Linear Distances between Markov Chains

We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by temporal logics or automata. We investigate which of the distances can be approximated by observing the systems, i.e. by black-box testing or simulation, and we provide both negative and positive results

Similarity quantification for linear stochastic systems as a set-theoretic control problem

For the formal verification and design of control systems, abstractions with quantified accuracy are crucial. Such similarity quantification is hindered by the challenging computation of approximate stochastic simulation relations. This is especially the case when considering accurate deviation bounds between a stochastic continuous-state model and its finite-state abstraction. In this work, we give a comprehensive computational approach and analysis for linear stochastic systems. More precisely, we develop a computational method that characterizes the set of possible simulation relations and optimally trades off the error contributions on the system's output with deviations in the transition probability. To this end, we establish an optimal coupling between the models and simultaneously solve the approximate simulation relation problem as a set-theoretic control problem using the concept of invariant sets. We show the variation of the guaranteed satisfaction probability as a function of the error trade-off in a case study where a formal specification is given as a temporal logic formula.Comment: 16 pages, 9 figures, submitted to Automatic

IST Austria Thesis

This dissertation concerns the automatic verification of probabilistic systems and programs with arrays by statistical and logical methods. Although statistical and logical methods are different in nature, we show that they can be successfully combined for system analysis. In the first part of the dissertation we present a new statistical algorithm for the verification of probabilistic systems with respect to unbounded properties, including linear temporal logic. Our algorithm often performs faster than the previous approaches, and at the same time requires less information about the system. In addition, our method can be generalized to unbounded quantitative properties such as mean-payoff bounds. In the second part, we introduce two techniques for comparing probabilistic systems. Probabilistic systems are typically compared using the notion of equivalence, which requires the systems to have the equal probability of all behaviors. However, this notion is often too strict, since probabilities are typically only empirically estimated, and any imprecision may break the relation between processes. On the one hand, we propose to replace the Boolean notion of equivalence by a quantitative distance of similarity. For this purpose, we introduce a statistical framework for estimating distances between Markov chains based on their simulation runs, and we investigate which distances can be approximated in our framework. On the other hand, we propose to compare systems with respect to a new qualitative logic, which expresses that behaviors occur with probability one or a positive probability. This qualitative analysis is robust with respect to modeling errors and applicable to many domains. In the last part, we present a new quantifier-free logic for integer arrays, which allows us to express counting. Counting properties are prevalent in array-manipulating programs, however they cannot be expressed in the quantified fragments of the theory of arrays. We present a decision procedure for our logic, and provide several complexity results