Nested Words Research Articles

Games for Active XML Revisited

The article studies the rewriting mechanisms for intensional documents in the Active XML framework, abstracted in the form of active context-free games. The safe rewriting problem studied in this article is to decide whether the first player, JULIET, has a winning strategy for a given game and (nested) word; this corresponds to a successful rewriting strategy for a given intensional document. The article examines several extensions of active context-free games. The primary extension allows for more expressive schemas (namely XML schemas and regular nested word languages) for both target and replacement languages and has the effect that games are played on nested words instead of (flat) words as in previous studies. Other extensions consider validation of input parameters of web services, and an alternative semantics based on insertion of values returned by the services. In general, the complexity of the safe rewriting problem is highly intractable (doubly exponential time), but the article identifies relevant tractable cases.

Weighted automata and logics for infinite nested words

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Büchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.

WEIGHTED NESTED WORD AUTOMATA AND LOGICS OVER STRONG BIMONOIDS

Nested words have been introduced by Alur and Madhusudan as a model for e.g. recursive programs or XML documents and have received much recent interest. In this paper, we investigate a quantitative automaton model and a quantitative logic for nested words. The behavior resp. the semantics map nested words to weights which are taken from a strong bimonoid. Strong bimonoids can be viewed as semirings without requiring the distributivity assumption which was essential in the classical theory of formal power series; strong bimonoids include e.g. all bounded lattices and many other structures from multi-valued logics. Our main results show that weighted nested word automata and suitable weighted MSO logics are expressively equivalent. This extends the classical Büchi-Elgot result from words to a weighted setting for nested words.

Queries on Xml streams with bounded delay and concurrency

Query answering algorithms on Xml streams check answer candidates on the fly in order to avoid the unnecessary buffering whenever possible. The delay and concurrency of a query are two measures for the degree of their streamability. They count the maximal number of stream elements during the life time for some query answer, and respectively, the maximal number of simultaneously alive answer candidates of a query. We study queries defined by deterministic nested word automata, which subsume large streamable fragments of XPath subject to schema restrictions by DTDs modulo P-time translations. We show that bounded and k-bounded delay and concurrency of such automata-defined queries are all decidable in polynomial time in the size of the automaton. Our results are obtained by P-time reduction to the bounded valuedness problem for recognizable relations between unranked trees, a problem that we show to be decidable in P-time.

Limitations of lower bound methods for deterministic nested word automata

Finite automata operating on nested words were introduced by Alur and Madhusudan in 2006. While nested word automata retain many of the desirable properties of ordinary finite automata, there is no known efficient minimization algorithm for deterministic nested word automata and, interestingly, state complexity bounds for nested word automata turn out to differ significantly from the corresponding bounds for ordinary finite automata. Consequently lower bounds for the state complexity of nested word languages need to rely on fooling set type techniques. We discuss limitations of the techniques and show that, even in the deterministic case, the bounds given by the lower bound methods may be arbitrarily far away from the actual state complexity of the nested word language.

Regular Languages of Nested Words: Fixed Points, Automata, and Synchronization

Nested words provide a natural model of runs of programs with recursive procedure calls. The usual connection between monadic second-order logic (MSO) and automata extends from words to nested words and gives us a natural notion of regular languages of nested words.In this paper we look at some well-known aspects of regular languages—their characterization via fixed points, deterministic and alternating automata for them, and synchronization for defining regular relations—and extend them to nested words. We show that mu-calculus is as expressive as MSO over finite and infinite nested words, and the equivalence holds, more generally, for mu-calculus with past modalities evaluated in arbitrary positions in a word, not only in the first position. We introduce the notion of alternating automata for nested words, show that they are as expressive as the usual automata, and also prove that Muller automata can be determinized (unlike in the case of visibly pushdown languages). Finally we look at synchronization over nested words. We show that the usual letter-to-letter synchronization is completely incompatible with nested words (in the sense that even the weakest form of it leads to an undecidable formalism) and present an alternative form of synchronization that gives us decidable notions of regular relations.

Adding nesting structure to words

We propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include executions of structured programs, annotated linguistic data, and HTML/XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata —finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-*, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over. The linear encodings of nested words give the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics. We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata.