Cost Preserving Bisimulations for Probabilistic Automata

Abstract

Abstract. Probabilistic automata constitute a versatile and elegant model for concurrent probabilistic systems. They are equipped with a compositional theory supporting abstraction, enabled by weak probabilistic bisimulation serving as the reference notion for summarising the effect of abstraction.This paper considers probabilistic automata augmented with costs. It extends the notions of weak transitions in probabilistic automata in such a way that the costs incurred along a weak transition are captured. This gives rise to cost-preserving and cost-bounding variations of weak probabilistic bisimilarity, for which we establish compositionality properties with respect to parallel composition. Furthermore, polynomial-time decision algorithms are proposed, that can be effectively used to compute reward-bounding abstractions of Markov decision processes in a compositional manner.

Compositional Reasoning for Interval Markov Decision Processes

“…By definition of synchronous product, it follows that for each i ∈ I, (s 2 , s 3 ) −→ µ 2,i × µ 3 ∈ T 2,3 , hence we have the combined transition (s 2 , s 3 ) −→ c µ 2 × µ 3 . By standard properties of lifting (see, e.g., [23]), it follows that µ 1 ×µ 3 L(R ) µ 2 ×µ 3 , as required. It is worthy to note that the unfolding mapping might transform an IMDP to a PA with an exponentially larger size.…”