New Type of Coding Problem Motivated by Database Theory

The present paper is intended to survey the interaction between relational database theory and coding theory. In particular it is shown how an extremal problem for relational databases gives rise to a new type of coding problem. The former concerns minimal representation of branching dependencies that can be considered as a data mining type question. The extremal configurations involve d-distance sets in the space of disjoint pairs of k-element subsets of an n-element set X. Let X be an n-element finite set, 0 < k < n/2 an integer. Suppose that {A(1), B-1} and {A(2), B-2} are pairs of disjoint k-element subsets of X (that is, \A(1)\ = \B-1\ = \A(2)\ = \B-2\ = k, A(1) boolean AND B-1 = 0, A(2) boolean AND B-2 = 0). Define the distance of these pairs by d({A(1), B-1}, {A(2), B-2}) = min{\A(1) - A(2)\ + \B-1 - B-2\, \A(1) - B-2\ + \B-1 - A(2)\). (C) 2004 Elsevier B.V. All rights reserved

Forbidden Families of Minimal Quadratic and Cubic Configurations

A matrix is \emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a \emph{configuration}, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch

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

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

Strongly possible functional dependencies for SQL

Missing data is a large-scale challenge to research and investigate. It reduces the statistical power and produces negative consequences that may introduce selection bias on the data. Many approaches to handle this problem have been introduced. The main approaches suggested are either missing values to be ignored (removed) or imputed (filled in) with new values. This paper uses the second method. Possible worlds and possible and certain keys were introduced in Köhler et.al., and by Levene et.al. Köhler and Link introduced certain functional dependencies (c-FD) as a natural complement to Lien's class of possible functional dependencies (p-FD). Weak and strong functional dependencies were studied by Levene and Loizou. We introduced the intermediate concept of strongly possible worlds that are obtained by imputing values already existing in the table in a preceding paper. This results in strongly possible keys (spKey's) and strongly possible functional dependencies (spFD's). We give a polynomial algorithm to verify a single spKey and show that in general, it is NP-complete to verify an arbitrary collection of spKeys. We give a graph-theoretical characterization of the validity of a given spFD X →sp Y. We show, that the complexity to verify a single strongly possible functional dependency is NP-complete in general, then we introduce some cases when verifying a single spFD can be done in polynomial time. As a step forward axiomatization of spFD's, the rules given for weak and strong functional dependencies are checked. Appropriate weakenings of those that are not sound for spFD's are listed. The interaction between spFD's and spKey's and certain keys is studied. Furthermore, a graph theoretical characterization of implication between singular attribute spFD's is given