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 on his paper.^[1] The axioms are sound in that they generate 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 < $R$ ( $U$ ), $F$ > 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 $F^{%2B}$ 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:

$F^{*}_{A} \subseteq F^{%2B}$

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:

$F^{%2B} \subseteq F^{*}_{A}$

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

1 Axioms
2 Additional rules
3 External links
4 References

Axioms

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 $X \cup Y$

Axiom of reflexivity

If $Y \subseteq X$ , then $X \to Y$

Axiom of augmentation

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

Axiom of transitivity

If $X \to Y$ and $Y \to Z$ , then $X \to Z$

Additional rules

Union

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

Decomposition

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

Pseudo Transitivity

If $A \to B$ and $BC \to D$ then $AC \to D$

External links

References

^ William Ward Armstrong: Dependency Structures of Data Base Relationships, page 580-583. IFIP Congress, 1974.

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