Extensible Proof Systems for Infinite-State Systems

This paper revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus. Our results rely on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reconstruct the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableaux method is sound and complete for timed transition systems.