Some results about normal forms for functional dependency in the relational datamodel

Abstract

In this paper we present some characterizations of relation schemes in second normal form (2NF), third normal form (3NF) and Boyce-Codd normal form (BCNF). It is known [6]that the set of minimal keys of a relation scheme is a Sperner system (an antichain) and for an arbitrary Sperner system there exists a relation scheme the set of minimal keys of which is exactly the given Sperner system. We investigate families of 2NF, 3NF and BCNF relation schemes where the sets of minimal keys are given Sperner systems. We give characterizations of these families. The minimal Armstrong relation has been investigated in the literature [3, 7, 11, 15, 18]. This paper gives new bounds on the size of minimal Armstrong relations for relation schemes. We show that given a relation scheme s such that the set of minimal keys is the Sperner system K, the number of antikeys (maximal nonkeys) of K is polynomial in the number of attributes iff so is the size of minimal Armstrong relation of s. We give a new characterization of relations and relation schemes that are uniquely determined by their minimal keys. From this characterization we give a polynomial-time algorithm deciding whether an arbitrary relation is uniquely determined by its set of all minimal keys. We present a new polynomial-time algorithm testing BCNF property of a given relation scheme.