Compositional Verification for Large-Scale Systems via Closure Certificates

Closure certificates (CCs), function analogs of transition invariants, provide a framework to verify discrete-time dynamical systems against
$\omega $
-regular specifications. Such certificates are similar to barrier certificates (BCs) yet are less conservative than BCs when leveraged to verify
$\omega -$
regular properties. However, CCs are defined over pairs of states of the system rather than over the state of the system, and seek to overapproximate the transitive closure of the transition relation. Thus, finding these certificates is often harder and computationally more demanding than BCs, especially for large-scale systems. To address this challenge, we propose a dissipativity-inspired approach to construct closure certificates for interconnected systems. In such a setting, we assume our large-scale system to be an interconnection of subsystems under a linear map. We then find local certificates for these subsystems. These local certificates are then composed to form a closure certificate for the interconnected system, acting as proof of the satisfaction of a desired
$\omega $
-regular specification. Finally, we illustrate our approach with a numerical simulation.