On weighted first-order logics with discounting

Abstract

We introduce a linear temporal logic and a first-order logic in the weighted setup of the max-plus semiring with discounting parameters in $$[0,1)$$ [ 0 , 1 ) . Furthermore, we define $$\omega \hbox {-}d$$ ? - d -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 $$\omega \hbox {-}d$$ ? - d -star-free series, and a subclass of $$\omega \hbox {-}d$$ ? - d -counter-free series coincide. This extends a fundamental result, for first-order logic theory, to series over the max-plus semiring with discounting.