Abstract
Probabilistic automata (PA), also known as probabilistic nondeterministic labelled transition systems, combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of distribution bisimilarity, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.
Highlights
Probabilistic automata (PA), closely related to Markov decision processes (MDPs), have been used along the years in various areas of verification [LMOW08, KNP11, KNSS02, BK08], machine learning [GMR+12, MCJ+16], and semantics [WJV+15, SK12]
Probabilistic automata are understood as transformers of belief states, labeled transition systems (LTSs) having as states probability distributions, see e.g. [DvGHM08, DvGHM09, KVAK10, AAGT12, DH13, FZ14, DMS14]
We show that the result of the whole procedure is exactly the expected belief-state transformer and that the induced notion of behavioural equivalence coincides with a canonical distribution bisimilarity present in the literature [DvGHM08, Hen[12], FZ14, HKK14]
Summary
Probabilistic automata (PA), closely related to Markov decision processes (MDPs), have been used along the years in various areas of verification [LMOW08, KNP11, KNSS02, BK08], machine learning [GMR+12, MCJ+16], and semantics [WJV+15, SK12]. In the case of NDA, the functor F X = 2 × XL can be lifted to the category of join-semilattices (algebras for P) and the coalgebra c : PS → 2 × (PS)L resulting from this construction turns out to be exactly the standard determinised automaton. This is not the case with probabilistic automata: because of the lack of a suitable distributive law of D over P [VW06], it is impossible to suitably lift F X = (PX)L to the category of convex algebras (algebras for the monad D).
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