Synchronization trees

Abstract

Synchronization trees are a concrete underlying model for much of the work on concurrency. They are trees with labelled arcs: the nodes represent states, the arcs occurrences of events and their labels how the events can synchronize with other events in the environment. The many different ways in which events are allowed to synchronize are captured abstractly by the concept of a synchronization algebra. It says which pairs of labelled events can combine to form an event of synchronization and what label the synchronization event carries. Synchronization trees are trees with arcs labelled by elements of a synchronization algebra. Our approach is based on a natural definition of morphism of trees which essentially expresses how the occurrence of events in one process imply the synchronized occurrence of events in another. Well-known operations on trees arise as categorical constructions. For example, a sum construction is a coproduct on synchronization trees while many familiar parallel compositions of synchronization trees are restrictions of the product in the underlying category of trees. The constructions are continuous with respect to a natural complete partial order structure on trees so one obtains denotational semantics as synchronization trees to a wide range of parallel programming languages, based on the constructions with recursion, in a routine manner by varying the synchronization algebra. Isomorphism of synchronization trees induces a basic congruence on terms of the language. We present a complete proof system for the congruence restricted to nonrecursive terms. The categories of trees are generalized to categories of transition systems. The pleasant categorical set-up which exists between the categories of trees and transition systems makes possible a smooth translation between operational semantics expressed in terms of transition systems and denotational semantics expressed in terms of trees.