Recursive queries and context-free graph grammars

Graph Operations, Graph Transformations and Monadic Second-Order Logic A survey

Graph Operations, Graph Transformations and Monadic Second-Order Logic:: A survey

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

Adaptive star grammars and their languages

Vigna and Ghezzi showed that the language of grid graphs could not be constructed by their context-free graph grammars [1]. In this paper, we construct a context-sensitive graph grammar for the two-dimensional grid graphs.

In this paper we discuss the use of views to address the problem of providing useful answers to provenance queries while ensuring that privacy concerns are met. In particular, we propose a hierarchical workflow model, based on context-free graph grammars, in which fine-grained dependencies between the inputs and outputs of a module are explicitly specified. Using this model, we examine how privacy concerns surrounding data, module function, and workflow structure can be addressed.

On the Complexity of Pattern Matching for Highly Compressed Two-Dimensional Texts

A complex network models relationships among data elements in several applications, such as social networks, urban street organization, disease monitoring and treatment co-occurrences, and communication and transport infrastructures. To store and manage large volumes of data, the Relational Database Management Systems (RDBMSs) provide efficient disk-based data management, being present in the most diverse application domains, including enterprise information, finance, medical or clinical support, and in airline reservations and scheduling. However, RDBMSs still face challenges to efficiently process queries that "navigate" over complex networks represented as graphs. Those queries, named in this work as graph queries, are often expressed by recursive queries, which execute multiple join operations between an internal, temporary table that stores intermediary results and the table that stores the graph's edges. This Ph.D. dissertation focuses on developing methods and specialized data structures based on adjacency-lists aiming at improving graph query performance in an RDBMS through recursive queries. This project's main goal targets the question: "How can we improve the performance of queries over graph data executed in a RDBMS?". To answer it, we proposed specializations over each table involved in a recursive query: the table of edges E, and the temporary result table T. The modifications in table T consists of adding extra measurement columns to enable employing it in multiple graph queries executed in a single database connection. In its turn, table E is specialized to allow storing multiple edges into a single row optimized for the execution of each join operation of the recursive query. Experiments using data from RDBMSs modeled as a graph allow concluding that: (i) enriching table T provides increased efficiency to execute multiple graph queries, such as the query types Single-Source Shortest Path (SSSP), Connected Components (CC), and PageRank (PR); (ii) grouping multiple edges from table E into a single row significantly improves the performance of individual recursive queries, reducing not only their elapsed time but also the required size to store the graph data; and (iii) employing Clustered Tables to group the edges into a single page also enables improved performance for recursive queries, with the advantage of requiring just a few changes in the SQL statements. Overall, specializing both tables T and E helps diminishing the gap of efficiently executing graph queries in a RDBMS.

The relational model is the basis for most modern databases, while SQL is the most commonly used query language. However, there are data structures and computational problems that cannot be expressed using SQL-92 queries. Among them are those concerned with the bill-of-material and corporate hierarchies. A newer standard, called the SQL-99, introduced recursive queries which can be used to solve such tasks. Yet, only recently recursive queries have been implemented in most of the leading relational databases. In this paper we have reviewed and compared implementations of the recursive queries defined by SQL:1999 through SQL:2008 and offered by leading vendors of DBMSs. Our comparison concerns features, syntax and performance.

Recently, several researchers realized that the modelling capabilities of traditional relational data base systems are rather limited for advanced applications. Among others, two main research areas can be identified: One area of research is concerned with removing the limitation of flat tuples by supporting structured objects instead. Another area of research deals with the limitation of traditional query languages. Already in the late seventies it was proven that so called relationally complete query languages do not allow to express the computation of the transitive closure of a relation. Most researchers solve the problem by introducing some kind of recursion. The work presented in this paper is an attempt to combine both areas of research. It proposes the integration of recursive queries and nested relations by providing a recursion mechanism which can be nested according to the structure of a non first normal form relation. It is shown by examples that this method can be used to formulate queries which are rather cumbersome if only flat relations are used. Especially, in some cases the problem of performing a selection on a recursively defined relation disappears. This question has been the topic of several research papers recently. Moreover, many queries can be formulated in a more natural way than in the world of flat tuples. Therefore, nested recursive queries in combination with non first normal form relations provide a contribution to the problem of modelling complex data.

In each graph-grammar approach it is defined how and under which conditions graph productions can be applied to a given graph in order to obtain a derived graph. The conditions under which productions can be applied are called application conditions. Although the generative power of most of the known general graph-grammar approaches is sufficient to generate any recursively enumerable set of graphs, it is often convenient to have specific application conditions for each production. Such application conditions, on the one hand, include context conditions like the existence or non-existence of nodes, edges, or certain subgraphs in the given graph as well as embedding restrictions concerning the morphisms from the left-hand side of the production to the given graph. In this paper, the concept of application conditions introduced by Ehrig and Habel is restricted to contextual conditions, especially negative ones. In addition to the general concept, we state local confluence and the Parallelism Theorem for derivations with application conditions. Finally we study context-free graph grammars with application conditions with respect to their generative power.

Given a sample from an unknown probability distribution over strings, there exist algorithms for inferring the structure and parameters of stochastic grammatical representations of the unknown distribution, i.e. string grammars. Despite the fact that research on grammatical representations of sets of graphs has been conducted since the late 1960’s, almost no work has considered the possibility of stochastic graph grammars and no algorithms exist for inferring stochastic graph grammars from data. This paper presents PEGG, an algorithm for estimating the parameters of stochastic context-free graph grammars given a sample from an unknown probability distribution over graphs. It is established formally that for a certain class of graph grammars PEGG finds parameter estimates in polynomial time that maximize the likelihood of the data, and preliminary empirical results demonstrate that the algorithm performs well in practice.KeywordsGraph TransformationGraph GrammarMaximum Likelihood ParameterStart SymbolEstimate Maximum Likelihood ParameterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Several query mapping algorithms have been proposed to translate XML queries into SQL queries for a schema-based relational XML storage. However, existing query mapping algorithms only support single-valued mapping schemes, in which each XML element type is mapped to exactly one relation, and do not support multi-valued mapping schemes, in which each XML element type can be mapped to multiple relations. In this paper, we propose a generic query mapping algorithm, ID-XMLToSQL, for a schema-based relational XML storage. To the best of our knowledge, our algorithm provides the first generic solution to the XML-to-Relational query mapping problem that is applicable to both single-valued and multi-valued mapping schemes. Moreover, our algorithm also provides an elegant solution to the query mapping problem in the presence of recursive XML schemas and recursive queries. While existing algorithms need special recursion operators, our algorithm only requires the traditional relational operators and thus, can work with all relational databases.

Semantic Querying (SQ) is emerging as an attractive approach for retrieval of data from relational and other conceptually similar databases, targeting users with limited or no technical expertise. Using SQ queries can be formulated using terminologies from a specific domain, which are then either translated in real time into the equivalent SQL queries, or executed against a materialised semantic database obtained by transforming the source relational data. This approach is suitable for non-technical users who are familiar with describing a domain using the terminologies in an ontology but lack expertise in writing SQL queries over complex relational schemas. As an alternative to the existing methods, we implement the vision of SQ on relational databases by deploying Semantic Web services over one or more databases and querying them with SPARQL queries. The approach based on manual Semantic Web service writing under-utilises easily manageable declarative mappings between source data schemas and domain ontologies. In this paper we introduce the Valet SADI framework to automate the creation of SADI Semantic Web services from declarative mappings, and demonstrate that Valet SADI, together with SADI query engines, establishes semantic querying as a viable, economical and user friendly way of querying relational databases.

Layout graph grammars are extensions of context-free graph grammars and are introduced as a tool for syntax directed constructions of graph layouts. The constructions are based on a layout specification of the productions, which are consistently transferred to the derivations. The layout specification consists of rules for a placement of the vertices and a partial routing of the edges. It specifies minimal distances between the vertices in X- or Y-dimension. These distances can be optimized according to some formal cost measures.