Parameterized Model-checking of Discrete-Timed Networks and Symmetric-Broadcast Systems

We study the complexity of the model-checking problem for parameterized discrete-timed systems with arbitrarily many anonymous and identical processes, with and without a distinguished "controller", and communicating via synchronous rendezvous. Our framework extends the seminal work from German and Sistla on untimed systems by adding discrete-time clocks to processes. For the case without a controller, we show that the systems can be efficiently simulated -- and vice versa -- by systems of untimed processes that communicate via rendezvous and symmetric broadcast, which we call "RB-systems". Symmetric broadcast is a novel communication primitive that allows all processes to synchronize at once; however, it does not distinguish between sending and receiving processes. We show that the parameterized model-checking problem for safety specifications is pspace-complete, and for liveness specifications it is decidable in exptime. The latter result is proved using automata theory, rational linear programming, and geometric reasoning for solving certain reachability questions in a new variant of vector addition systems called "vector rendezvous systems". We believe these proof techniques are of independent interest and will be useful in solving related problems. For the case with a controller, we show that the parameterized model-checking problems for RB-systems and systems with asymmetric broadcast as a primitive are inter-reducible. This allows us to prove that for discrete timed-networks with a controller the parameterized model-checking problem is undecidable for liveness specifications. Our work exploits the intimate connection between parameterized discrete-timed systems and systems of processes communicating via broadcast, providing a rare and surprising decidability result for liveness properties of parameterized timed-systems, as well as extend work from untimed systems to timed systems.