XPL: An extended probabilistic logic for probabilistic transition systems

Generalized Probabilistic Logic (GPL) is a temporal logic, based on the modal mu-calculus, for specifying properties of reactive probabilistic systems. We explore XPL, an extension to GPL allowing the semantics of nondeterminism present in Markov decision processes (MDPs). XPL is expressive enough that a number of independently studied problems--- such as termination of Recursive MDPs (RMDPs), PCTL* model checking of MDPs, and reachability for Branching MDPs--- can all be cast as model checking over XPL. Termination of multi-exit RMDPs is undecidable; thus, model checking in XPL is undecidable in general. We define a subclass, called separable XPL, for which model checking is decidable. Decidable problems such as termination of 1-exit RMDPs, PCTL* model checking of MDPs, and reachability for Branching MDPs can be reduced to model checking separable XPL. Thus, XPL forms a uniform framework for studying problems involving systems with non-deterministic and probabilistic behaviors, while separable XPL provides a way to solve decidable fragments of these problems.