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### 4NF & Multivalued Dependency

By Kristina Miguel

Review

- Superkey – a set of attributes which will uniquely identify each tuple in a relation
- Candidate key – a minimal superkey
- Primary key – a chosen candidate key
- Secondary key – all the rest of candidate keys
- Prime attribute – an attribute that is a part of a candidate key (key column)
- Non-prime attribute – a non-key column

Review (cont.)

- 1NF
- Eliminate repeating groups.
- Make a separate table for each set of related attributes, and give each table a primary key.
- 2NF
- Eliminate redundant data.
- Each attribute must be functionally dependent on the primary key.
- If an attribute depends on only part of a multi-valued key, remove it to a separate table.
- 3NF
- Eliminate columns not dependent on key.
- If attributes do not contribute to a description of the key, remove them to a separate table.
- Any transitive dependencies are moved into a smaller table.
- BCNF
- Every determinant in the table is a candidate key.
- If there are non-trivial dependencies between candidate key attributes, separate them out into distinct tables.
- All normal forms are additive, in that if a model is in 3NF, it is by definition also in 2NF and 1NF.

Multivalued Dependency (MVD)

- A MVD XY,
- Holds for some relation R, so that when you fix the values for one set of attributes, then the values in certain other attributes are independent of the values of all the other attributes in the relation.
- Is an assertion that two attributes or sets of attributes are independent of one another.
- For each value of X, the values of Y are independent of the values of R-X-Y.

MVD (cont.)

- More precisely, for MVD AB
- For each pair of tuples t and u of relation R that agree on all the A’s, we can find in R some tuple v that agrees
- With both t and u on the A’s,
- With t on the B’s, and
- With u on all attributes of R that are not among the A’s or B’s.

MVD Example

Drinkers(name, addr, phones, beersLiked)

- A drinker’s phones are independent of the beers they like.
- namephones and namebeersLiked.
- Thus, each of a drinker’s phones appears with each of the beers they like in all combinations.

MVD Example (cont.)

- Tuples Implied by namephones
- If we have tuples:
- Then these tuples must also be in the relation.

name addr phones beersLiked

sue a p1 b1

sue a p2 b2

sue a p2 b1

sue a p1 b2

4NF

- Definition
- A relation R is in 4NF if and only if, for every one of its non-trivial multivalued dependencies XY, X is a superkey—that is, X is either a candidate key or a superset thereof.
- NontrivialMVD means that:
- Y is not a subset of X, and
- X and Y are not, together, all the attributes.

Decomposition into 4NF

- If XY is a 4NF violation for relation R, we can decompose R using the same technique as for BCNF.
- XY is one of the decomposed relations.
- All but Y – X is the other.

Decomposition into 4NF Method

- Find a 4NF violation in R, say AB, where A is not a superkey.
- If there is such a 4NF violation, break the schema for the relation R that has the 4NF violation into two schemas.
- R1, whose schema is A’s and B’s.
- R2, whose schema is the A’s and all attributes of R that are not among the A’s or B’s.
- Find the FD’s and MVD’s that hold in R1 and R2. Recursively decompose R1 and R2 with respect to their projected dependencies.

4NF Decomposition Example

Drinkers(name, addr, phones, beersLiked)

FD: nameaddr

MVD’s: namephones

namebeersLiked

- Key is {name, phones, beersLiked}.
- All dependencies violate 4NF.

4NF Decomposition Example (cont.)

- Decompose using nameaddr:
- Drinkers1(name, addr)
- In 4NF; only dependency is nameaddr.
- Drinkers2(name, phones, beersLiked)
- Not in 4NF. MVD’s namephones and namebeersLiked apply. No FD’s, so all three attributes form the key.

4NF Decomposition Example (cont.)

- Decompose Drinkers2
- Either MVD name ->-> phones or name ->-> beersLiked tells us to decompose to:
- Drinkers3(name, phones)
- Drinkers4(name, beersLiked)

Summary

- A multivalued dependency is a statement that two sets of attributes in a relation have sets of values that appear in all possible combinations.
- If a relation is in 4NF, then every nontrivial MVD is really an FD with a superkey on the left.

References

- http://www.datamodel.org/NormalizationRules.html
- http://www.almaden.ibm.com/cs/people/fagin/tods77.pdf
- http://www.bkent.net/Doc/simple5.htm
- http://infolab.stanford.edu/~ullman/dscb.html

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