Pattern logics and auxiliary relations

A common theme in the study of logics over finite structures is adding auxiliary predicates to enhance expressiveness and convey additional information. Examples include adding an order or arith-metic for capturing complexity classes, or the power of real-life declarative languages. A recent trend is to add a data-value com-parison relation to words, trees, and graphs, for capturing modern data models such as XML and graph databases. Such additions often result in the loss of good properties of the underlying logic. Our goal is to show that such a loss can be avoided if we use pattern-based logics, standard in XML and graph data querying. The essence of such logics is that auxiliary relations are tested locally with respect to other relations in the structure. These logics are shown to admit strong versions of Hanf and Gaif-man locality theorems, which are used to prove a homomorphism preservation theorem, and a decidability result for the satisfiability problem. We discuss applications of these results to pattern logics over data forests, and consequently to querying XML data

Preservation Theorems Through the Lens of Topology

In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures the essence of their proofs

Model Theory of XPath on Data Trees: Part I: Bisimulation and Characterization

We investigate model theoretic properties of XPath with data (in)equality tests over the class of data trees, i.e., the class of trees where each node contains a label from a finite alphabet and a data value from an infinite domain.We provide notions of (bi)simulations for XPath logics containing the child, descendant, parent and ancestor axes to navigate the tree. We show that these notions precisely characterize the equivalence relation associated with each logic. We study formula complexity measures consisting of the number of nested axes and nested subformulas in a formula; these notions are akin to the notion of quantifier rank in first-order logic. We show char- acterization results for fine grained notions of equivalence and (bi)simulation that take into account these complexity measures. We also prove that positive fragments of these logics correspond to the formulas preserved under (non-symmetric) simulations. We show that the logic including the child axis is equivalent to the fragment of first-order logic invariant under the corresponding notion of bisimulation. If upward navigation is allowed the characterization fails but a weaker result can still be established. These results hold both over the class of possibly infinite data trees and over the class of finite data trees.Besides their intrinsic theoretical value, we argue that bisimulations are useful tools to prove (non)expressivity results for the logics studied here, and we substantiate this claim with examples.Fil: Figueira, Diego. Centre National de la Recherche Scientifique; FranciaFil: Figueira, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Areces, Carlos Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin