Model Checking Probabilistic Pushdown Automata

We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for PCTL and the subclass of stateless pPDA. Finally, we consider the class of omega-regular properties and show that both qualitative and quantitative model checking for pPDA is decidable

Model checking of continuous-time Markov chains against timed automata specifications

We study the verification of a finite continuous-time Markov chain (CTMC) C against a linear real-time specification given as a deterministic timed automaton (DTA) A with finite or Muller acceptance conditions. The central question that we address is: what is the probability of the set of paths of C that are accepted by A, i.e., the likelihood that C satisfies A? It is shown that under finite acceptance criteria this equals the reachability probability in a finite piecewise deterministic Markov process (PDP), whereas for Muller acceptance criteria it coincides with the reachability probability of terminal strongly connected components in such a PDP. Qualitative verification is shown to amount to a graph analysis of the PDP. Reachability probabilities in our PDPs are then characterized as the least solution of a system of Volterra integral equations of the second type and are shown to be approximated by the solution of a system of partial differential equations. For single-clock DTA, this integral equation system can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations. As the coefficients are in fact transient probabilities in CTMCs, this result implies that standard algorithms for CTMC analysis suffice to verify single-clock DTA specifications