Sven Hartmann

Efficient reasoning about a robust XML key fragment

We review key constraints in the context of XML as introduced by Buneman et al. We demonstrate that: (1) one of the proposed inference rules is not sound in general, and (2) the inference rules are incomplete for XML key implication, even for nonempty sets of simple key paths. This shows, in contrast to earlier statements, that the axiomatizability of XML keys is still open, and efficient algorithms for deciding their implication still need to be developed. Solutions to these problems have a wide range of applications including consistency validation, XML schema design, data exchange and integration, consistent query answering, XML query optimization and rewriting, and indexing. In this article, we investigate the axiomatizability and implication problem for XML keys with nonempty sets of simple key paths. In particular, we propose a set of inference rules that is indeed sound and complete for the implication of such XML keys. We demonstrate that this fragment is robust by showing the duality of XML key implication to the reachability problem of fixed nodes in a suitable digraph. This enables us to develop a quadratic-time algorithm for deciding implication, and shows that reasoning about this XML key fragment is practically efficient. Therefore, XML applications can be unlocked effectively since they benefit not only from those XML keys specified explicitly by the data designer but also from those that are specified implicitly.

More Functional Dependencies for XML

In this paper, we present a new approach towards functional dependencies in XML documents based on homomorphisms between XML data trees and XML schema graphs. While this approach allows us to capture functional dependencies similar to those recently studied by Arenas/Libkin and by Lee/Ling/Low, it also gives rise to a further class of functional dependencies in XML documents. We address some essential differences between the two classes of functional dependencies under discussion resulting in different expressiveness and different inference rules. Examples demonstrate that both classes of functional dependencies appear quite naturally in practice and, thus, should be taken into consideration when designing XML documents.

On the implication problem for cardinality constraints and functional dependencies

In database design, integrity constraints are used to express database semantics. They specify the way by that the elements of a database are associated to each other. The implication problem asks whether a given set of constraints entails further constraints. In this paper, we study the finite implication problem for cardinality constraints. Our main result is a complete characterization of closed sets of cardinality constraints. Similar results are obtained for constraint sets containing cardinality constraints, but also key and functional dependencies. Moreover, we construct Armstrong databases for these constraint sets, which are of special interest for example-based deduction in database design.

Design by example for SQL table definitions with functional dependencies

A database is C-Armstrong for a given set of constraints in a class C if it satisfies every constraint of the set and violates every constraint in C not implied by the set. Therefore, Armstrong databases are test data that perfectly illustrate the current perceptions about the semantics of a schema. We extend the existing theory of Armstrong relations to a toolbox of Armstrong tables. That is, we investigate structural and computational properties of Armstrong tables for the class of functional dependencies (FDs) over SQL tables. Relations are special instances of SQL tables with no duplicate rows and no null value occurrences. While FDs do not enjoy Armstrong tables, the combined class of standard FDs and NOT NULL constraints does enjoy Armstrong tables. The problem of finding an Armstrong table is shown to be precisely exponential for this combined class. However, we establish an algorithm that computes Armstrong tables with a size at most quadratic in that of a minimum-sized Armstrong table. Our resulting toolbox of Armstrong tables can be applied by data engineers to concisely visualize constraints on SQL data. Such support can lead to designs that guarantee efficient data management in practice.

On Orthogonal Double Covers of Graphs

An orthogonal double cover (ODC) is a collection of n spanning subgraphs(pages) of the complete graph Kn such that they cover every edge of the completegraph twice and the intersection of any two of them contains exactly one edge. If all the pages are isomorphic tosome graph G, we speak of an ODC by G. ODCs have been studied for almost 25 years, and existenceresults have been derived for many graph classes. We present an overview of the current state of research alongwith some new results and generalizations. As will be obvious, progress made in the last 10 years is in many waysrelated to the work of Ron Mullin. So it is natural and with pleasure that we dedicate this article to Ron, on theoccasion of his 65th birthday.

The implication problem of data dependencies over SQL table definitions: Axiomatic, algorithmic and logical characterizations

We investigate the implication problem for classes of data dependencies over SQL table definitions. Under Zaniolos “no information” interpretation of null markers we establish an axiomatization and algorithms to decide the implication problem for the combined class of functional and multivalued dependencies in the presence of NOT NULL constraints. The resulting theory subsumes three previously orthogonal frameworks. We further show that the implication problem of this class is equivalent to that in a propositional fragment of Schaerf and Cadolis [1995] family of para-consistent S-3 logics. In particular, S is the set of variables that correspond to attributes declared NOT NULL. We also show how our equivalences for multivalued dependencies can be extended to Delobels class of full first-order hierarchical decompositions, and the equivalences for functional dependencies can be extended to arbitrary Boolean dependencies. These dualities allow us to transfer several findings from the propositional fragments to the corresponding classes of data dependencies, and vice versa. We show that our results also apply to Codds null interpretation “value unknown at present”, but not to Imielinskis [1989] or-relations utilizing Levene and Loizous weak possible world semantics [Levene and Loizou 1998]. Our findings establish NOT NULL constraints as an effective mechanism to balance not only the certainty in database relations but also the expressiveness with the efficiency of entailment relations. They also control the degree by which the implication of data dependencies over total relations is soundly approximated in SQL table definitions.

Numerical constraints on XML data

Boundaries occur naturally in everyday life. This paper introduces numerical constraints into the framework of XML to take advantage of the benefits that result from the explicit specification of such boundaries. Roughly speaking, numerical constraints restrict the number of elements in an XML data fragment based on the data values of selected subelements. Efficient reasoning about numerical constraints provides effective means for predicting the number of answers to XQuery and XPath queries, the number of updates when using the XQuery update facility, and the number of encryptions or decryptions when using XML encryption. Moreover, numerical constraints can help to optimise XQuery and XPath queries, to exclude certain choices of indices from the index selection problem, and to generate views for efficient processing of common queries and updates. We investigate decision problems associated with numerical constraints in order to capitalise on the range of applications in XML data processing. To begin with we demonstrate that the implication problem is strongly coNP-hard for several classes of numerical constraints. These sources of potential intractability direct our attention towards the class of numerical keys that permit the specification of positive upper bounds. Numerical keys are of interest as they are reminiscent of cardinality constraints that are widely used in conceptual data modelling. At the same time, they form a natural generalisation of XML keys that are popular in XML theory and practice. We show that numerical keys are finitely satisfiable and establish a finite axiomatisation for their implication problem. Finally, we propose an algorithm that decides numerical key implication in quadratic time using shortest path methods.

Characterising nested database dependencies by fragments of propositional logic

Abstract We extend the earlier results on the equivalence between the Boolean and the multivalued dependencies in relational databases and fragments of the Boolean propositional logic. It is shown that these equivalences are still valid for the databases that store complex data elements obtained from the recursive nesting of record, list, set and multiset constructors. The major proof argument utilises properties of Brouwerian algebras. The equivalences have several consequences. Firstly, they provide new insights into databases that are not in first normal form. Secondly, they characterise the implication of data dependencies in nested databases in purely logical terms. The database designer can take advantage of these equivalences to reduce database design problems to well-studied problems in Boolean propositional logic. Furthermore, relational database design solutions can be reused to solve problems for nested databases.

Axiomatisations of functional dependencies in the presence of records, lists, sets and multisets

We investigate functional dependencies in databases that support complex values such as records, lists, sets anu multisets. Therefore, an abstract algebraic framework is proposed that classifies data models according to the underlying types they support. This allows to emphasise the impact of the data types rather than the specifics of a particular data model.The main results are finite, minimal, sound and complete sets of inference rules for the implication of functional dependencies in the presence of records and all combinations of lists, sets and multisets. The inference rules are similar to Armstrongs original axioms for the relational data model, thanks to the algebraic framework. The completeness result, however, requires a deep analysis in the case of sets and, in particular, multisets.

On Codd Families of Keys over Incomplete Relations

Keys allow a database management system to uniquely identify tuples in a database. Consequently, the class of keys is of great significance for almost all data processing tasks. In the relational model of data, keys have received considerable interest and are well understood. However, for efficient means of data processing most commercial relational database systems deviate from the relational model. For example, tuples may contain only partial information in the sense that they contain so-called null values to represent incomplete information. Codds principle of entity integrity says that every tuple of every relation must not contain a null value on any attribute of the primary key. Therefore, a key over partial relations enforces both uniqueness and totality of tuples on the attributes of the key. On the basis of these two requirements, we study the resulting class of keys over relations that permit occurrences of Zaniolos null value ‘no-information’. We show that the interaction of this class of keys is different from the interaction of the class of keys over total relations. We establish a finite ground axiomatization, and an algorithm for deciding the associated implication problem in linear time. Further, we characterize Armstrong relations for an arbitrarily given sets of keys; that is, we give a sufficient and necessary condition for a partial relation to satisfy a key precisely when it is implied by a given set of keys. We also establish an algorithm that computes an Armstrong relation for an arbitrarily given set of keys. While the problem of finding an Armstrong relation for a given key set is precisely exponential in general, our algorithm returns an Armstrong relation whose size is at most quadratic in the size of a minimal Armstrong relation. Finally, we settle various questions related to the maximal size of a family of non-redundant key sets. Our results help to bridge the gap between the existing theory of database constraints and database practice.