), which decides whether there is a reachable marking covering at least the tokens in the given marking. For live and bounded free-choice nets as well as for sound free-choice workflow nets, both problems are polynomial in their computational complexity. This paper refines this complexity for the class of sound acyclic free-choice workflow nets to a quadratic polynomial, more specifically to $O(P^2 + T^2)$. Furthermore, this paper shows the feasibility of accurately explaining why a given marking is or is not reachable. This can be achieved by three new concepts: admissibility, maximum admissibility, and diverging transitions. Admissibility requires that all places in a given marking are pairwise concurrent. Maximum admissibility states that adding a marked place to an admissible marking would make it inadmissible. A diverging transition is a transition which originally “produces” the concurrent tokens that lead to a given marking. In this paper, we provide algorithms for all these concepts and explain their computation in detail by basing them on the concepts of concurrency and post-dominance frontiers - a well known concept from compiler construction. In doing this, we present straight-forward implementations for solving (sub-marking) reachability.

💡 Research Summary

The paper addresses two fundamental decision problems in Petri net theory—reachability (whether a given marking can be reached from the initial marking) and the covering problem (whether a marking that dominates a given marking is reachable)—specifically for the class of sound acyclic free‑choice workflow nets (AFC‑WF‑Nets). While reachability for general Petri nets is Ackermann‑complete and for safe nets PSPACE‑complete, prior work has shown that for live and bounded free‑choice nets, as well as for sound free‑choice workflow nets, both problems are polynomial‑time solvable. However, the exact polynomial degree remained unknown.

The authors refine this result by proving that, for sound AFC‑WF‑Nets, both reachability and sub‑marking (covering) can be decided in quadratic time, O(P² + T²), where P is the number of places and T the number of transitions. The key contribution is a set of diagnostic concepts that not only decide the problems but also explain why a marking is reachable or not.

Admissibility is defined as the property that all places marked in a candidate marking are pairwise concurrent. Concurrency means that there exists a reachable marking containing tokens on both places simultaneously. In sound AFC‑WF‑Nets, safety guarantees that a place cannot be concurrent with itself, and Lemma 3.1 shows that two concurrent places cannot be connected by a directed path. Consequently, checking admissibility reduces to consulting a pre‑computed concurrency matrix, which can be built in O(P²) time.

Maximum admissibility strengthens admissibility: an admissible marking is maximally admissible if adding a token to any additional place would break concurrency. The authors prove that every reachable marking in a sound AFC‑WF‑Net must be maximally admissible, while every reachable sub‑marking must be admissible. Thus, failure of maximum admissibility immediately yields a concrete diagnostic: the set of places that are missing tokens required for maximality.

Admissibility and maximum admissibility are necessary but not sufficient conditions for reachability. To bridge the gap, the paper introduces diverging transitions. A diverging transition is a transition with multiple output places; firing it creates several tokens at once, potentially satisfying the concurrent token pattern required by the target marking. The authors employ the concept of post‑dominance frontiers—borrowed from compiler construction—to efficiently identify, for each transition, the set of places that are guaranteed to be marked after the transition dominates all paths to the sink. By intersecting these frontiers with the target marking, the algorithm can decide whether a suitable diverging transition exists without enumerating concrete firing sequences.

The overall algorithm proceeds as follows:

1. Pre‑processing – Compute the concurrency relation for all place pairs (O(P²)) and the post‑dominance frontier for each transition (O(T²)).
2. Admissibility test – Verify that the input marking is admissible; if not, report conflicting place pairs as a diagnostic.
3. Maximum admissibility test – If admissible, check whether it is maximal; if not, list the places lacking tokens.
4. Diverging transition search – Scan all transitions; for each, check whether its frontier covers the entire target marking. If a covering transition is found, the marking is reachable and the transition (plus its frontier) is returned as an explanatory witness. If none is found, the marking is declared unreachable, and the algorithm explains which diverging points are missing.

Because each step relies only on static structural information (no state‑space exploration), the worst‑case runtime stays within O(P² + T²). The authors also discuss how the method extends to extended free‑choice nets via a linear‑time transformation to simple free‑choice form, preserving the quadratic bound.

The paper situates its contribution in two practical motivations. First, many industrial business‑process models are naturally represented as free‑choice workflow nets, and soundness is a standard quality criterion. Empirical studies show that a large fraction of real‑world models are acyclic, making the proposed approach directly applicable. Second, recent work on “home‑cluster” nets and perpetual nets suggests that any perpetual net can be unrolled into a collection of acyclic sound free‑choice workflow nets; thus, the technique could eventually be combined with loop‑decomposition methods to handle a broader class of processes.

Experimental evaluation (briefly reported) confirms that the quadratic algorithm outperforms traditional reachability analysis based on exhaustive firing sequence generation, especially on medium‑sized models with hundreds of places and transitions. Moreover, the diagnostic output—conflicting places, missing tokens, and required diverging transitions—provides actionable insight for process designers, enabling them to pinpoint structural flaws or to adjust the model to achieve the desired behavior.

In summary, the paper delivers a theoretically tight quadratic‑time algorithm for reachability and covering in sound acyclic free‑choice workflow nets, enriches the decision procedure with concrete diagnostics based on admissibility, maximum admissibility, and diverging transitions, and demonstrates the practical relevance of these ideas for industrial process modeling and verification.