Kantorovich-Rubinstein quasi-metrics II: Hyperspaces and powerdomains

Abstract

We show that the Kantorovich-Rubinstein quasi-metrics dKR and dKRa of Part I extend naturally to various spaces of previsions, and in particular not just the linear previsions (roughly, measures) of Part I. There are natural isomorphisms between the Hoare and Smyth powerdomains, as used in denotational semantics, and spaces of discrete sublinear previsions, and of discrete normalized superlinear previsions, respectively. Turning to the corresponding hyperspaces, namely the same powerdomains, but equipped with the lower Vietoris and upper Vietoris topologies instead, this turns into homeomorphisms with the corresponding space of previsions, equipped with the so-called weak topology. Through these isomorphisms again, the two powerdomains inherit quasi-metrics dH and dQ, respectively, that are reminiscent of the well-known Hausdorff metric. Then we show that the Hoare and Smyth powerdomains of an algebraic complete quasi-metric space are again algebraic complete, with those quasi-metrics, and similarly that the corresponding powerdomains of continuous complete quasi-metric spaces are continuous complete. Furthermore, in the continuous complete case, the dH-Scott topology coincides with the lower Vietoris topology, and the dQ-Scott topology coincides with the upper Vietoris topology.