🤖 AI Summary
This paper investigates the model-checking complexity for discrete-time parametric systems comprising arbitrarily many anonymous, isomorphic processes, supporting both synchronous rendezvous and a newly introduced symmetric broadcast communication primitive; it distinguishes architectures with and without a distinguished controller. Methodologically, it introduces symmetric broadcast as a primitive and establishes a bisimulation between discrete-time systems and RB-systems; it proposes Vector Rendezvous Systems (VRS), integrating automata theory, rational linear programming, and geometric reasoning. Key contributions are: (1) safety verification is PSPACE-complete; (2) for systems without a controller, liveness verification is EXPTIME-decidable—the first such decidability result for liveness in parameterized real-time systems; (3) liveness becomes undecidable when a controller is present; (4) it provides a unifying semantic characterization linking discrete-time dynamics and broadcast communication at a fundamental level.
📝 Abstract
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.
