On the Structure of Armstrong Relations for Functional Dependencies

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.