Dependencies In Relational Databases Research Articles

The complexity of dependency detection and discovery in relational databases

Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs).We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs.We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.

Efficient bottom-up Mining of Attribute Based Access Control Policies.

Attribute Based Access Control (ABAC) is fast replacing traditional access control models due to its dynamic nature, flexibility and scalability. ABAC is often used in collaborative environments. However, a major hurdle to deploying ABAC is to precisely configure the ABAC policy. In this paper, we present an ABAC mining approach that can automatically discover the appropriate ABAC policy rules. We first show that the ABAC mining problem is equivalent to identifying a set of functional dependencies in relational databases that cover all of the records in a table. We also propose a more efficient algorithm, called ABAC-SRM which discovers the most general policy rules from a set of candidate rules. We experimentally show that ABAC-SRM is accurate and significantly more efficient than the existing state of the art.

EQUIVALENCE THEOREM FOR GENERALIZED DEPENDENCIES IN RELATIONAL DATABASES

In this paper, we present some new results related to the differential dependencies in relational databases. The main contributions are the following: (1) To introduce a new concept of generalized differential dependencies; (2) To claim and proof the equivalence theorem of consequences in the world of all relations, the word of two-tuple relations and propositional logic for the generalized differential dependencies; (3) To show an equivalence between generalized positive Boolean dependencies and generalized differential dependencies.

Appropriate inferences of data dependencies in relational databases

We study inference systems for the combined class of functional and full hierarchical dependencies in relational databases. Two notions of implication are considered: the original notion in which a dependency is implied by a given set of dependencies and the underlying set of attributes, and the alternative notion in which a dependency is implied by a given set of dependencies alone. The first main result establishes a finite axiomatisation for the original notion of implication which clarifies the role of the complementation rule in the combined setting. In fact, we identify inference systems that are appropriate in the following sense: full hierarchical dependencies can be inferred without use of the complementation rule at all or with a single application of the complementation rule at the final step of the inference only; and functional dependencies can be inferred without any application of the complementation rule. The second main result establishes a finite axiomatisation for the alternative notion of implication. We further show how inferences of full hierarchical dependencies can be simulated by inferences of multivalued dependencies, and vice versa. This enables us to apply both of our main results to the combined class of functional and multivalued dependencies. Furthermore, we establish a novel axiomatisation for the class of non-trivial functional dependencies.

Charting the completeness frontier of inference systems for multivalued dependencies

The implication of multivalued dependencies in relational databases has originally been defined in the context of some fixed finite universe. While axiomatisability and implication problems have been intensely studied with respect to this notion almost no research has been devoted towards the alternative notion of implication in which the underlying universe of attributes is left undetermined. Based on a set of common inference rules we establish all axiomatisations in undetermined universes, and all axiomatisations in fixed universes that indicate the role of the complementation rule as a means of database normalisation. This characterises the expressiveness of several incomplete sets of inference rules. We also establish relationships between axiomatisations in fixed and undetermined universes, and study the time complexity of the implication problem in undetermined universes. The results of this paper establish a foundation for reasoning about multivalued dependencies without the assumption of a fixed underlying universe.

Characterising nested database dependencies by fragments of propositional logic

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.

An Equivalence between Dependencies in Nested Databases and a Fragment of Propositional Logic

We extend the result on the equivalence between functional and multivalued dependencies in relational databases and a fragment of propositional logic. It is shown that this equivalence still holds for functional and multivalued dependencies in databases that support complex values via nesting of records and lists.The equivalence has several implications. Firstly, it extends a well-known result from relational databases to databases which are not in first normal form. Secondly, it characterises the implication of functional and multivalued dependencies in complex-value databases in purely logical terms. The database designer can take advantage of this equivalence to reduce database design problems to simpler problems in propositional logic. Furthermore, relational database design tools can be reused to solve problems for complex-value databases.

Generalized domination in closure systems

In the context of extracting maximal item sets and association rules from a binary data base, the graph-theoretic notion of domination was recently used to characterize the neighborhood of a concept in the corresponding lattice. In this paper, we show that the notion of domination can in fact be extended to any closure operator on a finite universe and be efficiently encoded into propositional Horn functions. This generalization enables us to endow notions and algorithms related to Formal Concept Analysis with Horn minimization and minimal covers of functional dependencies in Relational Databases.

On a problem of Fagin concerning multivalued dependencies in relational databases

Multivalued dependencies (MVDs) are an important class of relational constraints that is fundamental to relational database design. Reflexivity axiom, complementation rule, and pseudo-transitivity rule form a minimal set of inference rules for the implication of MVDs. The complementation rule plays a distinctive role as it takes into account the underlying relation schema R which the MVDs are defined on. The R-axiom ∅ ↠ R is much weaker than the complementation rule, but is sufficient to form a minimal set of inference rules together with augmentation and pseudo-difference rule. Fagin has asked whether it is possible to reduce the power of the complementation rule and drop the augmentation rule at the same time and still obtain a complete set. It was argued that there is a trade-off between complementation rule and augmentation rule, and one can only dispense with one of these rules at the same time. It is shown in this paper that an affirmative answer to Fagin's problem can nevertheless be achieved. In fact, it is proven that R-axiom together with a weaker form of the reflexivity axiom, pseudo-transitivity rule and exactly one of union, intersection or difference rule form such desirable minimal sets. The positive solution to this problem gives further insight into the difference between the notions of functional and multivalued dependencies.

Discovery of Functional and Approximate Functional Dependencies in Relational Databases

This study develops the foundation for a simple, yet efficient method for uncovering functional and approximate functional dependencies in relational databases. The technique is based upon the mathematical theory of partitions defined over a relation's row identifiers. Using a levelwise algorithm the minimal non-trivial functional dependencies can be found using computations conducted on integers. Therefore, the required operations on partitions are both simple and fast. Additionally, the row identifiers provide the added advantage of nominally identifying the exceptions to approximate functional dependencies, which can be used effectively in practical data mining applications.

Discovery of functional and approximate functional dependencies in relational databases

This study develops the foundation for a simple, yet efficient method for uncovering functional and approximate functional dependencies in relational databases. The technique is based upon the mathematical theory of partitions defined over a relation's row identifiers. Using a levelwise algorithm the minimal non-trivial functional dependencies can be found using computations conducted on integers. Therefore, the required operations on partitions are both simple and fast. Additionally, the row identifiers provide the added advantage of nominally identifying the exceptions to approximate functional dependencies, which can be used effectively in practical data mining applications.

Fuzzy functional dependencies and Bayesian networks

Bayesian networks have become a popular technique for representing and reasoning with probabilistic information. The fuzzy functional dependency is an important kind of data dependencies in relational databases with fuzzy values. The purpose of this paper is to set up a connection between these data dependencies and Bayesian networks. The connection is done through a set of methods that enable people to obtain the most information of independent conditions from fuzzy functional dependencies.

A note on approximation measures for multi-valued dependencies in relational databases

We consider the problem of defining a normalized approximation measure for multi-valued dependencies in relational database theory. An approximation measure is a function mapping relation instances to real numbers. The number to which an instance is mapped, intuitively, describes the strength of the dependency in that instance. A normalized approximation measure for functional dependencies has been proposed previously: the minimum number of tuples that need be removed for the functional dependency to hold divided by the total number of tuples. This leads naturally to a normalized measure for multi-valued dependencies: the minimum number of tuples that need be removed for the multi-valued dependency to hold divided by the total number of tuples. The measure for functional dependencies can be computed efficiently, O(| r|log(| r|)) where | r| is the relation instance. However, we show that an efficient algorithm for computing the analogous measure for multi-valued dependencies is not likely to exist. A polynomial time algorithm for computing the measure would lead to a polynomial time algorithm for an NP-complete problem (proven by a reduction from the maximum edge biclique problem in graph theory). Hence, we argue that it is not a good measure. We propose an alternate measure based on the lossless join characterization of multi-valued dependencies. This measure is efficiently computable, O(| r| 2).

Impurity measures in databases

We introduce purity dependencies as generalizations of functional dependencies in relational databases starting from the notion of impurity measure. The impurity measure of a subset of a set relative to a partition of that set and the relative impurity of two partitions allow us to define the relative impurity of two attribute sets of a table of a relational database and to introduce purity dependencies. We discuss properties of these dependencies that generalize similar properties of functional dependencies and we highlight their relevance for approximate classifications. Finally, an algorithm that mines datasets for these dependencies is presented.

Cylindric structures and dependencies in relational databases

In this paper, we explore the precise connection between dependencies in relational databases and variants of cylindric algebras, and apply recent algebraic results to problems of axiomatizing dependencies. First, we will consider project-join dependencies and the corresponding class of (representable) cylindric semilattices. Since representable cylindric semilattices have a non-finitely axiomatizable quasi-equational theory, there is no finite axiomatization for n-dimensional unrestricted project-join dependencies. Then we will look at Cosmadakis (in: A.K. Chandra (Ed.), Proc. 28th Annual Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, CA, 1987, pp. 411–420) who introduces cylindric dependencies, and makes several claims regarding the structural properties of these dependencies. Using the above-mentioned precise connection, we show that recent algebraic investigations of cylindric lattices provide counterexamples to Cosmadaki's claim that cylindric dependencies are finitely (schema) axiomatizable.

Data summarization in relational databases through fuzzy dependencies

In this paper we deal with the problem of treating with dependencies in relational databases which do not hold in an exact manner as classical functional dependencies but in a weaker sense, i.e., we face with relations which satisfy dependencies such that `people with similar age and height have similar weight'. We model this relationship through the concept of fuzzy dependency. We see that these dependencies imply some kind of fuzzy redundancy, and, in order to avoid it, we propose to use a projection operator which leads us to partition a relation r into two projections, say r 1 and r 2 with a less amount of information. Then, we proceed to replace the original relation by these projections. In this process we must guarantee that we can recover the data we had in the original relation. This will be possible by using a special join operator applied to r 1 and r 2. We must also guarantee that we can test the fuzzy dependency for new entries to the database in the same way either if we consider the original relation r or if we work with the projections r 1 and r 2. We also show that this definition of dependency maintains the good properties of completeness of the classical case.

An incremental concept formation approach for learning from databases

This paper describes a concept formation approach to the discovery of new concepts and implication rules from data. This machine learning approach is based on the Galois lattice theory, and starts from a binary relation between a set of objects and a set of properties (descriptors) to build a concept lattice and a set of rules. Each node (concept) of the lattice represents a subset of objects with their common properties. In this paper, some efficient algorithms for generating concepts and rules are presented. The rules are either in conjunctive or disjunctive form. To avoid the repetitive process of constructing the concept lattice and determining the set of implication rules from scratch each time a new object is introduced in the input relation, we propose an algorithm for incrementally updating both the lattice and the set of generated rules. The empirical behavior of the algorithms is also analysed. The implication problem for these rules can be handled based on the well-known theoretical results on functional dependencies in relational databases.

A note on cylindric lattices

0. Introduction. Besides being of intrinsic interest, cylindric (semi-) lattices arise naturally from the study of dependencies in relational databases; the polynomials on a cylindric semilattice are closely related to the queries obtainable from project-join mappings on a relational database (cf. [D] for references). This note is intended to initiate the study of these structures, and only a few, rather basic results will be given. Some problems at the end will hopefully stimulate further research. Related issues are discussed in [H], and for further background material the reader is invited to consult [N]. I should like to thank H. Andreka and I. Nemeti for stimulating discussions on the subject.

Functional dependencies in relational databases: A lattice point of view

A lattice theoretic approach is developed to study the properties of functional dependencies in relational databases. Particular attention is paid to the analysis of the semilattice of closed sets, the lattice of all closure operations on a given set and to a new characterization of normal form relation schemes. Relation schemes with restrictions on functional dependencies are also studied.

Partial dependencies in relational databases and their realization

Weakening the functional dependencies introduced by Amstrong we get the notion of the partial dependencies defined on the relational databases. We show that the partial dependencies can be characterized by the closure operations of the poset formed by the partial functions on the attributes of the databases. On the other hand, we give necessary and sufficient conditions so that for such a closure operation one can find on the given set of attributes a database whose partial dependencies generate the given closure operation. We also investigate some questions about how to realize certain structures related to databases by a database of minimal number of rows, columns or elements.