Reconciling Event Structures and Higher Dimensional Automata

Addressing the problem in the title, we diagnose the essential diiculty as the implicit assumption of two-valued logic in the event structure view of event occurrence as a true-or-false proposition. Taking the 3-valued logic of before-during-after as the common underlying logic of both event structures and higher dimensional automata (HDA's) uniies them into a single model having the same process algebra, linear logic, and schedule-automaton duality as had previously been achieved for event structures, but not previously for HDA's. The crucial tool is Chu spaces, whose application to concurrency has previously limited itself to Chu spaces over 2, and which here is taken over 3. This model is simpliied by the absence from the deenition of Chu spaces of any a priori logical structure on the set 3, allowing a process on a set A of events to be deened as simply any subset of the set 3 A. The structure required for our model of HDA's is a speciic 3-element quantale, which however is not built into the basic model but made an after-the-fact interpretation of it; remarkably, Chu transforms respect whatever structure that interpretation gives rise to, and no more. 1 The Problem At question time after my presentation of higher dimensional automata (HDA) Pra91], Boris Trakhtenbrot asked, what is the relationship between HDA's and event structures? 1 Despite having addressed the issue in section 7, focusing on persistence of connict, I had no good answer to the question, a connict that persisted for me until the solution presented here struck me late last year. The general problem here is to reconcile the respective motivating intuitions and supporting formalisms underlying the event-based and geometry-based approaches to concurrency. Event structures replace the traditional state-based view of computation by the view of akb as a set fa; bg representing two events unconstrained as to order. In contrast the geometry-based approach retains the state-based view and models akb as the evident four-state \square" automaton accepting ab+ba but with its square interior lled in as a two-dimensional surface representing the overlapped execution of a and b. 1 More precisely, he asked about the relationship with event spaces Pra92], but that distinction is a negligible one compared with the diierence with HDA's.