Weighted Specifications over Nested Words

Abstract

Abstract. This paper studies several formalisms to specify quantitative properties of finite nested words (or equivalently finite unranked trees). These can be used for XML documents or recursive programs: for instance, counting how often a given entry occurs in an XML document, or computing the memory required for a recursive program execution. Our main interest is to translate these properties, as efficiently as possible, into an automaton, and to use this computational device to decide problems related to the properties (e.g., emptiness, model checking, simulation) or to compute the value of a quantitative specification over a given nested word. The specification formalisms are weighted regular expressions (with forward and backward moves following linear edges or call-return edges), weighted first-order logic, and weighted temporal logics. We introduce weighted automata walking in nested words, possibly dropping/lifting (reusable) pebbles during the traversal. We prove that the evaluation problem for such automata can be done very efficiently if the number of pebble names is small, and we also consider the emptiness problem.

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.…”

Logical characterization of weighted pebble walking automata

“…In this paper, we consider particular classes of graphs, namely searchable classes of graphs, inspired by [13]. 3 Definition 3. A class of graphs G ⊆ G(A, D) is searchable if there exists a deterministic walking automaton A G = (Q, A, D, {q i }, ∆, {q o }) such that q o has no outgoing transition and for every graph G = (V, (E d ) d∈D , λ, ι) ∈ G, there exists a linear order over V , with minimal element ι, such that for every vertex v ∈ V , A G admits an accepting run over G from vertex v ending either at the direct successor of v for , if this successor exists, or at ι if it does not exist.…”

“…The syntax of [8] is purely quantitative, though Boolean connectives can be expressed indirectly. As it may be somewhat confusing to interpret purely logical formulae in a weighted manner, we slightly modify the original syntax, as already presented in [3], and later in [18], by clearly separating the Boolean and the quantitative parts: our weighted logic consists of a Boolean first-order kernel augmented with quantitative operators (addition, multiplication, sum and product, and weighted transitive closure). Thus, our language allows us to test explicitly for Boolean properties, and to perform computations.…”

“…They have proved useful for the specification and evaluation of quantitative properties over words [4,17], nested words [3], and, more generally, graphs [20]. Like their Boolean counterpart, wPWA traverse a graph along its edges.…”