Event structures : maps, monads and spans : invited talk

Event structures are a model of computational processes. They represent a process as a set of event occurrences with relations to express how events causally depend on others, or exclude other events from occurring. In one of their simpler forms they consist of a set of events on which there is a consistency relation expressing when events can occur together in a history and a partial order of causal dependency— writing e′ ≤ e if the occurrence of e depends on the previous occurrence of e′. In detail, an event structure comprises (E,Con,≤), consisting of a set E, of events which are partially ordered by ≤, the causal dependency relation, and a predicate Con consisting of finite subsets of E, the consistency relation, which satisfy {e′ | e′ ≤ e} is finite for all e ∈ E, {e} ∈ Con for all e ∈ E, Y ⊆ X ∈ Con ⇒ Y ∈ Con, and X ∈ Con & e ≤ e′ ∈ X ⇒ X ∪ {e} ∈ Con. Our understanding of the consistency predicate and the enabling relation are expressed in the notion of configuration (or state or history) we adopt for event structures. The events are to be thought of as event occurrences; in any history an event is to appear at most once. A configuration is a set of events which have occurred by some stage in a process. According to our understanding of the consistency predicate and causal dependency relations a configuration should be consistent and such that if an event appears in a configuration then so do all the events on which it causally depends. A configuration of an event structure E is a subset x ⊆ E which is Consistent: ∀X ⊆ x. X ∈ Con, and The configurations of an event structure are ordered by inclusion, where x ⊆ x′, i.e. x is a sub-configuration of x′, means that x is a sub-history of x′. This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs