QRB-Domains and the Probabilistic Powerdomain

TL;DR

This paper introduces omega-QRB-domains, a class of quasi-continuous dcpos, demonstrating their closure properties under probabilistic powerdomains and other constructions, but also their limitations regarding Cartesian closure.

Contribution

It defines omega-QRB-domains and proves their closure under key domain-theoretic operations, advancing the understanding of probabilistic semantics in domain theory.

Findings

omega-QRB is closed under probabilistic powerdomain functor

omega-QRB is closed under finite products and bilimits

omega-QRB is not Cartesian closed

Abstract

Is there any Cartesian-closed category of continuous domains that would be closed under Jones and Plotkin's probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. We introduce a natural class of such quasi-continuous dcpos, the omega-QRB-domains. We show that they form a category omega-QRB with pleasing properties: omega-QRB is closed under the probabilistic powerdomain functor, under finite products, under taking bilimits of expanding sequences, under retracts, and even under so-called quasi-retracts. But... omega-QRB is not Cartesian closed. We conclude by showing that the QRB domains are just one half of an FS-domain, merely lacking control.