Compositionality of Systems and Partially Ordered Runs

In the late 1970s, C.A. Petri introduced partially ordered event occurrences (runs), then called \emph{processes}, as the appropriate model to describe the individual evolutions of distributed systems. Here, we present a unified framework for handling Petri nets and their runs, specifically to compose and decompose them. It is shown that, for nets $M$ and $N$, the set of runs of the composed net $M \bullet N$ equals the composition of the runs of $M$ and $N$.

💡 Research Summary

The paper revisits the original notion of “processes” introduced by C.A. Petri in the late 1970s—partially ordered event occurrences that capture causal rather than temporal relationships. While the traditional interleaving semantics of Petri nets linearises events in time, it obscures true concurrency and makes composition of systems cumbersome. Existing work on partially ordered runs has struggled to provide a clean compositional framework, especially when trying to combine or refine nets without resorting to ad‑hoc shuffle operators.

To address these gaps, the authors propose a unified, modular framework built around net modules. A net module consists of a standard place/transition graph (P, T, E) together with a left and a right interface. Interfaces are finite, ordered, and labelled sets; the ordering reflects a vertical arrangement (top‑to‑bottom) and the labels identify the type of interaction (e.g., “aide busy”, “aide free”). Two interface elements are matching partners if they share the same label and the same degree (the number of preceding elements with the same label). Elements without a partner are called match‑free.

Composition of two modules M and N, written M · N, is defined purely in terms of these interfaces:

• The interior of M · N is the union of the interiors of M and N plus a new set of internal elements that correspond to each pair of matching partners from the right interface of M and the left interface of N.
• The left interface of the composite consists of the left interface of M together with the match‑free part of N’s left interface (placed after M’s elements). Symmetrically, the right interface consists of N’s right interface followed by the match‑free part of M’s right interface.
• All original arcs are retained, and each match (a, b) inherits the arcs incident to a or b, thereby wiring the two modules together.

This definition yields an identity element (the empty module) and satisfies associativity, giving a genuine algebraic structure for net composition.

The paper then defines steps and runs for modules. A step is a labelled module that represents the firing of a single transition t; its surrounding places become the step’s left (consumed) and right (produced) interfaces. A run is a finite, acyclic composition of steps, respecting the rule that no place appears with more than one incoming and one outgoing arc, which guarantees a true partial‑order (causal) structure.

The central technical contribution is the Composition Theorem: \