Weighted Automata and Expressions over Pre-Rational Monoids

TL;DR

This paper extends the Kleene theorem to weighted automata and expressions over pre-rational monoids, unifying various automata types and addressing infinite sums with rationally additive semirings.

Contribution

It introduces a generalized Kleene theorem for weighted automata over pre-rational monoids, encompassing two-way and tree-walking automata.

Findings

Generalization of Kleene theorem to pre-rational monoids

Equivalence of certain expressions and weighted automata

Handling infinite sums with rationally additive semirings

Abstract

The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata.