Infinite Concurrent Systems — I. The Relationship between Metric and Order Convergence

Abstract

In recent years, Mazurkiewicz trace theory has become increasingly popular as a model of semantics for non-interleaving concurrency, and the theory is here further extended to allow descriptions of infinite behaviours based on alphabets of arbitrary cardinality. We describe an unconventional but nonetheless natural, ordering on trace space, and show that under this ordering, every nonempty set of traces has a greatest lower bound. Consequently, the ordering is consistently complete, i.e. if a set of traces has an upper bound, it has a least such bound. This parallels the result that traces over finite alphabets form a domain.

Kwiatkowska has demonstrated an ultrametric, definable on (standard) trace spaces, under which they become compact, and hence complete, topological spaces. For finite alphabets, thus ultrametric essentially agrees with that of Comyn and Dauchet, but for infinite alphabets they differ in behaviour. We investigate the structure of trace space under various topologies, demonstrate the relationship between order convergence and the various metric convergence regimes, and thereby explain apparent behaviourial anomalies.