János Demetrovics

Functional dependencies in relational databases: a lattice point of view

Abstract 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.

Design type problems motivated by database theory

Abstract Let k⩽n, p⩽q • for any choice of k distinct columns c1,c2,…,ck, there are q+1 rows such that the number of different entries in c i (1⩽i⩽k−1) in these rows is at most p, while all q+1 entries of ck in these rows are different; • this is true for no choice of k+1 distinct columns. We review results minimizing m, given n,p,q,k. Two of the results are new. The optimal or nearly optimal constructions can be considered as n partitions of the m-element set satisfying certain conditions. This version leads to the orthogonal double covers, also surveyed here.

The poset of closures as a model of changing databases

The combinatorial properties of the poset of closures are studied, especially the degrees in the Hasse diagram.

Contribution to the theory of data base relations

After a summary of basic definitions and results concerning data base relations, an algebraic method is worked out which enables us to prove some new results of combinatorial type in this subject.

The characterization of branching dependencies

Abstract A new type of dependencies in a relational database model is introduced. If b is an attribute, A is a set of attributes then it is said that b (p,q)-depends on A, in notation A (p,q)→ b, in a database r if there are no q + 1 rows in r such that they have at most p different values in A, but q + 1 different values in b. (1,1)-dependency is the classical functional dependency. Let I (A) denote the set{b: A(p,q)→ b. The set function I (A) is characterized if p=1, 1

Asymptotic properties of keys and functional dependencies in random databases

Abstract Practical database applications give the impression that sets of constraints are rather small and that large sets are unusual and are caused by bad design decisions. Theoretical investigations, however, show that minimal constraint sets are potentially very large. Their size can be estimated to be exponential in terms of the number of attributes. The gap between observation in practice and theory results in the rejection of theoretical results. However, practice is related to average cases and is not related to worst cases. The theory used until now considered the worst-case complexity. This paper aims to develop a theory for the average-case complexity . Several probabilistic models and asymptotics of corresponding probabilities are investigated for random databases formed by independent random tuples with a common discrete distribution. Poisson approximations are studied for the distributions of some characteristics for such databases where the number of tuples is sufficiently large. We intend to prove that the exponential complexity of key sets and sets of functional dependencies is rather unusual and almost all minimal keys in a relation have a length which depends mainly on the size of the relation.

A note on minimal matrix representation of closure operations

A matrixM withn columns represents a closure operationF(A), (A⊂X, |X|=n) if for anyA, any two rows equal in the columns corresponding toA are also equal inF(A). Letm(F) be the minimum number of rows of the matrices representingF. Lower and upper estimates on maxm(F) are given where max runs over the set of all closure operations onn elements.

On the number of databases and closure operations

Closure operations are considered as models of databases. Estimates on the number of closure operations on n elements (or equivalently, on the number of databases with n attributes) are given.

Functional dependencies and the semilattice of closed classes

In this paper we present the semilattice-theoretical approach to the investigation of the relational datamodels. It is known that the poset of closures is a model of changing databases. We show that it is in fact a lattice isomorphic to the subsemilattice-lattice of free semilattice, and the arbitrary subsemilattice-lattices can be considered as a model of restrictedly changing databases. The properties of these lattices in the context of database problems are studied. We also discuss the separation and dimension problems and establish some estimates.

Algebraic Properties of Crowns and Fences

In this paper we study the clones of all order preserving operations for crowns and fences. By presenting explicit generating sets, we show that these clones are finitely generated. The case of crowns is particularly interesting because they admit no order preserving near unanimity operations. Various related questions are also discussed. For example, we give a new proof of a theorem of Duffus and Rival which states that crowns are irreducible.