omega-QRB-Domains and the Probabilistic Powerdomain

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. These retain many nice properties from continuous dcpos. 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, has all finite products, all bilimits, and is stable under retracts, and even under so-called quasi-retracts. But... omega-QRB is not cartesian closed.

[1]  Klaus Keimel,et al.  The probabilistic powerdomain for stably compact spaces , 2004, Theor. Comput. Sci..

[2]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[3]  J E A N G O U B A U L T-L A.  De Groot Duality and Models of Choice : Angels , Demons , and Nature , 2009 .

[4]  A. Jung,et al.  Cartesian closed categories of domains , 1989 .

[5]  Alex Simpson,et al.  Two Probabilistic Powerdomains in Topological Domain Theory , 2006 .

[6]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[7]  Michael W. Mislove,et al.  Topology, domain theory and theoretical computer science , 1998 .

[8]  Jean Goubault-Larrecq,et al.  Continuous Previsions , 2007, CSL.

[9]  M. Andrew Moshier,et al.  Stably Compact Spaces and Closed Relations , 2001, MFPS.

[10]  Matthias Schröder,et al.  Compactly generated domain theory , 2006, Mathematical Structures in Computer Science.

[11]  Matthias Schröder,et al.  A Convenient Category of Domains , 2007, Computation, Meaning, and Logic.

[12]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[13]  Michael B. Smyth,et al.  The Largest Cartesian Closed Category of Domains , 1983, Theor. Comput. Sci..

[14]  P. Venugopalan,et al.  Quasicontinuous posets , 1990 .

[15]  Jimmie D. Lawson.  The Versatile Continuous Order , 1987, MFPS.

[16]  Jean Goubault-Larrecq,et al.  Continuous Capacities on Continuous State Spaces , 2007, ICALP.

[17]  Achim Jung,et al.  The troublesome probabilistic powerdomain , 1997, COMPROX.

[18]  Klaus Keimel.  Topological Cones: Foundations for a Domain Theoretical Semantics Combining Probability and Nondeterminism , 2005, MFPS.

[19]  Abbas Edalat.  Domain Theory and Integration , 1995, Theor. Comput. Sci..