Thomas Schwentick

XPath Containment in the Presence of Disjunction, DTDs, and Variables

XPath is a simple language for navigating an XML tree and returning a set of answer nodes. The focus in this paper is on the complexity of the containment problem for various fragments of XPath. In addition to the basic operations (child, descendant, filter, and wildcard), we consider disjunction, DTDs and variables. W.r.t. variables we study two semantics: (1) the value of variables is given by an outer context; (2) the value of variables is defined existentially. We establish an almost complete classification of the complexity of the containment problem w.r.t. these fragments.

Finite state machines for strings over infinite alphabets

Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specifically, we consider register and pebble automata, and extensions of first-order logic and monadic second-order logic. For each type of automaton we consider one-way and two-way variants, as well as deterministic, nondeterministic, and alternating control. We investigate the expressiveness and complexity of the automata and their connection to the logics, as well as standard decision problems. Some of our results answer open questions of Kaminski and Francez on register automata.

XPath query containment

Consider an XML publish-subscribe scenario with hundreds of subscribers and tens of thousands of XML documents to be delivered per day. Subscribers specify the documents in which they are interested in by means of XPath [8] expressions. If an expression matches a (part of a) document it is delivered to the subscriber. Naturally, it is desired that the decision to which subscriber a document must be sent should be taken quickly. Although the test whether a single XPath expression matches can be done in polynomial time, it is not efficient to test every such expression for every document. Fortunately, there is a partial order on expressions, i.e., for some expressions p, q it might hold that whenever a document matches p it also matches q (denoted p ⊆0 q). If we already know that a document matches p, we do not need to test q anymore, as it matches automatically. Correspondingly, if we know that q does not match then p will not match either. Hence, the inclusion structure of the XPath expressions should be computed in advance to decrease online computation time. This leads to the algorithmic problem of XPath Query Containment, i.e., checking whether p ⊆0 q (for a different, indexbased approach see, e.g., [6]). The main idea of this article is to describe some of the main algorithmic techniques that have been proposed for XPath Query Containment. These techniques are described in Section 5. Before that, in Sections 2 and 3 the basic definitions on XPath and the

Automata for XML---A survey

Automata play an important role for the theoretical foundations of XML data management, but also in tools for various XML processing tasks. This survey article aims to give an overview of fundamental properties of the different kinds of automata used in this area and to relate them to the four key aspects of XML processing: schemas, navigation, querying and transformation.

On The Complexity Of Xpath Containment In The Presence Of Disjunction, Dtds, And Variables

XPath is a simple language for navigating an XML-tree and returning a set ofnanswer nodes. The focus in this paper is on the complexity of the containmentnproblem for various fragments of XPath. We restrict attention to the mostncommon XPath expressions which navigate along the child and/or descendant axis.nIn addition to basic expressions using only node tests and simple predicates,nwe also consider disjunction and variables (ranging over nodes). Further, weninvestigate the containment problem relative to a given DTD. With respect tonvariables we study two semantics, (1) the original semantics of XPath, wherenthe values of variables are given by an outer context, and (2) an existentialnsemantics introduced by Deutsch and Tannen, in which the values of variablesnare existentially quantified. In this framework, we establish an exactnclassification of the complexity of the containment problem for many XPathnfragments.

Counting in trees for free

It is known that MSO logic for ordered unranked trees is undecidable if Presburger constraints are allowed at children of nodes. We show here that a decidable logic is obtained if we use a modal fixpoint logic instead. We present a characterization of this logic by means of deterministic Presburger tree automata and show how it can be used to express numerical document queries. Surprisingly, the complexity of satisfiability for the extended logic is asymptotically the same as for the original fixpoint logic. The non-emptiness for Presburger tree automata (PTA) is pspace-complete, which is moderate given that it is already pspace-hard to test whether the complement of a regular expression is non-empty. We also identify a subclass of PTAs with a tractable non-emptiness problem. Further, to decide whether a tree t satisfies a formula ϕ is polynomial in the size of ϕ and linear in the size of t.

Numerical document queries

A query against a database behind a site like Napster may search, e.g., for all users who have downloaded more jazz titles than pop music titles. In order to express such queries, we extend classical monadic second-order logic by Presburger predicates which pose numerical restrictions on the children (content) of an element node and provide a precise automata-theoretic characterization. While the existential fragment of the resulting logic is decidable, it turns out that satisfiability of the full logic is undecidable. Decidable satisfiability and a querying algorithm even with linear data complexity can be obtained if numerical constraints are only applied to those contents of elements where ordering is irrelevant. Finally, it is sketched how these techniques can be extended also to answer questions like, e.g., whether the total price of the jazz music downloaded so far exceeds a users budget.

Two-variable logic on data trees and XML reasoning

Motivated by reasoning tasks in the context of XML languages, the satisfiability problem of logics on data trees is investigated. The nodes of a data tree have a label from a finite set and a data value from a possibly infinite set. It is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result decidability of satisfiability and containment for a data-aware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

On winning Ehrenfeucht games and monadic NP

Abstract Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that • — Graph Connectivity is not expressible in existential monadic second-order logic (MonNP), even in the presence of a built-in linear order, • — Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n 0(1) , and • — the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation.

Complexity of Decision Problems for Simple Regular Expressions.

We study the complexity of the inclusion, equivalence, and intersection problem for simple regular expressions arising in practical XML schemas. These basically consist of the concatenation of factors where each factor is a disjunction of strings possibly extended with ‘*’ or ‘?’. We obtain lower and upper bounds for various fragments of simple regular expressions. Although we show that inclusion and intersection are already intractable for very weak expressions, we also identify some tractable cases. For equivalence, we only prove an initial tractability result leaving the complexity of more general cases open. The main motivation for this research comes from database theory, or more specifically XML and semi-structured data. We namely show that all lower and upper bounds for inclusion and equivalence, carry over to the corresponding decision problems for extended context-free grammars and single-type tree grammars, which are abstractions of DTDs and XML Schemas, respectively. For intersection, we show that the complexity only carries over for DTDs.