Automata Games for Multiple-model Checking

3-valued models have been advocated as a means of system abstraction such that verifications and refutations of temporal-logic properties transfer from abstract models to the systems they represent. Some application domains, however, require multiple models of a concrete or virtual system. We build the mathematical foundations for 3-valued property verification and refutation applied to sets of common concretizations of finitely many models. We show that validity checking for the modal mu-calculus has the same cost (EXPTIME-complete) on such sets as on the set of all 2-valued models, provide an efficient algorithm for checking whether common concretizations exist for a fixed number of models, and propose using parity games on variants of tree automata to efficiently approximate validity checks of multiple models. Structural properties of a universal topological model confirm that such approximations are reasonably precise only for tree-automata-like models.