Adding Pebbles to Weighted Automata

Abstract

Abstract. We extend weighted automata and weighted rational expressions with 2-way moves and (reusable) pebbles. We show with examples from natural language modeling and quantitative model-checking that weighted expressions and automata with pebbles are more expressive and allow much more natural and intuitive specifications than classical ones. We extend Kleene-Schützenberger theorem showing that weighted expressions and automata with pebbles have the same expressive power. We focus on an efficient translation from expressions to automata. We also prove that the evaluation problem for weighted automata can be done very efficiently if the number of (reusable) pebbles is low.

Weighted Specifications over Nested Words

“…In a sense, these are extensions of the tree-walking automata with invisible pebbles, introduced in [11], to the weighted setting and to nested words. We extend results over words stated in [12], namely a KleeneSchützenberger theorem showing correspondence between weighted expressions with pebbles and layered weighted automata with pebbles (i.e., those that can only use a bounded number of pebbles). We also show how to efficiently compute the value associated to a given nested word in a weighted automaton with pebbles, and prove decidability (not surprinsingly, with non-elementary complexity) of the emptiness problem in case the underlying weight structure has no zero divisor.…”

“…An efficient algorithm constructing an automaton from an expression uses standard automata (which has as variants Berry-Sethi algorithm, or Glushkov algorithm). In [12], we extended this algorithm to deal with pebbles and two-way navigation in (non-nested) words. It is not difficult -and not surprising -to see that this extension holds in the context of nested words too.…”

“…It is not difficult -and not surprising -to see that this extension holds in the context of nested words too. For the converse translation, weighted versions of the state elimination method of Brzozowski-McCluskey, or the procedure of McNaughton-Yamada can easily be applied to our two-way/pebble setting as previously stated in [12] over (non-nested) words. We now study the evaluation problem of a K-layered pebWA A: given a nested word W over A and a valuation σ : Peb → pos(W ), compute […”

A Probabilistic Kleene Theorem

“…Note that (non-probabilistic) two-way automata are in fact an appropriate machine model for compiling XPath queries [5]. The concept presented in this paper may therefore constitute a first step towards probabilistic database query languages: an expression is considered as a query, and an equivalent automaton can be used as a tool for evaluating queries efficiently (see [16] for recent developments on weighted query evaluation).…”

Pebble Weighted Automata and Weighted Logics

“…This evaluation procedure can be improved with some efforts as presented in [Gastin and Monmege 2012]. In particular, lifted to our framework, this permits to evaluate a P1WA with O(|Q| 3 · |u| max(p,1) ) scalar operations (sum and product of elements of the semiring).…”