Causal Unfoldings

Authors Marc de Visme, Glynn Winskel

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Author Details

Marc de Visme
  • Univ Lyon, ENS de Lyon, CNRS, UCB Lyon 1, LIP, France
Glynn Winskel
  • Computer Laboratory, University of Cambridge, UK


Thanks to the anonymous referees. Thanks to Simon Castellan, Pierre Clairambault, Ioana Cristescu, Mai Gehrke, Jonathan Hayman, Tamas Kispeter, Jean Krivine, Martin Hyland and Daniele Varacca for discussions, advice and encouragement; to ENS Paris for supporting Marc de Visme’s internship; and to the ERC for Advanced Grant ECSYM.

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Marc de Visme and Glynn Winskel. Causal Unfoldings. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Event Structures
  • Parallel Causes
  • Causal Unfolding
  • Probability


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