Weighted restarting automata

Abstract

Restarting automata have been introduced as a formal model for the linguistic technique of analysis by reduction, which can be used to check the correctness of natural language sentences. In order to study quantitative aspects of restarting automata, we introduce the concept of a weighted restarting automaton. Such an automaton is given through a pair $(M,\omega )$, where M is a restarting automaton on some input alphabet $\Sigma$, and $\omega$ is a weight function that assigns an element of a given semiring S to each transition of M. Thus, $(M,\omega )$ defines a function $f_\omega ^M:\Sigma ^*\rightarrow S$ that associates an element of S to each input word over $\Sigma$. By looking at different semirings S and different weight functions $\omega$, various quantitative aspects of the behavior of M can be expressed through these functions. We are interested in the syntactic and semantic properties of these functions, e.g., their growth rates and the closure properties under various operations.