Model Checking for a Probabilistic Branching

We consider concurrent probabilistic systems, based on probabilistic automata of Segala & Lynch 55], which allow non-deterministic choice between probability distributions. These systems can be decomposed into a collection of \computation trees" which arise by resolving the non-deterministic, but not probabilistic, choices. The presence of non-determinism means that certain liveness properties cannot be established unless fairness is assumed. We introduce a probabilistic branching time logic PBTL, based on the logic TPCTL of Hansson 30] and the logic PCTL of 55], resp. pCTL of 14]. The formulas of the logic express properties such as \every request is eventually granted with probability at least p". We give three interpretations for PBTL on concurrent probabilistic processes: the rst is standard, while in the remaining two interpretations the branching time quantiiers are taken to range over a certain kind of fair computation trees. We then present a model checking algorithm for verifying whether a concurrent probabilistic process satisses a PBTL formula assuming fairness constraints. We also propose adaptations of existing model checking algorithms for pCTL 14, 4] to obtain procedures for PBTL under fairness constraints. The techniques developed in this paper have applications in automatic veriication of randomized distributed systems.