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...
Gespeichert in:
Hauptverfasser: | , |
---|---|
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | 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. |
---|---|
DOI: | 10.48550/arxiv.1007.1045 |