Logical definability and query languages over ranked and unranked trees

We study relations on trees defined by first-order constraints over a vocabulary that includes the tree extension relation T ≺ T ′ , holding if and only if every branch of T extends to a branch of T ′, unary node-tests, and a binary relation checking if the domains of two trees are equal. We consider both ranked and unranked trees. These are trees with and without a restriction on the number of children of nodes. We adopt the model-theoretic approach to tree relations and study relations definable over the structure consisting of the set of all trees and the above predicates. We relate definability of sets and relations of trees to computability by tree automata. We show that some natural restrictions correspond to familiar logics in the more classical setting, where every tree is a structure over a fixed vocabulary, and to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of trees, and obtain quantifier-restriction results that give us bounds on the expressive power and complexity. As unrestricted relational calculi can express problems complete for each level of the polynomial hierarchy, we look at their restrictions, corresponding to the restricted logics over the family of all unranked trees, and find several calculi with low (NC 1) data complexity, while still expressing properties important for database an

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

Deciding definability in FO2(<h,<v) on trees

We provide a decidable characterization of regular forest languages definable in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first order logic built from the descendant relation and the following sibling relation. In terms of expressive power it corresponds to a fragment of the navigational core of XPath that contains modalities for going up to some ancestor, down to some descendant, left to some preceding sibling, and right to some following sibling. We also show that our techniques can be applied to other two variable first-order logics having exactly the same vertical modalities as FO2(<h,<v) but having different horizontal modalities

Recognisable languages over monads

The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propose monads as a unifying framework for discussing existing algebras and designing new algebras

Piecewise testable tree languages

This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma_1 sentences. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma_1 sentences if and only if its syntactic monoid is J-trivial

Modulo Counting on Words and Trees

We consider the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees. We prove a small-model property of this logic, which gives a technique for deciding the satisfiability problem. In the case of words this gives a new proof of EXPSPACE upper bound, and in the case of trees it gives a 2EXPTIME algorithm. This algorithm is optimal: we prove a matching lower bound by a generic reduction from alternating Turing machines working in exponential space; the reduction involves a development of a new version of tiling games

Monadic Queries over Tree-Structured Data

Monadic query languages over trees currently receive considerable interest in the database community, as the problem of selecting nodes from a tree is the most basic and widespread database query problem in the context of XML. Partly a survey of recent work done by the authors and their group on logical query languages for this problem and their expressiveness, this paper provides a number of new results related to the complexity of such languages over so-called axis relations (such as "child" or "descendant") which are motivated by their presence in the XPath standard or by their utility for data extraction (wrapping)

Transitive closure logic, nested tree walking automata, and XPath

International audienceWe study FO(MTC), first-order logic with monadic transitive closure, a logical formalism in between FO and MSO on trees. We characterize the expressive power of FO(MTC) in terms of nested tree-walking automata. Using the latter, we show that FO(MTC) is strictly less expressive than MSO, solving an open problem. We also present a temporal logic on trees that is expressively complete for FO(MTC), in the form of an extension of the XML document navigation language XPath with two operators: the Kleene star for taking the transitive closure of path expressions, and a subtree relativisation operator, allowing one to restrict attention to a specific subtree while evaluating a subexpression. We show that the expressive power of this XPath dialect equals that of FO(MTC) for Boolean, unary and binary queries. We also investigate the complexity of the automata model as well as the XPath dialect. We show that query evaluation be done in polynomial time (combined complexity), but that emptiness (or, satisfiability) is 2ExpTime-complete

XML Database Transformations

Database transformations provide a unifying umbrella for queries and updates. In general, they can be characterised by five postulates, which constitute the database analogue of Gurevich's sequential ASM thesis. Among these postulates the background postulate supposedly captures the particularities of data models and schemata. For the characterisation of XML database transformations the natural first step is therefore to define the appropriate tree-based backgrounds, which draw on hereditarily finite trees, tree algebra operations, and extended document type definitions. This defines a computational model for XML database transformation using a variant of Abstract State Machines. Then the incorporation of weak monadic second-order logic provides an alternative computational model called XML machines. The main result is that these two computational models for XML database transformations are equivalent