The Troublesome Probabilistic Powerdomain

AbstractIn [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerdomain construction can be restricted to RB.In this paper, we give a counterexample to Graham's proof and describe our own attempts at proving a closure result for the probabilistic powerdomain construction. We have positive results for finite trees and finite reversed trees. These illustrate the difficulties we face, rather than being a satisfying answer to the question of whether the probabilistic powerdomain and function spaces can be reconciled.We are more successful with coherent or Lawson-compact domains. These form a category with many pleasing properties but they fall short of supporting function spaces.Along the way, we give a new proof of Jones' Splitting Lemma.A number of people have been subjected to our (mostly flawed) attempts to prove a general closure theorem for the probabilistic powerdomain construction, and we gratefully acknowledge the patience and encouragement of the research teams in Darmstadt and Birmingham.We would also like to thank two anonymous referees for their comments and Mathias Kegelmann for very careful reading and many suggestions for improvement.The result in Section 4 was obtained while the first author visited the Isaac Newton Institute in Cambridge