Quantitative Continuous Domains

Abstract

We relate two approaches to Quantitative Domain Theory, the partial metrics of Steve Matthews and the measurements of Keye Martin, by showing that stable partial metrics are in one-to-one correspondence to weakly modular measurements. It is shown that every ω-algebraic domain admits a partial metric whose associated measurement assigns zero precisely to the set of maximal elements, a condition which features prominently in Martin's work. A partial metric gives rise to a metric in a standard way; we study the conditions under which the resulting space is complete and show that every ω-algebraic domain admits a partial metric of this kind. We discuss a number of examples and counterexamples to locate the strength of our results more exactly.