Polynomial time decision algorithms for probabilistic automata

Deciding in an efficient way weak probabilistic bisimulation in the context of probabilistic automata is an open problem for about a decade. In this work we close this problem by proposing a procedure that checks in polynomial time the existence of a weak combined transition satisfying the step condition of the bisimulation. This enables us to arrive at a polynomial time algorithm for deciding weak probabilistic bisimulation, and also branching probabilistic bisimulation. We furthermore present several extensions to interesting related problems, in particular weak and branching probabilistic simulation, setting the ground for the development of more effective and compositional analysis algorithms for probabilistic systems.