Metric semantics from partial order semantics

Abstract

In dealing with denotational semantics of programming languages partial orders resp. metric spaces have been used with great benefit in order to provide a meaning to recursive and repetitive constructs. This paper presents two methods to define a metric on a subset $M$ of a complete partial order $D$ such that $M$ is a complete metric spaces and the metric semantics on $M$ coincides with the partial order semantics on $D$ when the same semantic operators are used. The first method is to add a ‘length’ on a complete partial order which means a function $\rho : D \to {\Bbb N} \cup \{\infty\}$ of increasing power. The second is based on the ideas of [11] and uses pseudo rank orderings, i.e. monotone sequences of monotone functions $\pi_n : D \to D$ . We show that SFP domains can be characterized as special kinds of rank orderded cpo's. We also discuss the connection between the Lawson topology and the topology induced by the metric.