# Armstrong's axioms

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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 on 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 ${\displaystyle F}$, 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

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

### Axiom of augmentation

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

### Axiom of transitivity

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

These rules can be derived from above axioms.

### Union

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

### Decomposition

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

### Pseudo transitivity

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

## 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]