Connections Between Two Theories of Concurrency: Metric Spaces and Synchronization Trees

A connection is established between the semantic theories of concurrency and communication in the works of de Bakker and Zucker, who develop a denotational semantics of concurrency using metric spaces instead of complete partial orders, and Milner, who develops an algebraic semantics of communication based upon observational equivalence between processes. His rigid synchronization trees (RSTs) are endowed with a simple pseudometric distance induced by Milner's weak equivalence relation and the quotient space is shown to be complete. An isometry between this space and the solution to a domain equation of de Bakker and Zucker is established, presenting their solution in a conceptually simpler framework. Under an additional assumption, the equivalence between the weak equivalence relation over RSTs and the elementary equivalence relation induced by the sentences of a modal logic due to Hennessy and Milner is established.