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The troublesome probabilistic powerdomain

By Achim Jung and Regina Tix


In [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. Inthispaper, wegiveacounterexampletoGraham’sproofanddescribe 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.

Year: 1998
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