Skew and infinitary formal power series

We investigate finite-state systems with weights. Departing from the classical theory, in this paper the weight of an action does not only depend on the state of the system, but also on the time when it is executed; this reflects the usual human evaluation practices in which later events are considered less urgent and carry less weight than close events. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Schützenberger.Secondly, we deal with nonterminating behaviors and their weights. This includes an extension of the Büchi-acceptance condition from finite automata to weighted automata and provides a characterization of these nonterminating behaviors in terms of ω-rational formal power series. This generalizes a classical theorem of Büchi.