Multivalued Dependencies

Definition of MVD

A multivalued dependency (MVD) on R, X ->->Y

• says that if two tuples of R agree on all the attributes of X, then their components in Y may be swapped（多值依赖讲的是当X中的值全相等，那么Y的值是可交换的）
• the result will be two tuples that are also in the relation（交换后的元组也在关系中）
• for each value of X, the values of Y are independent of the values of R-X-Y（Y的值是独立于其他属性的）

Example: MVD

Drinkers(name, addr, phones, beersLiked)

A drinker’s phones are independent of the beers they like.（phones是独立于喜爱的啤酒的）

name->->phones and name ->->beersLiked.

Thus, each of a drinker’s phones appears with each of the beers they like in all combinations.（饮酒者的电话号码与他们喜欢的啤酒自由组合）

This repetition is unlike FD redundancy.

name->addr is the only FD.

MVD Rules

Every FD is an MVD (promotion ).

• If X ->Y, then swapping Y ’s between two tuples that agree on X doesn’t change the tuples.
• Therefore, the “new” tuples are surely in the relation, and we know X ->->Y.

Complementation : If X ->->Y, and Z is all the other attributes, then X ->->Z.（多值依赖可以理解为相互独立）

Fourth Normal Form

• The redundancy（冗余） that comes from MVD’s is not removable by putting the database schema in BCNF.
• There is a stronger normal form, called 4NF

that (intuitively) treats MVD’s as FD’s when it comes to decomposition, but not when determining keys of the relation.

4NF Definition

• A relation R is in 4NF if: whenever X ->->Y is a nontrivial MVD, then X is a superkey.（当X非平凡多值依赖与Y时，X一定是超键）
• Nontrivial MVD means that:

1. Y is not a subset of X, and X and Y are not, together, all the attributes.

• Note that the definition of “superkey” still depends on FD’s only.（注意的是超键的定义依然取决于函数依赖）

BCNF Versus 4NF

• Remember that every FD X ->Y is also an MVD, X ->->Y
• Thus, if R is in 4NF, it is certainly in BCNF.（模式R满足4NF一定满足BCNF）

1. Because any BCNF violation is a 4NF violation (after conversion to an MVD).
2. But R could be in BCNF and not 4NF, because MVD’s are “invisible” to BCNF.

Decomposition and 4NF

If X ->->Y 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.

Example: 4NF Decomposition

Drinkers(name, addr, phones, beersLiked)

FD: name -> addr MVD’s: name ->-> phones 、name ->-> beersLiked Key is {name, phones, beersLiked}.

• Decompose using name -> addr: Drinkers1(name, addr) In 4NF;

only dependency is name -> addr.

• Drinkers2(name, phones, beersLiked) Not in 4NF.
• MVD’s name ->-> phones and name ->-> beersLiked apply.

No FD’s, so all three attributes form the key.

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