This paper proposes a novel abstraction technique for fully probabilistic systems. The models of our study are classical discrete-time and continuous-time Markov chains (DTMCs and CTMCs, for short). A DTMC is a Kripke structure in which each transition is equipped with a discrete probability; in a CTMC, in addition, state residence times are governed by negative exponential distributions. Our abstraction technique fits within the realm of three-valued abstraction methods that have been used successfully for traditional model checking. The key ingredients of our technique are a partitioning of the state space combined with an abstraction of transition probabilities by intervals. It is shown that this provides a conservative abstraction for both negative and affirmative verification results for a three-valued semantics of PCTL (Probabilistic Computation Tree Logic). In the continuous-time setting, the key idea is to apply abstraction on uniform CTMCs which are readily obtained from general CTMCs. In a similar way as for the discrete case, this is shown to yield a conservative abstraction for a three-valued semantics of CSL (Continuous Stochastic Logic). Abstract CTMCs can be verified by computing time-bounded reachability probabilities in continuous-time MDPs.
The research has been partially funded by the DFG Research Training Group 1298 (AlgoSyn), the Cluster of Excellence on Multimodal Computing and Interaction at Saarland University and the EU Project Quasimodo under FP-ICT-2007-1 Grant 214755.