Abstract. Given a parametric Markov model, we consider the problem of computing the rational function expressing the probability of reaching a given set of states. To attack this principal problem, Daws has suggested to first convert the Markov chain into a finite automaton, from which a regular expression is computed. Afterwards, this expression is evaluated to a closed form function representing the reachability probability. This paper investigates how this idea can be turned into an effective procedure. It turns out that the bottleneck lies in the growth of the regular expression relative to the number of states (n Θ(log n) ). We therefore proceed differently, by tightly intertwining the regular expression computation with its evaluation. This allows us to arrive at an effective method that avoids this blow up in most practical cases. We give a detailed account of the approach, also extending to parametric models with rewards and with non-determinism. Experimental evidence is provided, illustrating that our implementation provides meaningful insights on non-trivial models.

The Markov Reward Model Checker (MRMC) is a software tool for verifying properties over probabilistic models. It supports PCTL and CSL model checking, and their reward extensions. Distinguishing features of MRMC are its support for computing time-and reward-bounded reachability probabilities, (property-driven) bisimulation minimization, and precise on-the-fly steady-state detection. Recent tool features include time-bounded reachability analysis for uniform CTMDPs and CSL model checking by discrete-event simulation. This paper presents the tool's current status and its implementation details. This research was performed as part of the MC=MC project financed by the Netherlands Organization for Scientific Research (NWO) and the DFG Research Training Group 623 on Leistungsgarantien für Rechnersysteme. We thank Maneesh Khattri (Oxford Univ.), Christina Jansen (RWTH Aachen), and Tim Kemna (Univ. Twente) for their implementation efforts.

We provide the first solution for model-free reinforcement learning of ω-regular objectives for Markov decision processes (MDPs). We present a constructive reduction from the almost-sure satisfaction of ω-regular objectives to an almostsure reachability problem, and extend this technique to learning how to control an unknown model so that the chance of satisfying the objective is maximized. A key feature of our technique is the compilation of ω-regular properties into limitdeterministic Büchi automata instead of the traditional Rabin automata; this choice sidesteps difficulties that have marred previous proposals. Our approach allows us to apply model-free, off-the-shelf reinforcement learning algorithms to compute optimal strategies from the observations of the MDP. We present an experimental evaluation of our technique on benchmark learning problems.An ω-word w on an alphabet Σ is a function w : N → Σ. We abbreviate w(i) by w i . The set of ω-words on Σ is written Σ ω and a subset of Σ ω is an ω-language on Σ.A probability distribution over a finite set S is a function d : S→[0, 1] such that s∈S d(s) = 1. Let D(S) denote the set of all discrete distributions over S. We say a distribution d ∈ D(S) is a point distribution if d(s)=1 for some s ∈ S. For a distribution d ∈ D(S) we write supp(d) def = {s ∈ S : d(s) > 0}.

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