1 research outputs found
Weighted Automata and Recurrence Equations for Regular Languages
Let be the semiring of languages, and consider its
subset . In this paper we define the language recognized
by a weighted automaton over and a one-letter alphabet.
Similarly, we introduce the notion of language recognition by linear recurrence
equations with coefficients in . As we will see, these two
definitions coincide. We prove that the languages recognized by linear
recurrence equations with coefficients in 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 th
cross-section contains all the words of length 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 the number of words of length
in the language. We also show how to count the number of successful paths
of a weighted automaton.Comment: 14 pages, 6 figure