QRB-Domains and the Probabilistic Powerdomain

• Published in: Logical Methods in Computer Science
• Main Author: Jean Goubault-Larrecq
• Format: Article
• Language: English
• Published: Logical Methods in Computer Science e.V. 2012-02-01
• Subjects: computer science - programming languages; d.3.1, f.1.2, f.3.2
• Online Access: https://lmcs.episciences.org/956/pdf

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.