Causal Unfoldings and Disjunctive Causes
Marc de Visme, Glynn Winskel
TL;DR
The paper tackles representing causal histories with disjunctive and parallel causes in distributed systems, where traditional prime event structures fall short for probabilistic reasoning. It introduces prime event structures with an equivalence relation (ese) and equivalence families (ef), and develops causal unfoldings as a pseudo-right adjoint to the embedding of prime-like models into broader configurations, using extremal realisations to capture minimal causal histories. A network of (pseudo)adjunctions connects ese to equivalence families and to general event structures, with a key subcategory Edc ($\mathcal{E}_\equiv^1$) that supports parallel causes, hiding, and pullbacks for probabilistic distributed strategies. The framework provides a robust semantic foundation for composing strategies with parallel causes, enabling probabilistic reasoning about causal histories and offering tools for applications in reversible computing and biochemical pathway analysis. Overall, the causal unfolding and its adjunctions unify disparate models of concurrency under a shared, extensible theory that supports both parallel causes and probabilistic reasoning.
Abstract
In the simplest form of event structure, a prime event structure, an event is associated with a unique causal history, its prime cause. However, it is quite common for an event to have disjunctive causes in that it can be enabled by any one of multiple sets of causes. Sometimes the sets of causes may be mutually exclusive, inconsistent one with another, and sometimes not, in which case they coexist consistently and constitute parallel causes of the event. The established model of general event structures can model parallel causes. On occasion however such a model abstracts too far away from the precise causal histories of events to be directly useful. For example, sometimes one needs to associate probabilities with different, possibly coexisting, causal histories of a common event. Ideally, the causal histories of a general event structure would correspond to the configurations of its causal unfolding to a prime event structure; and the causal unfolding would arise as a right adjoint to the embedding of prime in general event structures. But there is no such adjunction. However, a slight extension of prime event structures remedies this defect and provides a causal unfolding as a universal construction. Prime event structures are extended with an equivalence relation in order to dissociate the two roles, that of an event and its enabling; in effect, prime causes are labelled by a disjunctive event, an equivalence class of its prime causes. With this enrichment a suitable causal unfolding appears as a pseudo right adjoint. The adjunction relies critically on the central and subtle notion of extremal causal realisation as an embodiment of causal history. Finally, we explore subcategories which support parallel causes as well the key operations needed in developing probabilistic distributed strategies with parallel causes.