Logical characterization of weighted pebble walking automata

Abstract

Weighted automata are a conservative quantitative extension of finite automata that enjoys applications, e.g., in language processing and speech recognition. Their expressive power, however, appears to be limited, especially when they are applied to more general structures than words, such as graphs. To address this drawback, weighted automata have recently been generalized to weighted pebble walking automata, which proved useful as a tool for the specification and evaluation of quantitative properties over words and nested words. In this paper, we establish the expressive power of weighted pebble walking automata in terms of transitive closure logic, lifting a similar result by Engelfriet and Hoogeboom from the Boolean case to a quantitative setting. This result applies to general classes of graphs, including all the aforementioned classes.

Weighted Automata and Expressions over Pre-Rational Monoids

“…A final direction would be to use logics instead of expressions, to describe in a less operational way the behaviour of weighted automata over monoids. Promising results have already been obtained in specific contexts, like non-looping automata walking (with pebbles) on words, trees or graphs [3], but a cohesive point of view via monoids is still lacking.…”

“…We define its indexed expression I(W ) as the Kleene expression over an alphabet being a finite subset of (K ∪ M ) × N, obtained by replacing each of its atomic subexpression ∈ K ∪ M by a letter ( , i) ∈ (K ∪ M ) × N where i is a unique index (starting from 0 for the leftmost one) associated with each atomic subexpression. For instance, with the weighted expression W = (2 (3,7)). From the indexed expression, one can recover the initial expression, by forgetting about the index.…”

Two-way pebble transducers for partial functions and their composition

“…Other work on nested pebbles has appeared in, e.g., [2][3][4][5]13,14,[16][17][18][19]22,[26][27][28]30,32]. All results stated in this paper are effective.…”