Synthesizing Petri Nets from Labelled Petri Nets using Token Trail Regions

Synthesis automatically generates a process model from a behavioural specification. When the target model is a Petri net, we address synthesis through region theory. Researchers have studied region-based synthesis extensively for state-based specifications, such as transition systems and step-transition systems, as well as for language-based specifications. Accordingly, in literature, region theory is divided into two main branches: state-based regions and language-based regions. Using state-based regions, the behavioural specification is a set of global states and related state-transitions. This representation can express conflicts and the merging of global states naturally. However, it suffers from state explosion and can not express concurrency explicitly. Using language-based regions, the behavioural specification is a set of example runs defined by partially or totally ordered sets of events. This representation can express concurrency and branching naturally. However, it grows rapidly with the number of choices and can not express merging of conflicts. So far, synthesis requires a trade-off between these two approaches. Both region definitions have fundamental limitations, and synthesis therefore always involves a compromise. In this paper, we lift these limitations by introducing a new region theory that covers both state-based and language-based input. We prove that the new definition is a region meta theory that combines both concepts. It uses specifications given as a set of labelled nets, which allow us to express conflicts, concurrency and merging of local states naturally, and synthesizes a Petri net that simulates all labelled nets of the input specification.

💡 Research Summary

The paper addresses a long‑standing limitation in Petri‑net synthesis: existing region‑theoretic approaches are split into two distinct families. State‑based region theory takes a global state space (a transition system) as input, which naturally captures conflicts and state merging but suffers from state‑explosion and cannot represent concurrency explicitly. Language‑based region theory, on the other hand, uses sets of runs or partially ordered event sets, which express concurrency and branching well, yet requires enumerating every alternative execution to represent conflicts, leading to a rapid growth of the specification. In practice, modelers must choose one representation early, and the choice heavily influences both modelling effort and the quality of the synthesized net.

To overcome this dichotomy, the authors propose a meta‑region theory that can handle both kinds of specifications simultaneously. The key innovation is to use labelled Petri nets as the specification language. A labelled net is a Petri net where each transition carries a label; the same label may appear multiple times (label splitting) to denote repeated occurrences of the same activity. Such nets can express conflicts, concurrency, and merging of local states within a single compact structure, thereby unifying the expressive power of state graphs and partial‑order runs.

The new region definition builds on the token‑trail semantics introduced in earlier work. A token trail records, for a given firing sequence, the multiset of tokens that travel along arcs. Using token trails, the authors define a region as a multiset of places that respects token conservation across all token trails of the input labelled nets. Crucially, the synthesis requirement is simulation rather than bisimulation: the resulting Petri net must be able to simulate every input net (i.e., there exists a linear mapping from each reachable state of an input net to a reachable state of the synthesized net that preserves initial markings and step transitions). This weaker requirement is appropriate because each input net typically describes only a fragment of the intended behaviour. When the specification consists solely of state graphs, the simulation mapping becomes the identity, reproducing classic state‑based synthesis. When the specification consists solely of marked graphs (partial orders), the simulation condition coincides with the language‑inclusion condition of language‑based synthesis.

Algorithmically, the approach extracts token‑trail constraints from each labelled net and encodes them as an integer linear program (ILP). Variables represent potential places and arc weights of the target net; constraints enforce token conservation, label consistency, and linearity of the simulation mapping. The objective function usually minimises the number of places or the total arc weight, yielding a compact net. Solving the ILP produces a Petri net that simultaneously simulates all input nets.

The authors demonstrate the method on a running workflow example. Four labelled nets are combined: (1) a simple state graph (conflict only), (2) a marked graph (concurrency only), (3) a labelled net containing both conflict and concurrency with two occurrences of an activity, and (4) a similar net with three occurrences. This mixed specification is far smaller than the full state‑space or the exhaustive set of runs. Using their prototype tool, the synthesis reproduces the original workflow net (Figure 1) and, unlike the pure state‑based specification, also discovers a short loop place. Experiments show that pure language‑based specifications lead to longer computation times, while pure state‑based specifications miss certain structural features (e.g., short loops).

The paper’s contributions are:

1. Definition of a meta‑region theory that subsumes both state‑based and language‑based region concepts.
2. Introduction of labelled Petri nets as a flexible, compact behavioural specification formalism.
3. Development of a token‑trail‑based ILP synthesis algorithm and its implementation in a tool.
4. Empirical evidence that mixed specifications can be more concise and lead to higher‑quality synthesis results than either extreme.

Future work includes extending token‑trail semantics to inhibitor nets, exploring non‑linear optimisation criteria (e.g., structural complexity), and scaling the approach to large event‑log‑derived specifications. Overall, the paper provides a solid theoretical foundation and a practical method for synthesising Petri nets from heterogeneous behavioural descriptions, promising broader applicability in process discovery and model‑driven engineering.