Functional dependencies over XML documents with DTDs

In this article an axiomatisation for functional dependencies over XML documents is presented. The approach is based on a representation of XML document type definitions (or XML schemata) by nested attributes using constructors for records, disjoint unions and lists, and a particular null value, which covers optionality. Infinite structures that may result from referencing attributes in XML are captured by rational trees. Using a partial order on nested attributes we obtain non-distributive Brouwer algebras. The operations of the Brouwer algebra are exploited in the soundness and completeness proofs for derivation rules for functional dependencies

Keys and Armstrong databases in trees with restructuring

The definition of keys, antikeys, Armstrong-instances are extended to complex values in the presence of several constructors. These include tuple, list, set and a union constructor. Nested data structures are built using the various constructors in a tree-like fashion. The union constructor complicates all results and proofs significantly. The reason for this is that it comes along with non-trivial restructuring rules. Also, so-called counter attributes need to be introduced. It is shown that keys can be identified with closed sets of subattributes under a certain closure operator. Minimal keys correspond to closed sets minimal under set-wise containment. The existence of Armstrong databases for given minimal key systems is investigated. A sufficient condition is given and some necessary conditions are also exhibited. Weak keys can be obtained if functional dependency is replaced by weak functional dependency in the definition. It is shown, that this leads to the same concept. Strong keys are defined as principal ideals in the subattribute lattice. Characterization of antikeys for strong keys is given. Some numerical necessary conditions for the existence of Armstrong databases in case of degenerate keys are shown. This leads to the theory of bounded domain attributes. The complexity of the problem is shown through several examples

Weak functional dependencies on trees with restructuring

We present an axiomatisation for weak functional dependencies, i.e. disjunctions of functional dependencies, in the presence of several constructors for complex values. The investigated constructors capture records, sets, multisets, lists, disjoint union and optionality, i.e. the complex values are indeed trees. The constructors cover the gist of all complex value data models including object oriented databases and XML. Functional and weak functional dependencies are expressed on a lattice of subattributes, which even carries the structure of a Brouwer algebra as long as the union-constructor is absent. Its presence, however, complicates all results and proofs significantly. The reason for this is that the union-constructor causes non-trivial restructuring rules to hold. In particular, if either the set- or the the union-constructor is absent, a subset of the rules is complete for the implication of ordinary functional dependencies, while in the general case no finite axiomatisation for functional dependencies exists

Multi-valued Dependencies in the Presence of Lists

Multi-valued depdendencies (MVDs) are an important class of relational constraints. We axiomatise MVDs in data models that support nested list types. In order to capture different data models at a time, an abstract approach based on nested attributes is taken. The set of subattributes of some fixed nested attribute carries the structure of a co-Heyting algebra. This enables us to generalise significant features of MVDs from the relational data model to the presence of lists. It is shown that an MVD is satisfied by some instance exactly when this instance can be decomposed without loss of information. The full power of the algebraic framework allows to provide a sound and complete set of inference rules for the implication of MVDs in the context of lists. The presence of the list operator calls for a new inference rule which is not required in the relational data model. Further differences become apparant when the minimality of the inference rules is investigated. The extension of the relational theory of MVDs to the presence of lists allows to specify more real-world constraints and increases therefore the number of application domains