Weighted Automata and Recurrence Equations for Regular Languages

TL;DR

This paper establishes a novel connection between weighted automata, recurrence equations, and regular languages, demonstrating how recurrence equations can characterize regular languages and facilitate computations like word counts and density functions.

Contribution

It introduces an equivalence between weighted automata and recurrence equations over language semirings, providing new methods to analyze regular languages.

Findings

Linear recurrence equations with coefficients in language semirings recognize exactly the regular languages.

The approach allows partitioning languages into cross-sections based on word length.

Methods for calculating the density function and counting successful paths are developed.

Abstract

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.