  The compositional approach is important for reasoning about large and complex
systems. In this work, we address synchronous systems with hierarchical
structures, which are often used to model cyber-physical systems. We revisit
the theory of reactive modules and reformulate it based on hypergraphs to
clarify the parallel composition and the hierarchical description of modules.
Then, we propose an automatic verification method for hierarchical systems.
Given a system description annotated with assume-guarantee contracts, the
proposed method divides the system into modules and verifies them separately to
show that the top-level system satisfies its contract. Our method allows an
input to be a circular system in which submodules mutually depend on each
other. Experimental result shows our method can be effectively implemented
using an SMT-based model checker.
