Least upper bounds for probability measures and their applications to abstractions

Abstract

Least upper bounds play an important role in defining the semantics of programming languages, and in abstract interpretations. In this paper, we identify conditions on countable ordered measurable spaces that ensure the existence of least upper bounds for all sets of probability measures. These conditions are shown to be necessary as well — for any measurable space not satisfying these conditions, there are (finite) sets of probability measures for which no least upper bound exists. For measurable spaces meeting these conditions, the existence of least upper bounds is established constructively. Based on this least upper bound construction, we present a novel abstraction method applicable to Discrete Time Markov Chains (DTMCs), Markov Decision Processes (MDPs), and Continuous Time Markov Chains (CTMCs). The main advantage of the new abstraction techniques is that the resulting abstract models are purely probabilistic that may be more amenable to automated analysis than models with both nondeterministic and probabilistic transitions which arise from previously known abstraction techniques.