Maximal Structuring of Acyclic Process Models

This paper contributes to the solution of the problem of transforming a process model with an arbitrary topology into an equivalent structured process model. In particular, this paper addresses the subclass of process models that have no equivalent well-structured representation but which, nevertheless, can be partially structured into their maximally-structured representation. The structuring is performed under a behavioral equivalence notion that preserves observed concurrency of tasks in equivalent process models. The paper gives a full characterization of the subclass of acyclic process models that have no equivalent well-structured representation but do have an equivalent maximally-structured one, as well as proposes a complete structuring method.

💡 Research Summary

The paper addresses the problem of transforming acyclic process models that lack an equivalent well‑structured representation into a maximally‑structured form. A well‑structured model is one in which every split gateway has a matching join gateway, forming a single‑entry‑single‑exit (SESE) component. Many real‑world models, however, contain splits without corresponding joins and therefore cannot be fully structured under existing techniques. The authors introduce the notion of a maximally‑structured model: a model that is behaviorally equivalent (under fully concurrent bisimulation) to the original and contains the greatest possible number of SESE components among all equivalent models. In other words, no other model equivalent to it can have more SESE fragments.

The proposed solution consists of two main phases. First, the original process model is mapped to a workflow net (WF‑net), a special class of free‑choice Petri nets that captures the control‑flow semantics. From this net a complete prefix unfolding is constructed. An unfolding is a tree‑like occurrence net that explicitly enumerates all possible executions of the system. To keep the unfolding finite and manageable, the authors employ an “adequate order” to identify cut‑off events, which truncate the unfolding once further expansion would only repeat already explored markings. They refine this concept by defining a “proper complete prefix unfolding,” which requires cut‑off events to be “healthy”: the cut after the event must be identical to the cut after its corresponding event (corr(e)). This healthiness condition guarantees that every XOR split and join pair, as well as every AND split and join, appear together in the same branch of the unfolding, preserving the full concurrency information needed for later structuring.

Second, the structural information extracted from the proper unfolding is used to restructure the model. The authors employ the Refined Process Structure Tree (RPST), a hierarchical decomposition of a process model into four component types: Trivial (single arc), Polygon (sequence), Bond (well‑structured fragment with a single entry and exit), and Rigid (any fragment that does not fit the previous categories). Rigid components are precisely those that prevent a model from being well‑structured. For each Rigid node, the ordering relations among its child nodes—causal, inverse causal, conflict, and concurrency—are derived from the proper unfolding. If these relations match a pattern that can be expressed by a well‑structured sub‑tree (typically an XOR‑AND‑XOR pattern), the Rigid component is replaced by that sub‑tree. The replacement preserves the original behavior because the ordering relations are identical, and it increases the number of SESE components.

The overall algorithm proceeds as follows:

1. Convert the input process model into a WF‑net.
2. Generate a proper complete prefix unfolding using an adequate order that yields a compact yet information‑complete prefix.
3. Build the RPST of the original model and locate all Rigid components.
4. For each Rigid component, extract the ordering relations from the unfolding.
5. Synthesize a well‑structured fragment that reproduces those relations and substitute it for the Rigid component.
6. Verify that the resulting model is maximally‑structured by checking fully concurrent bisimulation with the original.

The paper demonstrates the technique on a representative unstructured model (Fig. 1(a)). Although the model cannot be fully structured, the method produces a maximally‑structured version (Fig. 1(b)) that contains the maximal number of SESE fragments while preserving observed concurrency. The authors also discuss the impact of the chosen adequate order on the size of the prefix unfolding; newer orders lead to significantly smaller prefixes, making the approach practical for larger models.

In conclusion, the work provides a rigorous, behavior‑preserving method to push acyclic process models as far toward structuredness as possible, even when a fully well‑structured equivalent does not exist. By leveraging proper complete prefix unfoldings and RPST‑based analysis, the method guarantees maximal SESE coverage, retains full concurrency semantics, and offers a solid foundation for automated tooling in process model verification, optimization, and execution.