Why the Closure Algorithm Works

In this section, we shall show why the closure algorithm correctly decides whether or not a FD A1A2 . . . An → B follows from a given set of FDs S. There are two parts to the proof:

Decomposition into BCNF

By frequently selecting appropriate decompositions, we can break any relation schema into a collection of subsets of its attributes with the following important properties:

Recovering Information from a Decomposition

Let us now turn our attention to the question of why the decomposition algorithm of "Decomposition into BCNF" preserves the information that was included in the original relation. The idea is that if we follow this algorithm, then the projections of the original tuples can be "joined"

Fourth Normal Form

The redundancy that we found in "Multivalued Dependencies" to be caused by MVD's can be removed if we use these dependencies in a new decomposition algorithm for relations. In this section we shall introduce a new normal form, called "fourth normal form". In this normal form,

Decomposition into Fourth Normal Form

The 4NF decomposition algorithm is quite similar to the BCNF decomposition algorithm. We find a 4NF violation, say A1A2 . . . An → B1B2 . . . Bm , where {A1, A2, . . . , An} is not a superkey. Note this MVD could be a true MVD, or it could be derived from the equivalent FD A1A2 . . . An

Relationships Among Normal Forms

As we have pointed out, 4NF implies BCNF, which in turn implies 3NF. Therefore, the sets of relation schemas (including dependencies) satisfying the three normal forms are related as in the following figure. That is, if a relation with certain dependencies is in 4NF, it is also in

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