On Improving Dependency Implication Algorithms

Let 2 be a finite set of join dependencies (JDs), multivalued dependencies (MVDs) and functional dependencies (IDS). To test whether .% implies an MVD X -+ Y or an PD X -+ A, the procedure in [ 31 first generates the dependency basis DEP( X) of X with respect to 2. Then, X + Y if and only if Y is the union of some elements in DEP( X) , and X -+ A if and only if A E DEP( X), and there is an PD V --f W in 2 with AEW-V. In this paper, we present two algorithms, Algorithms 1 and 2, that test the same implication problems. However, these algorithms do not generate DEP(X) with respect to 2, but rather directly determine whether X + Y or X + Y without computing the basis for all right-hand sides Z for X -+ Z or X -++ Z as in [ 31. As part of proving these algorithms correct, we also prove a result that is interesting in its own right. We organize this paper as follows. In Section 2, we present the basic definitions and some results in the literature that we need. In Section 3, we prove a theo-