A Decidable Recursive Logic for Weighted Transition Systems

Labelled weighted transition systems (LWSs) are transition systems labelled with actions and real valued quantities representing the costs of transitions with respect to various resources. We introduce Recursive Weighted Logic (RWL) being a multi-modal logic that expresses not only qualitative, but also quantitative properties of LWSs by using first-order variables that measure local costs, similar to the clocks in timed logics. In addition, RWL is endowed with simultaneous recursive equations, which specify the weakest properties satisfied by the recursive variables. It is proved that the satisfiability problem for the logic is decidable by applying a variant of the region technique developed for timed automata. This result is in contrast to corresponding temporal logics for real-time systems, where satisfiability is known to be undecidable.