Weighted Automata and Recurrence Equations for Regular Languages

Let $\mathcal{P}(\Sigma^*)$ be the semiring of languages, and consider its subset $\mathcal{P}(\Sigma)$. In this paper we define the language recognized by a weighted automaton over $\mathcal{P}(\Sigma)$ and a one-letter alphabet. Similarly, we introduce the notion of language recognition by linear recurrence equations with coefficients in $\mathcal{P}(\Sigma)$. As we will see, these two definitions coincide. We prove that the languages recognized by linear recurrence equations with coefficients in $\mathcal{P}(\Sigma)$ are precisely the regular languages, thus providing an alternative way to present these languages. A remarkable consequence of this kind of recognition is that it induces a partition of the language into its cross-sections, where the $n$th cross-section contains all the words of length $n$ in the language. Finally, we show how to use linear recurrence equations to calculate the density function of a regular language, which assigns to every $n$ the number of words of length $n$ in the language. We also show how to count the number of successful paths of a weighted automaton.