Attractor Basins in Concurrent Systems
Giann Karlo Aguirre Samboni, Stefan Haar, Loic Paulevé, Stefan Schwoon, Nick Würdemann
TL;DR
This work addresses identifying long-run behaviours and doom avoidance in concurrent systems modeled by safe Petri nets, by mapping attraction basins $\mathcal{B}(\mathbf{A})$ and introducing cliff-edges via minimally doomed configurations $\check{\mathcal{D}}$ and a protection measure $\mathbf{prot}(\cdot)$. It advances the methodology by leveraging net unfoldings to enumerate configurations leading toward fatal basins, and employs complete prefixes $^\prec_0$ (Esparza prefixes) together with a new MinDoo algorithm to compute $\check{\mathcal{D}}$, enabling precise characterization of doom-avoidance opportunities. The approach is grounded in formal Petri-net theory, linking attractors to end-of-free-behaviour and providing a practical pipeline with implementation and experiments on biological models, while acknowledging computational bottlenecks such as McMillan’s cutoff. Overall, the work offers a principled framework to analyze and quantify the risk of doom in concurrent biological/ecological systems and lays groundwork for potential doom-avoidance control strategies using the protectedness metric.
Abstract
A crucial question in analyzing a concurrent system is to determine its long-run behaviour, and in particular, whether there are irreversible choices in its evolution, leading into parts of the reachability space from which there is no return to other parts. Casting this problem in the unifying framework of safe Petri nets, our previous work has provided techniques for identifying attractors, i.e. terminal strongly connected components of the reachability space. What we aim at is to determine the attraction basins associated to those attractors; that is, those states from where all infinite runs are doomed to end in the given attractor, as opposed to those that are free to evolve differently. Here, we provide a solution for the case of safe Petri nets. Our algorithm uses net unfoldings and provides a map of all of those configurations (concurrent executions of the system) that lead onto cliff-edges, i.e. any maximal extension for those configurations lies in some basin that is considered fatal.