Synthesis for Multi-weighted Games with Branching-Time Winning Conditions

We investigate the synthesis problem in a quantitative game-theoretic setting with branching-time objectives. The objectives are given in a recursive modal logic with semantics defined over a multi-weighted extension of a Kripke structure where each transition is annotated with multiple nonnegative weights representing quantitative resources such as discrete time, energy and cost. The objectives may express bounds on the accumulation of each resource both in a global scope and in a local scope (on subformulae) utilizing a reset operator. We show that both the model checking problem as well as the synthesis problem are decidable and that the model checking problem is EXPTIME-complete, while the synthesis problem is in 2-EXPTIME and is NEXPTIME-hard. Furthermore, we encode both problems to the calculation of maximal fixed points on dependency graphs, thus achieving on-the-fly algorithms with the possibility of early termination.