Functional dependency

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In relational database theory, a functional dependency is the following constraint between two attribute sets in a relation: Given a relation R and attribute sets X,Y ${\displaystyle \subseteq }$ R, X is said to functionally determine Y (written X → Y) if each X value is associated with precisely one Y value. R is then said to satisfy the functional dependency X → Y. Equivalently, the projection ${\displaystyle \Pi _{X,Y}R}$ is a function, that is, Y is a function of X.^[1][2] In simple words, if the values for the X attributes are known (say they are x), then the values for the Y attributes corresponding to x can be determined by looking them up in any tuple of R containing x. Customarily X is called the determinant set and Y the dependent set. A functional dependency FD: X → Y is called trivial if Y is a subset of X.

In other words, a dependency FD: X → Y means that the values of Y are determined by the values of X. Two tuples sharing the same values of X will necessarily have the same values of Y.

The determination of functional dependencies is an important part of designing databases in the

${\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)=R}$

from the system as possible.

A notion of

logical implication

is defined for functional dependencies in the following way: a set of functional dependencies ${\displaystyle \Sigma }$ logically implies another set of dependencies ${\displaystyle \Gamma }$, if any relation R satisfying all dependencies from ${\displaystyle \Sigma }$ also satisfies all dependencies from ${\displaystyle \Gamma }$; this is usually written ${\displaystyle \Sigma \models \Gamma }$. The notion of logical implication for functional dependencies admits a

Suppose one is designing a system to track vehicles and the capacity of their engines. Each vehicle has a unique vehicle identification number (VIN). One would write VIN → EngineCapacity because it would be inappropriate for a vehicle's engine to have more than one capacity. (Assuming, in this case, that vehicles only have one engine.) On the other hand, EngineCapacity → VIN is incorrect because there could be many vehicles with the same engine capacity.

This functional dependency may suggest that the attribute EngineCapacity be placed in a relation with candidate key VIN. However, that may not always be appropriate. For example, if that functional dependency occurs as a result of the transitive functional dependencies VIN → VehicleModel and VehicleModel → EngineCapacity then that would not result in a normalized relation.

This example illustrates the concept of functional dependency. The situation modelled is that of college students visiting one or more lectures in each of which they are assigned a teaching assistant (TA). Let's further assume that every student is in some semester and is identified by a unique integer ID.

We notice that whenever two rows in this table feature the same StudentID, they also necessarily have the same Semester values. This basic fact can be expressed by a functional dependency:

• StudentID → Semester.

If a row was added where the student had a different value of semester, then the functional dependency FD would no longer exist. This means that the FD is implied by the data as it is possible to have values that would invalidate the FD.

Other nontrivial functional dependencies can be identified, for example:

• {StudentID, Lecture} → TA
• {StudentID, Lecture} → {TA, Semester}

The latter expresses the fact that the set {StudentID, Lecture} is a superkey of the relation.

A classic example of functional dependency is the employee department model.

This case represents an example where multiple functional dependencies are embedded in a single representation of data. Note that because an employee can only be a member of one department, the unique ID of that employee determines the department.

• Employee ID → Employee Name
• Employee ID → Department ID

In addition to this relationship, the table also has a functional dependency through a non-key attribute

• Department ID → Department Name

This example demonstrates that even though there exists a FD Employee ID → Department ID - the employee ID would not be a logical key for determination of the department Name. The process of normalization of the data would recognize all FDs and allow the designer to construct tables and relationships that are more logical based on the data.

Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies. Among the most important are the following, usually called Armstrong's axioms:^[3]

• Reflexivity: If Y is a subset of X, then X → Y
• Augmentation: If X → Y, then XZ → YZ
• Transitivity: If X → Y and Y → Z, then X → Z

"Reflexivity" can be weakened to just ${\displaystyle X\rightarrow \varnothing }$, i.e. it is an actual

• Pseudotransitivity: If X → Y and YW → Z, then XW → Z[3]
• Composition: If X → Y and Z → W, then XZ → YW^[6]

One can also derive the union and decomposition rules from Armstrong's axioms:^[3][7]

X → Y and X → Z if and only if X → YZ

The closure of a set of values is the set of attributes that can be determined using its functional dependencies for a given relationship. One uses Armstrong's axioms to provide a proof - i.e. reflexivity, augmentation, transitivity.

Given ${\displaystyle R}$ and ${\displaystyle F}$ a set of FDs that holds in ${\displaystyle R}$: The closure of ${\displaystyle F}$ in ${\displaystyle R}$ (denoted ${\displaystyle F}$⁺) is the set of all FDs that are logically implied by ${\displaystyle F}$.^[8]

Closure of a set of attributes X with respect to ${\displaystyle F}$ is the set X⁺ of all attributes that are functionally determined by X using ${\displaystyle F}$⁺.

Imagine the following list of FDs. We are going to calculate a closure for A (written as A⁺) from this relationship.

1. A → B
2. B → C
3. AB → D

The closure would be as follows:

Therefore, A⁺= ABCD. Because A⁺ includes every attribute in the relationship, it is a superkey.

Definition: ${\displaystyle F}$ covers ${\displaystyle G}$ if every FD in ${\displaystyle G}$ can be inferred from ${\displaystyle F}$. ${\displaystyle F}$ covers ${\displaystyle G}$ if ${\displaystyle G}$⁺ ⊆ ${\displaystyle F}$⁺
Every set of functional dependencies has a canonical cover.

Two sets of FDs ${\displaystyle F}$ and ${\displaystyle G}$ over schema ${\displaystyle R}$ are equivalent, written ${\displaystyle F}$ ≡ ${\displaystyle G}$, if ${\displaystyle F}$⁺ = ${\displaystyle G}$⁺. If ${\displaystyle F}$ ≡ ${\displaystyle G}$, then ${\displaystyle F}$ is a cover for ${\displaystyle G}$ and vice versa. In other words, equivalent sets of functional dependencies are called covers of each other.

A set ${\displaystyle F}$ of FDs is nonredundant if there is no proper subset ${\displaystyle F'}$ of ${\displaystyle F}$ with ${\displaystyle F'}$ ≡ ${\displaystyle F}$. If such an ${\displaystyle F'}$ exists, ${\displaystyle F}$ is redundant. ${\displaystyle F}$ is a nonredundant cover for ${\displaystyle G}$ if ${\displaystyle F}$ is a cover for ${\displaystyle G}$ and ${\displaystyle F}$ is nonredundant.
An alternative characterization of nonredundancy is that ${\displaystyle F}$ is nonredundant if there is no FD X → Y in ${\displaystyle F}$ such that ${\displaystyle F}$ - {X → Y} ${\displaystyle \models }$ X → Y. Call an FD X → Y in ${\displaystyle F}$ redundant in ${\displaystyle F}$ if ${\displaystyle F}$ - {X → Y} ${\displaystyle \models }$ X → Y.

An important property (yielding an immediate application) of functional dependencies is that if R is a relation with columns named from some set of attributes U and R satisfies some functional dependency X → Y then ${\displaystyle R=\Pi _{XY}(R)\bowtie \Pi _{XZ}(R)}$ where Z = U − XY. Intuitively, if a functional dependency X → Y holds in R, then the relation can be safely split in two relations alongside the column X (which is a key for ${\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)}$) ensuring that when the two parts are joined back no data is lost, i.e. a functional dependency provides a simple way to construct a lossless join decomposition of R in two smaller relations. This fact is sometimes called Heaths theorem; it is one of the early results in database theory.^[9]

Heath's theorem effectively says we can pull out the values of Y from the big relation R and store them into one, ${\displaystyle \Pi _{XY}(R)}$, which has no value repetitions in the row for X and is effectively a

in the remainder of the big table ${\displaystyle \Pi _{XZ}(R)}$.

Functional dependencies however should not be confused with

. Enforcing referential constraints after relation schema decomposition (normalization) requires a new formalism, i.e. inclusion dependencies. In the decomposition resulting from Heath's theorem, there is nothing preventing the insertion of tuples in ${\displaystyle \Pi _{XZ}(R)}$ having some value of X not found in ${\displaystyle \Pi _{XY}(R)}$.

Normal forms

Normal forms are database normalization levels which determine the "goodness" of a table. Generally, the third normal form is considered to be a "good" standard for a relational database.^{[citation needed]}

Normalization aims to free the database from update, insertion and deletion anomalies. It also ensures that when a new value is introduced into the relation, it has minimal effect on the database, and thus minimal effect on the applications using the database.^{[citation needed]}

A set S of functional dependencies is irreducible if the set has the following three properties:

1. Each right set of a functional dependency of S contains only one attribute.
2. Each left set of a functional dependency of S is irreducible. It means that reducing any one attribute from left set will change the content of S (S will lose some information).
3. Reducing any functional dependency will change the content of S.

Sets of functional dependencies with these properties are also called canonical or minimal. Finding such a set S of functional dependencies which is equivalent to some input set S' provided as input is called finding a minimal cover of S': this problem can be solved in polynomial time.^[10]

• Chase (algorithm)
• Inclusion dependency
• Join dependency
• Multivalued dependency (MVD)
• Database normalization
• First normal form