Armstrong's axioms

Armstrong's axioms are a set of axioms (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong in his 1974 paper.^[1] The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as ${\displaystyle F^{+}}$) when applied to that set (denoted as ${\displaystyle F}$). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure ${\displaystyle F^{+}}$.

More formally, let ${\displaystyle \langle R(U),F\rangle }$ denote a relational scheme over the set of attributes ${\displaystyle U}$ with a set of functional dependencies ${\displaystyle F}$. We say that a functional dependency ${\displaystyle f}$ is logically implied by ${\displaystyle F}$, and denote it with ${\displaystyle F\models f}$ if and only if for every instance ${\displaystyle r}$ of ${\displaystyle R}$ that satisfies the functional dependencies in ${\displaystyle F}$, ${\displaystyle r}$ also satisfies ${\displaystyle f}$. We denote by ${\displaystyle F^{+}}$ the set of all functional dependencies that are logically implied by ${\displaystyle F}$.

Furthermore, with respect to a set of inference rules ${\displaystyle A}$, we say that a functional dependency ${\displaystyle f}$ is derivable from the functional dependencies in ${\displaystyle F}$ by the set of inference rules ${\displaystyle A}$, and we denote it by ${\displaystyle F\vdash _{A}f}$ if and only if ${\displaystyle f}$ is obtainable by means of repeatedly applying the inference rules in ${\displaystyle A}$ to functional dependencies in ${\displaystyle F}$. We denote by ${\displaystyle F_{A}^{*}}$ the set of all functional dependencies that are derivable from ${\displaystyle F}$ by inference rules in ${\displaystyle A}$.

Then, a set of inference rules ${\displaystyle A}$ is sound if and only if the following holds:

${\displaystyle F_{A}^{*}\subseteq F^{+}}$

that is to say, we cannot derive by means of ${\displaystyle A}$ functional dependencies that are not logically implied by ${\displaystyle F}$. The set of inference rules ${\displaystyle A}$ is said to be complete if the following holds:

${\displaystyle F^{+}\subseteq F_{A}^{*}}$

more simply put, we are able to derive by ${\displaystyle A}$ all the functional dependencies that are logically implied by ${\displaystyle F}$.

Axioms (primary rules)

Let ${\displaystyle R(U)}$ be a relation scheme over the set of attributes ${\displaystyle U}$. Henceforth we will denote by letters ${\displaystyle X}$, ${\displaystyle Y}$, ${\displaystyle Z}$ any subset of ${\displaystyle U}$ and, for short, the union of two sets of attributes ${\displaystyle X}$ and ${\displaystyle Y}$ by ${\displaystyle XY}$ instead of the usual ${\displaystyle X\cup Y}$; this notation is rather standard in database theory when dealing with sets of attributes.

Axiom of reflexivity

If ${\displaystyle X}$ is a set of attributes and ${\displaystyle Y}$ is a subset of ${\displaystyle X}$, then ${\displaystyle X}$ holds ${\displaystyle Y}$. Hereby, ${\displaystyle X}$ holds ${\displaystyle Y}$ [${\displaystyle X\to Y}$] means that ${\displaystyle X}$ functionally determines ${\displaystyle Y}$.

If ${\displaystyle Y\subseteq X}$ then ${\displaystyle X\to Y}$.

Axiom of augmentation

If ${\displaystyle X}$ holds ${\displaystyle Y}$ and ${\displaystyle Z}$ is a set of attributes, then ${\displaystyle XZ}$ holds ${\displaystyle YZ}$. It means that attribute in dependencies does not change the basic dependencies.

If ${\displaystyle X\to Y}$, then ${\displaystyle XZ\to YZ}$ for any ${\displaystyle Z}$.

Axiom of transitivity

If ${\displaystyle X}$ holds ${\displaystyle Y}$ and ${\displaystyle Y}$ holds ${\displaystyle Z}$, then ${\displaystyle X}$ holds ${\displaystyle Z}$.

If ${\displaystyle X\to Y}$ and ${\displaystyle Y\to Z}$, then ${\displaystyle X\to Z}$.

Additional rules (Secondary Rules)

These rules can be derived from the above axioms.

Decomposition

If ${\displaystyle X\to YZ}$ then ${\displaystyle X\to Y}$ and ${\displaystyle X\to Z}$.

Proof

1. ${\displaystyle X\to YZ}$	(Given)
2. ${\displaystyle YZ\to Y}$	(Reflexivity)
3. ${\displaystyle X\to Y}$	(Transitivity of 1 & 2)

Composition

If ${\displaystyle X\to Y}$ and ${\displaystyle A\to B}$ then ${\displaystyle XA\to YB}$.

Proof

1. ${\displaystyle X\to Y}$	(Given)
2. ${\displaystyle A\to B}$	(Given)
3. ${\displaystyle XA\to YA}$	(Augmentation of 1 & A)
4. ${\displaystyle YA\to YB}$	(Augmentation of 2 & Y)
5. ${\displaystyle XA\to YB}$	(Transitivity of 3 and 4)

Union

If ${\displaystyle X\to Y}$ and ${\displaystyle X\to Z}$ then ${\displaystyle X\to YZ}$.

Proof

1. ${\displaystyle X\to Y}$	(Given)
2. ${\displaystyle X\to Z}$	(Given)
3. ${\displaystyle X\to XZ}$	(Augmentation of 2 & X)
4. ${\displaystyle XZ\to YZ}$	(Augmentation of 1 & Z)
5. ${\displaystyle X\to YZ}$	(Transitivity of 3 and 4)

Pseudo transitivity

If ${\displaystyle X\to Y}$ and ${\displaystyle YZ\to W}$ then ${\displaystyle XZ\to W}$.

Proof

1. ${\displaystyle X\to Y}$	(Given)
2. ${\displaystyle YZ\to W}$	(Given)
3. ${\displaystyle XZ\to YZ}$	(Augmentation of 1 & Z)
4. ${\displaystyle XZ\to W}$	(Transitivity of 3 and 2)

Self determination

${\displaystyle I\to I}$ for any ${\displaystyle I}$. This follows directly from the axiom of reflexivity.

Extensivity

The following property is a special case of augmentation when ${\displaystyle Z=X}$.

If ${\displaystyle X\to Y}$, then ${\displaystyle X\to XY}$.

Extensivity can replace augmentation as axiom in the sense that augmentation can be proved from extensivity together with the other axioms.

Proof

1. ${\displaystyle XZ\to X}$	(Reflexivity)
2. ${\displaystyle X\to Y}$	(Given)
3. ${\displaystyle XZ\to Y}$	(Transitivity of 1 & 2)
4. ${\displaystyle XZ\to XYZ}$	(Extensivity of 3)
5. ${\displaystyle XYZ\to YZ}$	(Reflexivity)
6. ${\displaystyle XZ\to YZ}$	(Transitivity of 4 & 5)

Armstrong relation

Given a set of functional dependencies ${\displaystyle F}$, an Armstrong relation is a relation which satisfies all the functional dependencies in the closure ${\displaystyle F^{+}}$ and only those dependencies. Unfortunately, the minimum-size Armstrong relation for a given set of dependencies can have a size which is an exponential function of the number of attributes in the dependencies considered.^[2]

References

1. William Ward Armstrong: Dependency Structures of Data Base Relationships, page 580-583. IFIP Congress, 1974.
2. Beeri, C.; Dowd, M.; Fagin, R.; Statman, R. (1984). "On the Structure of Armstrong Relations for Functional Dependencies" (PDF). Journal of the ACM. 31: 30–46. CiteSeerX 10.1.1.68.9320. doi:10.1145/2422.322414. Archived from the original (PDF) on 2018-07-23.

External links

• UMBC CMSC 461 Spring '99
• CS345 Lecture Notes from Stanford University