Weighted recognizability over infinite alphabets

We introduce weighted variable automata over infinite alphabets and commutative semirings. We prove that the class of their behaviors is closed under sum, and under scalar, Hadamard, Cauchy, and shuffle products, as well as star operation. Furthermore, we consider rational series over infinite alphabets and we state a Kleene-Schützenberger theorem. We introduce a weighted monadic second order logic and a weighted linear dynamic logic over infinite alphabets and investigate their relation to weighted variable automata. An application of our theory, to series over the Boolean semiring, concludes to new results for the class of languages accepted by variable automata