Pebble Weighted Automata and Transitive Closure Logics

Abstract

We introduce new classes of weighted automata on words. Equipped with pebbles and a two-way mechanism, they go beyond the class of recognizable formal power series, but capture a weighted version of first-order logic with bounded transitive closure. In contrast to previous work, this logic allows for unrestricted use of universal quantification. Our main result states that pebble weighted automata, nested weighted automata, and this weighted logic are expressively equivalent. We also give new logical characterizations of the recognizable series.

Adding Pebbles to Weighted Automata

“…First, as shown in Section 2 for applications in natural language processing and quantitative model-checking, 2-way moves and pebbles allow more natural and more concise descriptions of the quantitative expressions we need to evaluate. Second, in the weighted case, 2-way and pebbles do increase the expressive power as already observed in [8] in relation with weighted logics or in [26] in the probabilistic setting. This is indeed in contrast with the Boolean case where 2-way and pebbles do not add expressive power over words (see, e.g., [19]) even though they allow more succinct descriptions (see, e.g., [4]).…”

A Probabilistic Kleene Theorem

“…We also raise the question of whether our technique can be used to obtain ω-expressions for probabilistic Büchi automata, which have attracted a lot of attention [2,1,8]. Just like classical finite automata, weighted automata over semirings enjoy characterizations in terms of monadic second-order logic [13,6]. Continuing this line of research, a recent paper establishes a logical characterization of probabilistic automata [30].…”

“…Continuing this line of research, a recent paper establishes a logical characterization of probabilistic automata [30]. It would be interesting to study whether alternative characterizations exist that use, for example, a transitive-closure operator [6].…”

Weighted First-Order Logics over Semirings

“…Next, we define a fragment of our logic. For this, we recall the notion of an FO-step formula from [4]. More precisely, a formula…”

“…We proceed by induction on m, hence, assume firstly that m = 1. Without any loss, we suppose the state sets Q i (1 ≤ i ≤ n) to be pairwise disjoint 4 . For every p, p ∈ P 1 and 2 ≤ i ≤ n − 1, we consider the simple cfwa…”