Multivalued dependency

In database theory, multivalued dependency is a full constraint between two sets of attributes in a relation.

In contrast to the functional dependency, the multivalued dependency requires that certain tuples be present in a relation. Therefore, a multivalued dependency is a special case of tuple-generating dependency. The multivalued dependency plays a role in the 4NF database normalization.

Formal definition

The formal definition is given as follows.

Let $R$ be a relation schema and let $\alpha \subseteq R$ and $\beta \subseteq R$ (subsets). The multivalued dependency
$\alpha \twoheadrightarrow \beta$
(which can be read as $\alpha$ multidetermines $\beta$) holds on $R$ if, in any legal relation $r(R)$, for all pairs of tuples $t _1$ and $t _2$ in $r$ such that $t _1[\alpha]=t _2[\alpha]$, there exist tuples $t _3$ and $t _4$ in $r$ such that
$t _1[\alpha] = t _2 [\alpha] = t _3 [\alpha] = t _4 [\alpha]$
$t _3[\beta] = t _1 [\beta]$
$t _3[R - \beta] = t _2 [R - \beta]$
$t _4[\beta] = t _2 [\beta]$
$t _4[R - \beta] = t _1 [R - \beta]$

In more simple words the above condition can be expressed as follows: if we denote by $(x,y,z)$ the tuple having values for $\alpha,$ $\beta,$ $R - \alpha - \beta$ collectively equal to $x,$ $y,$ $z,$ correspondingly, then whenever the tuples $(a,b,c)$ and $(a,d,e)$ exist in $r$, the tuples $(a,b,e)$ and $(a,d,c)$ should also exist in $r$.

Example

Consider this example of a database of teaching courses, the books recommended for the course, and the lecturers who will be teaching the course:

Teaching database

Course	Book	Lecturer
AHA	Silberschatz	John D
AHA	Nederpelt	William M
AHA	Silberschatz	William M
AHA	Nederpelt	John D
AHA	Silberschatz	Christian G
AHA	Nederpelt	Christian G
OSO	Silberschatz	John D
OSO	Silberschatz	William M

Because the lecturers attached to the course and the books attached to the course are independent of each other, this database design has a multivalued dependency; if we were to add a new book to the AHA course, we would have to add one record for each of the lecturers on that course, and vice versa.
Put formally, there are two multivalued dependencies in this relation: {course} $\twoheadrightarrow$ {book} and equivalently {course} $\twoheadrightarrow$ {lecturer}.
Databases with multivalued dependencies thus exhibit redundancy. In database normalization, fourth normal form requires that either every multivalued dependency X $\twoheadrightarrow$ Y is trivial or for every nontrivial multivalued dependency X $\twoheadrightarrow$ Y, X is a superkey.

Interesting properties

• If $\alpha \twoheadrightarrow \beta$, Then $\alpha \twoheadrightarrow R - \beta$
• If $\alpha \twoheadrightarrow \beta$ and $\gamma \subseteq \delta$, Then $\alpha \delta \twoheadrightarrow \beta \gamma$
• If $\alpha \twoheadrightarrow \beta$ and $\beta \twoheadrightarrow \gamma$, then $\alpha \twoheadrightarrow \gamma - \beta$

The following also involve functional dependencies:

• If $\alpha \rightarrow \beta$, then $\alpha \twoheadrightarrow \beta$
• If $\alpha \twoheadrightarrow \beta$ and $\beta \rightarrow \gamma$, then $\alpha \twoheadrightarrow \gamma - \beta$

The above rules are sound and complete.

• A decomposition of R into (X, Y) and (X, R − Y) is a lossless-join decomposition if and only if X $\twoheadrightarrow$ Y holds in R.

Definitions

full constraint

A constraint which expresses something about all attributes in a database. (In contrast to an embedded constraint.) That a multivalued dependency is a full constraint follows from its definition,as where it says something about the attributes $R - \beta$.

tuple-generating dependency

A dependency which explicitly requires certain tuples to be present in the relation.

trivial multivalued dependency 1

A multivalued dependency which involves all the attributes of a relation i.e.$R = \alpha \cup \beta$. A trivial multivalued dependency implies, for tuples $t _1$ and $t _2$, tuples $t _3$ and $t _4$ which are equal to $t _1$ and $t _2$.

trivial multivalued dependency 2

A multivalued dependency for which $\beta \subseteq \alpha$.

External links

• Multivalued dependencies and a new Normal form for Relational Databases (PDF) - Ronald Fagin, IBM Research Lab