Satisfiability for two-variable logic with two successor relations on finite linear orders

We study the finitary satisfiability problem for first order logic with two variables and two binary relations, corresponding to the induced successor relations of two finite linear orders. We show that the problem is decidable in NEXPTIME

Alternating register automata on finite words and trees

We study alternating register automata on data words and data trees in relation to logics. A data word (resp. data tree) is a word (resp. tree) whose every position carries a label from a finite alphabet and a data value from an infinite domain. We investigate one-way automata with alternating control over data words or trees, with one register for storing data and comparing them for equality. This is a continuation of the study started by Demri, Lazic and Jurdzinski. From the standpoint of register automata models, this work aims at two objectives: (1) simplifying the existent decidability proofs for the emptiness problem for alternating register automata; and (2) exhibiting decidable extensions for these models. From the logical perspective, we show that (a) in the case of data words, satisfiability of LTL with one register and quantification over data values is decidable; and (b) the satisfiability problem for the so-called forward fragment of XPath on XML documents is decidable, even in the presence of DTDs and even of key constraints. The decidability is obtained through a reduction to the automata model introduced. This fragment contains the child, descendant, next-sibling and following-sibling axes, as well as data equality and inequality tests

Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

Dickson's Lemma is a simple yet powerful tool widely used in termination proofs, especially when dealing with counters or related data structures. However, most computer scientists do not know how to derive complexity upper bounds from such termination proofs, and the existing literature is not very helpful in these matters. We propose a new analysis of the length of bad sequences over (N^k,\leq) and explain how one may derive complexity upper bounds from termination proofs. Our upper bounds improve earlier results and are essentially tight

Bottom-up automata on data trees and vertical XPath

A data tree is a finite tree whose every node carries a label from a finite alphabet and a datum from some infinite domain. We introduce a new model of automata over unranked data trees with a decidable emptiness problem. It is essentially a bottom-up alternating automaton with one register that can store one data value and can be used to perform equality tests with the data values occurring within the subtree of the current node. We show that it captures the expressive power of the vertical fragment of XPath - containing the child, descendant, parent and ancestor axes - obtaining thus a decision procedure for its satisfiability problem

Bisimulations on data graphs

Bisimulation provides structural conditions to characterize indistinguishability from an external observer between nodes on labeled graphs. It is a fundamental notion used in many areas, such as verification, graph-structured databases, and constraint satisfaction. However, several current applications use graphs where nodes also contain data (the so called “data graphs”), and where observers can test for equality or inequality of data values (e.g., asking the attribute ‘name’ of a node to be different from that of all its neighbors). The present work constitutes a first investigation of “data aware” bisimulations on data graphs. We study the problem of computing such bisimulations, based on the observational indistinguishability for XPath —a language that extends modal logics like PDL with tests for data equality— with and without transitive closure operators. We show that in general the problem is PSPACE-complete, but identify several restrictions that yield better complexity bounds (CO- NP, PTIME) by controlling suitable parameters of the problem, namely the amount of non-locality allowed, and the class of models considered (graphs, DAGs, trees). In particular, this analysis yields a hierarchy of tractable fragments.Fil: Abriola, Sergio Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación En Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación En Ciencias de la Computacion; ArgentinaFil: Barceló, Pablo. Universidad de Chile; ChileFil: Figueira, Diego. Centre National de la Recherche Scientifique; FranciaFil: Figueira, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación En Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación En Ciencias de la Computacion; Argentin

Closure Properties of Synchronized Relations

A standard approach to define k-ary word relations over a finite alphabet A is through k-tape finite state automata that recognize regular languages L over {1, ..., k} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the tuple (u_1, ..., u_k) in (A^*)^k in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of rational relations, enforcing restrictions on the reading regime from the tapes, which we call synchronization, yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language L onto {1, ..., k}. In this way, for each regular language C subseteq {1, ..., k}^*, one obtains a class Rel({C}) of relations. Synchronous, Recognizable, and Length-preserving rational relations are all examples of classes that can be defined in this way. We study basic properties of these classes of relations, in terms of closure under intersection, complement, concatenation, Kleene star and projection. We characterize the classes with each closure property. For the binary case (k=2) this yields effective procedures

Satisfiability of Downward XPath with Data Equality Tests

International audienceIn this work we investigate the satisfiability problem for the logic XPath(↓*, ↓, =), that includes all downward axes as well as equality and inequality tests. We address this problem in the absence of DTDs and the sibling axis. We prove that this fragment is decidable, and we nail down its complexity, showing the problem to be ExpTime-complete. The result also holds when path expressions allow closure under the Kleene star operator. To obtain these results, we introduce a new automaton model over data trees that captures XPath(↓*, ↓, =) and has an ExpTime emptiness problem. Furthermore, we give the exact complexity of several downward-looking fragments

Separating Automatic Relations

We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations R and R\u27, and asks if there exists a recognizable relation S that contains R and does not intersect R\u27. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation S with at most k products of regular languages that separates R from R\u27 is undecidable, for each fixed k ? 2. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages