DATABASE DESIGN -2 part2

Multi-valued Dependencies and Fourth Normal Form

(a) The EMP relation with two MVDs: ENAME —>> PNAME and ENAME —>> DNAME.

(b) Decomposing the EMP relation into two 4NF relations EMP_PROJECTS and

EMP_DEPENDENTS.

(d) Decomposing the relation SUPPLY into the 5NF relations R1, R2, and R3.

Multi-valued Dependencies and Fourth Normal Form

Definition:

A multi-valued dependency (MVD) X —>> Y specified on relation schema R, where X and Y are both subsets of R, specifies the following constraint on any relation state r of R: If two tuples t1 and t2 exist in r such that t1[X] = t2[X], then two tuples t3 and t4 should also exist in r with the following properties, where we use Z to denote

(R - (X υ Y)):
t3[X] = t4[X] = t1[X] = t2[X].
t3[Y] = t1[Y] and t4[Y] = t2[Y].
t3[Z] = t2[Z] and t4[Z] = t1[Z].

An MVD X —>> Y in R is called a trivial MVD if (a) Y is a subset of

X, or (b) X υ Y = R.

Inference Rules for Functional and Multi-valued Dependencies:

IR1 (reflexive rule for FDs): If X ⊇Y, then X –> Y.

IR2 (augmentation rule for FDs): {X –> Y} I= XZ –> YZ. IR3 (transitive rule for FDs): {X –> Y, Y –>Z} I= X –> Z.

IR4 (complementation rule for MVDs): {X —>> Y} I= X —>> (R – (X ∪Y))}.

IR5 (augmentation rule for MVDs): If X —>> Y and W ⊇Z then WX —>> YZ.

IR6 (transitive rule for MVDs): {X —>> Y, Y —>> Z} ⏐ = X —>> (Z 2 Y).

IR7 (replication rule for FD to MVD): {X –> Y} ⏐ = X —>> Y.

IR8 (coalescence rule for FDs and MVDs): If X —>> Y and there exists W with the properties that

(a) W ∩ Y is empty, (b) W –> Z, and (c) Y ⊇ Z, then X –> Z.

Definition:

A relation schema R is in 4NF with respect to a set of dependencies F (that includes functional dependencies and multivalued dependencies) if, for every nontrivial

multivalued dependency X —>> Y in F+, X is a superkey for R.

Note: F+ is the (complete) set of all dependencies (functional or multivalued) that will hold in every relation state r of R that satisfies F. It is also called the closure of F.

Decomposing a relation state of EMP that is not in 4NF:

(a) EMP relation with additional tuples.

(b) Two corresponding 4NF relations EMP_PROJECTS and EMP_DEPENDENTS.

Lossless (Non-additive) Join Decomposition into 4NF Relations:

PROPERTY LJ1’

The relation schemas R1 and R2 form a lossless (non-additive) join decomposition of R with respect to a set F of functional and multi-valued dependencies if and only if

(R1 ∩ R2) —>> (R1 - R2)

or by symmetry, if and only if (R1 ∩ R2) —>> (R2 - R1)).

Algorithm 11.5: Relational decomposition into 4NF relations with non-additive join property

Input: A universal relation R and a set of functional and multi-valued dependencies F.

1. Set D := { R };

2. While there is a relation schema Q in D that is not in 4NF do { choose a relation schema Q in D that is not in 4NF;

find a nontrivial MVD X —>> Y in Q that violates 4NF; replace Q in D by two relation schemas (Q - Y) and (X υ Y);

};

Join Dependencies and Fifth Normal Form

Definition:

A join dependency (JD), denoted by JD(R1, R2, ..., Rn), specified on relation schema R, specifies a constraint on the states r of R.

The constraint states that every legal state r of R should have a non-additive join decomposition into R1, R2, ..., Rn; that is, for every such r we have * (πR1(r), πR2(r),

..., πRn(r)) = r

Note: an MVD is a special case of a JD where n = 2.

A join dependency JD(R1, R2, ..., Rn), specified on relation schema R, is a trivial JD if one of the relation schemas Ri in JD(R1, R2, ..., Rn) is equal to R.

Definition:

A relation schema R is in fifth normal form (5NF) (or Project-Join Normal Form (PJNF)) with respect to a set F of functional, multivalued, and join dependencies if, for every nontrivial join dependency JD(R1, R2, ..., Rn) in F+ (that is, implied by F), every Ri is a superkey of R.

Relation SUPPLY with Join Dependency and conversion to Fifth Normal Form

Inclusion Dependencies

Definition:

An inclusion dependency R.X < S.Y between two sets of attributes—X of relation schema R, and Y of relation schema S—specifies the constraint that, at any specific time when r is a relation state of R and s a relation state of S, we must have πX(r(R)) ⊇ πY(s(S))

Note:

The ? (subset) relationship does not necessarily have to be a proper subset.

The sets of attributes on which the inclusion dependency is specified—X of R and Y of S—must have the same number of attributes.

In addition, the domains for each pair of corresponding attributes should be compatible.

Objective of Inclusion Dependencies:

To formalize two types of interrelational constraints which cannot be expressed using F.D.s or MVDs:

Referential integrity constraints

Class/subclass relationships

Inclusion dependency inference rules

IDIR1 (reflexivity): R.X < R.X.

IDIR2 (attribute correspondence): If R.X < S.Y

where X = {A1, A2 ,..., An} and Y = {B1, B2, ..., Bn} and Ai Corresponds-to Bi, then R.Ai < S.Bi for 1 ≤ i ≤ n.

IDIR3 (transitivity): If R.X < S.Y and S.Y < T.Z, then R.X < T.Z.

Other Dependencies and Normal Forms Template Dependencies:

Template dependencies provide a technique for representing constraints in relations that typically have no easy and formal definitions.

The idea is to specify a template—or example—that defines each constraint or dependency.

There are two types of templates:

- tuple-generating templates

-constraint-generating templates.

A template consists of a number of hypothesis tuples that are meant to show an example of the tuples that may appear in one or more relations. The other part of the template is the template conclusion.

Domain-Key Normal Form (DKNF):

Definition:

A relation schema is said to be in DKNF if all constraints and dependencies that should hold on the valid relation states can be

enforced simply by enforcing the domain constraints and key constraints on the relation.

The idea is to specify (theoretically, at least) the “ultimate normal form” that takes into account all possible types of dependencies and constraints. .

For a relation in DKNF, it becomes very straightforward to enforce all database constraints by simply checking that each attribute value in a tuple is of the appropriate domain and that every key constraint is enforced.

The practical utility of DKNF is limited

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