On probabilistic stable event structures

Concurrency and probability are both much studied extensions of sequential computation. Within concurrency theory, there is a broad divide between interleaving models and logics, which model concurrency by non-determinism, and `truly concurrent' models, in which concurrency (and/or causality) are directly represented. True concurrency gives much richer models and logics, but is also harder to work with, as the interactions of causality and concurrency can become complex. In probabilistic computation, we have by now many well understood probabilistic models and logics on interleaving models, developed over the last thirty years or so. However, for true concurrency, adding probability has been much harder, owing to the interactions between probabilistic independence and causal/concurrent independence, piling both conceptual and technical difficulties upon an already complex model. Various notions of, e.g., probabilistic Petri net have been introduced, but for simpler cases such as free-choice nets. A significant contribution in the last decade was Abbes and Benveniste 2006, which gave a rigorous foundation for probabilistic concurrency in terms of Winskel's event structures, which can be seen as the lowest level of concurrent/causal modelling. However, event structures are themselves not ideal for representing richer high-level models such as (arbitrary) Petri nets, as the concurrency is derived from a binary conflict relation. Here we develop an extension of the theory of Abbes and Benveniste to a more general class of models, a certain subclass of `stable event structures'. This class is sufficient to give a semantics to probabilistic Petri nets (including the notoriously awkward notion of confusion, which is forbidden by free-choice nets), and thereby allows the definition of general probabilistic logics for Petri nets.