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. 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 ⟨ 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 ⊨ 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 ⊢ 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 ∗ ⊆ 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 + ⊆ F A ∗
more simply put, we are able to derive by A all the functional dependencies that are logically implied by F .
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 X Y instead of the usual X ∪ Y ; this notation is rather standard in database theory when dealing with sets of attributes.
If Y ⊆ X then X → Y
If X → Y , then X Z → Y Z for any Z
If X → Y and Y → Z , then X → Z
These rules can be derived from the above axioms.
If X → Y and X → Z then X → Y Z
If X → Y Z then X → Y and X → Z
If X → Y and Y Z → W then X Z → W
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.
