# When are Stochastic Transition Systems Tameable?

A decade ago, Abdulla et al introduced the elegant concept of decisiveness for denumerable Markov chains [1]. Roughly decisiveness allows one to lift most good properties from finite Markov chains to denumerable ones, and therefore to adapt existing verification algorithms to infinite-state models. Denumerable Markov chains however do not encompass stochastic real-time systems, and general stochastic transition systems (STSs) are needed. In this paper, we provide a framework to perform both the qualitative and the quantitative analysis of STSs. We first define various notions of decisiveness (inherited from [1]), notions of fairness and of attractors for STSs, and explicit the relationships between them. Then we define a notion of abstraction, together with natural concepts of soundness and completeness, and we give general transfer properties, which will be central to several verification algorithms on STSs. Then we focus on qualitative analysis. Beyond (repeated) reachability properties for which our technics are strongly inspired by [1], we use abstractions to design algorithms for the qualitative model-checking of arbitrary omega-regular properties, when the STS admits a denumerable (sound and complete) abstraction with a finite attractor. We further design generic approximation procedures for quantitative analysis; in addition to extensions of [1] for general STSs, we design approximation algorithms for omega-regular properties (once again by means of specific abstractions). Last we instantiate our framework with stochastic timed automata and generalized semi-Markov processes, two models combining dense-time and probabilities. This allows to derive decidability and approximability results for those models. Some of these results were known from the literature, but our generic approach permits to view them in a unified framework. We also derive interesting new approximability results.

**Comments:**63 pages

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