Family of functional dependencies and its equivalent descriptions

The family of functional dependencies (FDs) was introduced by E.F. Codd. Equivalent descriptions of family of FDs play essential roles in the design and implementation of the relational datamodel. It is known that closure operations, meet-semilattices, families of members which are not intersections of two other members, give the equivalent descriptions of family of FDs, i.e., they and family of FDs determine each other uniquely. These equivalent descriptions were successfully applied to find many desirable properties of functional dependency. This paper introduces the concept of maximal family of attributes. The authors prove that this family is an equivalent description of family of FDs. The concept of nonredundant family of attributes is also introduced in this paper. The authors present some characterizations and desirable properties of these families. They prove that given a relation scheme, the time complexities of problems of finding nonredundant and maximal families of attributes are exponential in the number of attributes. However, this paper shows that if a relation scheme is changed to a relation then these problems are solved by polynomial time algorithms.