Time and Duration in Noninterleaving Concurrency

Abstract

The purpose of this paper is to present a real-timed concurrency theory in the noninterleaving tradition. The theory is based on the occurrences of actions; each occurrence or event has a start and a finish. Causality is modelled by assigning a strict partial order to these starts and finishes, while timing is modelled by giving them reals.

The theory is presented in some detail. All of the traditional notions found in concurrency theories (such as conflict, confusion, liveness, and so on) are found to be expressible. Four notions of causality arise naturally from the model, leading to notions of securing. Three of the notions give rise to underlying event structures, demonstrating that our model generalises Winskel’s.

Infinite structures are then analysed: a poset of finite structures is defined and suitably completed to give one containing infinite structures. These infinite structures are characterised as just those arising as limits of finite ones. Our technique here, which relies on the structure of time, is of independent interest.