Weighted automata with discounting

We investigate weighted automata with discounting and their behaviors over semirings and finitely generated graded monoids. We characterize the discounted behaviors of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene–Schützenberger. Here we show that the very special case of Schützenberger's result for free monoids over singleton alphabets suffices to deduce our generalization.