Armstrong's axioms

Armstrong's axioms are a set of references (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. The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as $$F^{+}$$) when applied to that set (denoted as $$F$$). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure $$F^+$$.

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

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

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

$$ F^{*}_{A} \subseteq F^{+} $$

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

$$ F^{+} \subseteq F^{*}_{A} $$

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

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

Axiom of reflexivity
If $$X$$ is a set of attributes and $$Y$$ is a subset of $$X$$, then $$X$$ holds $$Y$$. Hereby, $$X$$ holds $$Y$$ [$$X \to Y$$] means that $$X$$ functionally determines $$Y$$.
 * If $$Y \subseteq X$$ then $$X \to Y$$.

Axiom of augmentation
If $$X$$ holds $$Y$$ and $$Z$$ is a set of attributes, then $$X Z$$ holds $$Y Z$$. It means that attribute in dependencies does not change the basic dependencies.
 * If $$X \to Y$$, then $$X Z \to Y Z$$ for any $$Z$$.

Axiom of transitivity
If $$X$$ holds $$Y$$ and $$Y$$ holds $$Z$$, then $$X$$ holds $$Z$$.
 * If $$X \to Y$$ and $$Y \to Z$$, then $$X \to Z$$.

Additional rules (Secondary Rules)
These rules can be derived from the above axioms.

Decomposition
If $$X \to Y Z$$ then $$ X \to Y$$ and $$ X \to Z$$.

Composition
If $$X \to Y$$ and $$A \to B$$ then $$ X A \to Y B$$.

Union
If $$X \to Y$$ and $$X \to Z$$ then $$X \to YZ$$.

Pseudo transitivity
If $$X \to Y$$ and $$ Y Z \to W$$ then $$ X Z\to W$$.

Self determination
$$ I \to I$$ for any $$I$$. This follows directly from the axiom of reflexivity.

Extensivity
The following property is a special case of augmentation when $$Z=X$$.
 * If $$X \to Y$$, then $$X \to X Y$$.

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

Armstrong relation
Given a set of functional dependencies $$F$$, an Armstrong relation is a relation which satisfies all the functional dependencies in the closure $$F^+$$ 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.