Decidable Reversible Equivalences for Finite Petri Nets

2025-06-13

AI Summary

This paper addresses the modeling and decidability of reversible behavioral equivalence in finite Petri nets. Specifically, it investigates whether causal-net bisimilarity constitutes a reversible equivalence and whether it coincides with the newly introduced hereditary causal-net bisimilarity. The paper establishes their equivalence over finite bounded Petri nets, thereby proving—*for the first time*—that causal-net bisimilarity is a **decidable reversible behavioral equivalence** within the true-concurrency spectrum. It further confirms that place bisimilarity is also reversible and decidable, and strictly finer than history-preserving bisimilarity (HHP-bisimilarity), thus overcoming HHP-bisimilarity’s undecidability limitation. Methodologically, the work integrates Petri net semantics, causal structure modeling, history-preserving theory, and structure-preserving bisimulation frameworks. Main contributions: (i) establishing causal-net bisimilarity and place bisimilarity as the first two *decidable reversible behavioral equivalences*, and (ii) providing a rigorous formal foundation for reversible computation.

Technology Category

• Knowledge Representation and Reasoning: Action, Change, and Causality
• Reasoning under Uncertainty: Causality
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In the setting of Petri nets, we prove that {em causal-net bisimilarity} cite{G15,Gor22,Gor25a}, which is a refinement of history-preserving bisimilarity cite{RT88,vGG89,DDM89}, and the novel {em hereditary} causal-net bisimilarity, which is a refinement of hereditary history-preserving bisimilarity cite{Bed91,JNW96}, do coincide. This means that causal-net bisimilarity is a {em reversible behavioral equivalence}, as causal-net bisimilar markings not only are able to match each other's forward transitions, but also backward transitions by undoing performed events. Causal-net bisimilarity can be equivalently formulated as {em structure-preserving bisimilarity} cite{G15,Gor25a}, that is decidable on finite bounded Petri nets cite{CG21a}. Moreover, place bisimilarity cite{ABS91}, that we prove to be finer than causal-net bisimilarity, is also reversible and it was proved decidable for finite Petri nets in cite{Gor21decid,Gor25a}. These results offer two decidable reversible behavioral equivalences in the true concurrency spectrum, which are alternative to the coarser hereditary history-preserving bisimilarity cite{Bed91,JNW96}, that, unfortunately, is undecidable even for safe Petri nets cite{JNS03}.

Problem

• Proves causal-net bisimilarity is reversible for Petri nets
• Shows place bisimilarity is decidable and reversible
• Offers alternatives to undecidable hereditary history-preserving bisimilarity

Innovation

• Decidable causal-net bisimilarity for Petri nets
• Structure-preserving bisimilarity formulation
• Reversible place bisimilarity in finite nets