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. GPL is expressive enough that termination of Recursive Markov Chains (RMCs) can be cast as a model checking problem over GPL. In this paper, we explore XPL, an extension to GPL for expressing properties of transition systems with probabilistic and nondeterministic behavior, such as Recursive Markov Decision Processes (Recursive MDPs or RMDPs). However, since termination of multi-exit RMDPs is undecidable, model checking in XPL is undecidable as well. We provide a model checking procedure for XPL that always terminates: either successfully with the model checking result, or with failure. We define a sufficient condition for success via a syntactically-characterized class, called separable XPL formulae. We show that a large class of problems, including termination of 1-exit RMDPs and PCTL* model checking of MDPs, can be reduced to the model checking of separable XPL formulae.