On weighted first-order logics with discounting

We introduce a linear temporal logic and a first-order logic in the weighted setup of the max-plus semiring with discounting parameters in . Furthermore, we define -star-free series and counter-free weighted Buchi automata. We show that the classes of series definable in fragments of the weighted linear temporal logic and first-order logic, the class of -star-free series, and a subclass of -counter-free series coincide. This extends a fundamental result, for first-order logic theory, to series over the max-plus semiring with discounting.