Using Stochastic Comparison for Efficient Model Checking of Uncertain Markov Chains

We consider model checking of discrete time Markov Chains (DTMC) with transition probabilities which are not exactly known but lie in a given interval. Model checking a probabilistic computation tree logic (PCTL) formula for interval-valued DTMCs (IMC) has been shown to be NP hard and co-NP hard. Since the state space of a realistic DTMC is generally huge, these lower bounds prevent the application of exact algorithms for such models. Therefore we propose to apply the stochastic comparison method to check an extended version of PCTL for IMCs. More precisely, we first design linear time algorithms to quantitatively analyze IMCs. Then we develop an efficient, semi-decidable PCTL model checking procedure for IMCs. Furthermore, our procedure returns more refined answers than traditional ones: emph{YES, NO, DON´T KNOW}. Thus we may provide useful partial information for modelers in the `DON´T KNOW´ case.