Free Kleene algebras with domain

First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are 'regular' sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.Comment: 22 pages. Some proofs expande

An introduction to Graph Data Management

A graph database is a database where the data structures for the schema and/or instances are modeled as a (labeled)(directed) graph or generalizations of it, and where querying is expressed by graph-oriented operations and type constructors. In this article we present the basic notions of graph databases, give an historical overview of its main development, and study the main current systems that implement them

Nesting Depth of Operators in Graph Database Queries: Expressiveness Vs. Evaluation Complexity

Designing query languages for graph structured data is an active field of research, where expressiveness and efficient algorithms for query evaluation are conflicting goals. To better handle dynamically changing data, recent work has been done on designing query languages that can compare values stored in the graph database, without hard coding the values in the query. The main idea is to allow variables in the query and bind the variables to values when evaluating the query. For query languages that bind variables only once, query evaluation is usually NP-complete. There are query languages that allow binding inside the scope of Kleene star operators, which can themselves be in the scope of bindings and so on. Uncontrolled nesting of binding and iteration within one another results in query evaluation being PSPACE-complete. We define a way to syntactically control the nesting depth of iterated bindings, and study how this affects expressiveness and efficiency of query evaluation. The result is an infinite, syntactically defined hierarchy of expressions. We prove that the corresponding language hierarchy is strict. Given an expression in the hierarchy, we prove that it is undecidable to check if there is a language equivalent expression at lower levels. We prove that evaluating a query based on an expression at level i can be done in $\Sigma_i$ in the polynomial time hierarchy. Satisfiability of quantified Boolean formulas can be reduced to query evaluation; we study the relationship between alternations in Boolean quantifiers and the depth of nesting of iterated bindings.Comment: Improvements from ICALP 2016 review comment

Regular Queries on Graph Databases

Graph databases are currently one of the most popular paradigms for storing data. One of the key conceptual differences between graph and relational databases is the focus on navigational queries that ask whether some nodes are connected by paths satisfying certain restrictions. This focus has driven the definition of several different query languages and the subsequent study of their fundamental properties. We define the graph query language of Regular Queries, which is a natural extension of unions of conjunctive 2-way regular path queries (UC2RPQs) and unions of conjunctive nested 2-way regular path queries (UCN2RPQs). Regular queries allow expressing complex regular patterns between nodes. We formalize regular queries as nonrecursive Datalog programs with transitive closure rules. This language has been previously considered, but its algorithmic properties are not well understood. Our main contribution is to show elementary tight bounds for the containment problem for regular queries. Specifically, we show that this problem is 2EXPSPACE-complete. For all extensions of regular queries known to date, the containment problem turns out to be non-elementary. Together with the fact that evaluating regular queries is not harder than evaluating UCN2RPQs, our results show that regular queries achieve a good balance between expressiveness and complexity, and constitute a well-behaved class that deserves further investigation

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

Expressive Path Queries on Graph with Data

Graph data models have recently become popular owing to their applications, e.g., in social networks and the semantic web. Typical navigational query languages over graph databases - such as Conjunctive Regular Path Queries (CRPQs) - cannot express relevant properties of the interaction between the underlying data and the topology. Two languages have been recently proposed to overcome this problem: walk logic (WL) and regular expressions with memory (REM). In this paper, we begin by investigating fundamental properties of WL and REM, i.e., complexity of evaluation problems and expressive power. We first show that the data complexity of WL is nonelementary, which rules out its practicality. On the other hand, while REM has low data complexity, we point out that many natural data/topology properties of graphs expressible in WL cannot be expressed in REM. To this end, we propose register logic, an extension of REM, which we show to be able to express many natural graph properties expressible in WL, while at the same time preserving the elementariness of data complexity of REMs. It is also incomparable to WL in terms of expressive power.Comment: 39 page

Jumping Evaluation of Nested Regular Path Queries

International audienceNested regular path queries are used for querying graph databases and RDF triple stores. We pro- pose a new algorithm for evaluating nested regular path queries on a graph from a set of start nodes in combined linear time. We show that this complexity upper bound can be reduced by making it de- pendent on the size of the query’s top-down needed subgraph, a notion that we introduce. For many queries in practice, the top-down needed subgraph is way smaller than the whole graph. Our algo- rithm is based on a novel compilation schema from nested regular path queries to monadic datalog queries. Its complexity upper bound follows from known properties of top-down datalog evaluation. As an application, we show that our algorithm permits to reformulate in simple terms a variant of a very efficient automata-based algorithm proposed by Maneth and Nguyen that evaluates navigational path queries in datatrees based on indexes and jumping. Moreover, it overcomes some limitations of Maneth and Nguyen’s: it is not bound to trees and applies to graphs; it is not limited to forward navigational XPath but can treat any nested regular path query and it can be implemented efficiently without any dedicated techniques, by using any efficient top-down datalog evaluator. We confirm the efficiency of our algorithm experimentally based on an implementation with LogicBlox

Logics with rigidly guarded data tests

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi's approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata

Schema-Based Automata Determinization

International audienceWe propose an algorithm for schema-based determinization of finite automata on words and of stepwise hedge automata on nested words. The idea is to integrate schema-based cleaning directly into automata determinization. We prove the correctness of our new algorithm and show that it is always more efficient than standard determinization followed by schema-based cleaning. Our implementation permits to obtain a small deterministic automaton for an example of an XPath query, where standard determinization yields a huge stepwise hedge automaton for which schema-based cleaning runs out of memory