On the Structure of Armstrong Relations for Functional Dependencies

Abstract

Abstract. An Armstrong relation for a set of functional dependencies (FDs) is a relation that satisfies each FD implied by the set but no FD that is not implied by it. The structure and size (number of tuples) of Armstrong relatsons are investigated. Upper and lower bounds on the size of minimal-sized Armstrong relations are derived, and upper and lower bounds on the number of distinct entries that must appear m an Armstrong relation are given. It is shown that the time complexity of finding an Armstrong relation, gwen a set of functional dependencies, is precisely exponential in the number of attributes. Also shown ,s the falsity of a natural conjecture which says that almost all relations obeying a given set of FDs are Armstrong relations for that set of FDs. Finally, Armstrong relations are used to generahze a result, obtained by Demetrovics using quite complicated methods, about the possible sets of keys for a relauon.

Some properties of choice functions.

“…Those well-known concepts in relational database given in this section can be found in [2], [3], [4], [8], [10] and [20]. A relational database system of the scheme R( al, ... ,an) is considered as a table, where columns correspond to the attributes ai's while the row are n-tuples of relation r. Let X and Y be nonempty sets of attributes in R. We say that instance r of R satisfies the FD if two tuples agree on the values in attributes X, they must also agree on the values in attributes Y.…”

“…And the semilattice with greatest elements give an equivalent description of FDs. The closure operations, and other equivalent descriptions of family of FDs have been studied widely by Armstrong [2], Beeri, Dowd, Fagin and Statman [4], Mannila and Raiha [16].…”

Reformulating Positive Table Constraints Using Functional Dependencies

“…The reason is that finding the dependencies from the relation is itself an NP-Complete problem. However, given a set of functional dependencies δ, it can be shown that there always exists a relation on which only those functional dependencies hold; such relations are known as Armstrong relations [3].…”

“…, x m ). While constructing an Armstrong relation is exponential in the size of δ, upper and lower bounds on the minimal number of tuples in the relation are known [3]. The BSR problem contrary to BAR refers to a set of dependencies with the specific number of tuples in each projection of the relation on the scope of each dependency of δ.…”

Database sampling with functional dependencies

“…A database can be sampled according to many different criteria, such as for example random selection, or selecting data items in such a way that the resulting sample satisfies a set of integrity constraints. Functional dependencies are an example of such integrity constraints, which have been widely investigated in the context of relational databases [3,10,1]. They capture intuitive relationships between data items, and play a central role in the design of databases [16].…”