Characterizations for Functional Dependency and Boyce-CODD Normal Form Families

A functional dependency (fd) family was recently defined [20] as the set of all instances satisfying some set of functional dependencies. A Royce-Codd normal form, abbreviated RCNF. family is defined here as an fd-family specified by some RCNF set of functi!)nal dependem& . The purpose of this paper is to present set-theoretic/algebraic characterizations relating to both types of families. Two characterizations of ,7(.d), the smallcsr fd-family containing the farnil\ .3 of instances. are established. The first involves the notion of agreement. a concept relatecl to that of a closed set of attributes. The seconC describes .gc,B ) as the smallest family of instances containing .I/ and closed under four specific qjerations on instances. Companion results are also g.ven for HCNFfamilies. The remaining results concern characterizations involving the well-known operations of projection. join and union. Two chtiracterizations for when the projection of an fd-family is again an fd-family arc given. Several c,.~r4laries are obtained, including the effective decidability of whether a projection of an fd-family is an fd-family. The problem for RCNF-families disappears since it ih \hcxvn that the projection of a RCNF-family is alway\ a HC’NF-familv. An:~iogo~~ to result\ for fd-familic\ presented in [20], characterizations of when the join and union of RCNF-families arc‘ RC’NF-families are given. Finally, the collections of all fd-families and all HClJF-familic4 art’ characterized in terms of inverse projection operations and intersection.