Bisimilarity and Behaviour-Preserving Reconfigurations of Open Petri Nets

We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open systems which might interact with the surrounding environment and endowed with a colimit-based composition operation. We show that natural notions of bisimilarity over open nets are congruences with respect to the composition operation. The considered behavioural equivalences differ for the choice of the observations, which can be single firings or parallel steps. Additionally, we consider weak forms of such equivalences, arising in the presence of unobservable actions. We also provide an up-to technique for facilitating bisimilarity proofs. The theory is used to identify suitable classes of reconfiguration rules (in the double-pushout approach to rewriting) whose application preserves the observational semantics of the net.

💡 Research Summary

The paper introduces a formal framework for specifying and verifying behavior‑preserving reconfigurations of systems modeled as Petri nets. The authors start by extending ordinary Place/Transition (P/T) nets to open nets, a mild generalisation that makes the interface of a net explicit. Each place and transition can have designated input and output ports, allowing the net to interact with an external environment. This openness is captured categorically: objects are open nets, morphisms are partial net embeddings, and composition of nets is defined via a colimit construction that merges two nets along a shared interface. The colimit‑based composition provides a mathematically clean way to model modular system integration, dynamic plugging‑in of components, and the boundary handling required during reconfiguration.

On top of this structural foundation the authors define several notions of bisimilarity (behavioral equivalence). They distinguish observations based on single firings versus parallel steps and consider both strong (no abstraction of internal actions) and weak (τ‑actions are hidden) variants. Consequently four main equivalences arise: strong firing bisimilarity, strong step bisimilarity, weak firing bisimilarity, and weak step bisimilarity. For each, the paper proves a congruence theorem: if two open nets are equivalent under a given bisimilarity, then placing them in any identical context (i.e., composing them with the same surrounding net) yields equivalent composite nets. This property is essential for modular reasoning because it guarantees that local equivalence is preserved under global composition.

Recognising that direct bisimulation proofs can be cumbersome, the authors develop an up‑to technique. The up‑to methods (up‑to context, up‑to expansion, up‑to bisimulation) allow a relation to be closed under certain operations, thereby reducing the number of transition pairs that must be explicitly matched. In the setting of open nets, up‑to context is particularly powerful because it automatically accounts for the surrounding interface, simplifying proofs that involve reconfiguration where the interface may change.

The behavioral theory is then integrated with the double‑pushout (DPO) graph rewriting approach. Reconfiguration rules are expressed as spans of open nets: a left‑hand side (LHS) pattern, a right‑hand side (RHS) pattern, and a common interface. A rule can be applied when a match of the LHS into a host net respects the interface, and the pushout construction replaces the matched part with the RHS. The central contribution here is the identification of a class of observational‑semantics‑preserving rules. The authors prove that if the LHS and RHS are related by a chosen bisimilarity (strong or weak) and share the same interface, then applying the rule does not alter the observable behavior of the host net. This result provides a rigorous guarantee that dynamic structural changes—such as adding or removing components, rewiring connections, or scaling resources—can be performed without exposing any behavioral differences to the environment.

To demonstrate practicality, the paper presents case studies: a manufacturing line where machines are added or removed, a communication protocol with dynamic pipeline reconfiguration, and a cloud service that auto‑scales. In each scenario the system is modeled as an open net, reconfiguration rules are defined, and the up‑to bisimulation technique is used to certify that the rules preserve the chosen equivalence. The experiments show a substantial reduction in proof obligations and illustrate how the theory can be combined with automated tooling for near‑real‑time verification.

The discussion concludes with limitations and future work. Current results focus on untimed, deterministic Petri nets; extensions to timed, stochastic, or colored nets are left open. Moreover, the development of a dedicated software environment that supports rule specification, matching, pushout construction, and up‑to bisimulation checking is identified as a crucial next step. Overall, the paper delivers a robust categorical and behavioral foundation for modular, behavior‑preserving reconfiguration of open Petri nets, opening avenues for reliable dynamic system design in domains ranging from manufacturing to cloud computing.