The complexity of probabilistic verification

We determine the complexity of testing whether a finite state, sequential or concurrent probabilistic program satisfies its specification expressed in linear-time temporal logic. For sequential programs, we present an algorithm that runs in time linear in the program and exponential in the specification, and also show that the problem is in PSPACE, matching the known lower bound. For concurrent programs, we show that the problem can be solved in time polynomial in the program and doubly exponential in the specification, and prove that it is complete for double exponential time. We also address these questions for specifications described by ω-automata or formulas in extended temporal logic.