Reduction methods for probabilistic model checking

Model Checking is a fully automatic verification method that has undergone a vast development for almost 30 years now. In contrast to simulation and testing, model checking is a verification technique that explores all possible system states exhaustively and can therefore reveal errors that have not been discovered by testing or simulation. It thus is a prominent verification technique for safety-critical systems. However, exploring the entire state space makes model checking very sensitive to the size of the system to be verified. In this thesis, we address the issue of reduction techniques for probabilistic model checking. Taking probabilities into account in addition to nondeterministic behavior expands the possibilities of modeling certain aspects of the system under consideration. While nondeterministic systems are considered in connection to underspecification, interleaving of several processes and interaction with the specified system from the outside, the probabilities can be exploited to model a certain probability of error or other stochastic behavior both occurring in various real world applications, e.g. randomized algorithms or communication protocols over faulty media. In this thesis we restrict our investigations to models that are specified by Markov decision processes. On the one hand we study the applicability of partial order reduction methods on Markov decision processes. These allow to construct a submodel of the model to be verified and to model check the (smaller) submodel, yielding a valid answer also for the original model. We investigate Doron Peled’s ample set method in a probabilistic setting and point out that the classical conditions on the ample sets are not sufficient when dealing with Markov decision processes. We show a conservative extension of the classical conditions which makes the ample set method work for Markov decision processes with respect to lineartime properties. Here conservative means that the new stronger conditions are equivalent to the classical ones, if they are applied to non-probabilistic (classical) systems. We also show how to extend the classical conditions for branching time properties such that the ample set method works for Markov decision processes with respect to probabilistic branching time properties. In the context of automata-theoretic model checking another chance to enhance the performance is to generate a “small” automaton for the given specification that one wants to verify for a system. We introduce and investigate the concept of probabilistic ω-automata. It turned out that they do not apply to the model checking of MDPs as their emptiness problem is undecidable. Nevertheless they form an interesting field of research. We introduce probabilistic Büchi automata (PBA) as acceptors for languages of infinite words, where a word is accepted by a PBA if and only if the set of accepting runs for this word has a positive measure. We show that PBA strictly subsume the ω-regular languages and also study the efficiency (with respect to the size) of PBA. We show that PBA are closed under union, intersection and complementation. Moreover we prove that the emptiness problem is undecidable for PBA. This result implies the undecidability of some qualitative ω-regular properties for partially observable Markov decision processes. Furthermore we investigate