Petri Nets Enable Causal Reasoning in Dynamical Systems

Dynamical systems, e.g. economic systems or biomolecular signaling networks, are processes comprised of states that evolve in time. Causal models represent these processes, and support causal queries inferring outcomes of system perturbations. Unfortunately, Structural Causal Models, the traditional causal models of choice, require the system to be in steady state and don't extend to dynamical systems. Recent formulations of causal models with a compatible dynamic syntax, such as Probability Trees, lack a semantics for representing both states and transitions of a system, limiting their ability to fully represent the system and ability to encode the underlying causal assumptions. In contrast, Petri Nets are well-studied models of dynamical systems, with the ability to encode states and transitions. However, their use for causal reasoning has so far been under-explored. This manuscript expands the scope of causal reasoning in dynamical systems by proposing a causal semantics for Petri Nets. We define a pipeline of constructing a Petri Net model and calculating the fundamental causal queries: conditioning, interventions and counterfactuals. A novel aspect of the proposed causal semantics is an unwrapping procedure, which allows for a dichotomy of Petri Net models when calculating a query. On one hand, a base Petri Net model visually represents the system, implicitly encodes the traces defined by the system, and models the underlying causal assumptions. On the other hand, an unwrapped Petri Net explicitly represents traces, and answers causal queries of interest. We demonstrate the utility of the proposed approach on a case study of a dynamical system where Structural Causal Models fail.