Logics for Unranked Trees: An Overview

Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to relate complex properties of trees to the existence of tree automata for those properties. Furthermore, logics differ significantly in their model-checking properties, their automata models, and their behavior on ordered and unordered trees. In this paper we present a survey of logics for unranked trees

Reasoning about XML with temporal logics and automata

We show that problems arising in static analysis of XML specifications and transformations can be dealt with using techniques similar to those developed for static analysis of programs. Many properties of interest in the XML context are related to navigation, and can be formulated in temporal logics for trees. We choose a logic that admits a simple single-exponential translation into unranked tree automata, in the spirit of the classical LTL-to-Büchi automata translation. Automata arising from this translation have a number of additional properties; in particular, they are convenient for reasoning about unary node-selecting queries, which are important in the XML context. We give two applications of such reasoning: one deals with a classical XML problem of reasoning about navigation in the presence of schemas, and the other relates to verifying security properties of XML views

Combining Temporal Logics for Querying XML Documents

Abstract. Close relationships between XML navigation and temporal logics have been discovered recently, in particular between logics LTL and CTL ⋆ and XPath navigation, and between the µ-calculus and navigation based on regular expressions. This opened up the possibility of bringing model-checking techniques into the field of XML, as documents are naturally represented as labeled transition systems. Most known results of this kind, however, are limited to Boolean or unary queries, which are not always sufficient for complex querying tasks. Here we present a technique for combining temporal logics to capture nary XML queries expressible in two yardstick languages: FO and MSO. We show that by adding simple terms to the language, and combining a temporal logic for words together with a temporal logic for unary tree queries, one obtains logics that select arbitrary tuples of elements, and can thus be used as building blocks in complex query languages. We present general results on the expressiveness of such temporal logics, study their model-checking properties, and relate them to some common XML querying tasks.

Wreath Products of Forest Algebras, with Applications to Tree Logics

We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics

Order-Invariant Types and their Applications

Our goal is to show that the standard model-theoretic concept of types can be applied in the study of order-invariant properties, i.e., properties definable in a logic in the presence of an auxiliary order relation, but not actually dependent on that order relation. This is somewhat surprising since order-invariant properties are more of a combinatorial rather than a logical object. We provide two applications of this notion. One is a proof, from the basic principles, of a theorem by Courcelle stating that over trees, order-invariant MSO properties are expressible in MSO with counting quantifiers. The other is an analog of the Feferman-Vaught theorem for order-invariant properties

Mu-Calculus Based Resolution of XPath Decision Problems

XPath is the standard declarative notation for navigating XML data and returning a set of matching nodes. In the context of XSLT/XQuery analysis, query optimization, and XML type checking, XPath decision problems arise naturally. They notably include XPath containment (whether or not for any tree the result of a particular query is included in the result of a second one), and XPath satisfiability (whether or not an expression yields a non-empty result), in the presence (or the absence) of XML DTDs. In this paper, we propose a unifying logic for XML, namely the alternation-free modal mu-calculus with converse. We show how to translate major XML concepts such as XPath and DTDs into this logic. Based on these embeddings, we show how XPath decision problems can be solved using a state-of-the-art EXPTIME decision procedure for mu-calculus satisfiability. We provide preliminary experiments which shed light, for the first time, on the cost of solving XPath decision problems in practice

A Decision Procedure for XPath Containment

XPath is the standard language for addressing parts of an XML document. We present a sound and complete decision procedure for containment of XPath queries. The considered XPath fragment covers most of the language features used in practice. Specifically, we show how XPath queries can be translated into equivalent formulas in monadic second-order logic. Using this translation, we construct an optimized logical formulation of the containment problem, which is decided using tree automata. When the containment relation does not hold between two XPath expressions, a counter-example XML tree is generated. We provide a complexity analysis together with practical experiments that illustrate the efficiency of the decision procedure for realistic scenarios

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 eval-uated 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