Hierarchical games and their applications in reasoning about reactive systems

We present a unified game-based approach for branching-time model checking of hierarchical systems. Such systems are exponentially more succinct than standard state-transition graphs, as repeated sub-systems are described only once. Early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that "flattens" the system and removes the hierarchy. Given a hierarchical system S and a branching-time speci¯cation à for it, we reduce the model-checking problem (does S satisfy f?) to the problem of solving a hierarchical game obtained by taking the product of S with an alternating tree automaton Af for f. Our approach leads to clean, uniform, and improved model-checking algorithms for a variety of branching-time temporal logics. In particular, by improving the algorithm for solving hierarchical par- ity games, we are able to solve the model-checking problem for the mu-calculus in Pspace and time complexity that is only polynomial in the depth of the hierarchy. Our approach also leads to an abstraction-refinement paradigm for hierarchical systems. The abstraction maintains the hierarchy, and is obtained by merging both states and sub-systems into abstract states.