Model Checking and Synthesis for Branching Multi-Weighted Logics

We investigate the open synthesis problem in a quantitative game theoretic setting where the system model is annotated with multiple nonnegative weights representing quantitative resources such as energy, discrete time or cost. We consider system specifications expressed in the branching time logic CTL extended with bounds on resources. As our first contribution, we show that the model checking problem for the full logic is undecidable with already three weights. By restricting the bounds to constant upper or lower-bounds on the individual weights, we demonstrate that the problem becomes decidable and that the model checking problem is PSPACE-complete. As a second contribution, we show that by imposing upper-bounds on the temporal operators and assuming that the cost converges over infinite runs, the synthesis problem is also decidable. Finally, we provide an on-the-fly algorithm for the synthesis problem on an unrestricted model for a reachability fragment of the logic and we prove EXPTIME-completeness of the synthesis problem.