Armstrong Axioms

Definition

The term Armstrong axioms refers to the sound and complete set of inference rules or axioms, introduced by William W. Armstrong [2], that is used to test logical implication of functional dependencies.

Given a relation schema R[U] and a set of functional dependencies Σ over attributes in U, a functional dependency f is logically implied by Σ, denoted by Σ⊧f, if for every instance I of R satisfying all functional dependencies in Σ, I satisfies f. The set of all functional dependencies implied by Σ is called the closure of Σ, denoted by Σ⁺.

Key Points

Armstrong axioms consist of the following three rules:

• Reflexivity: If Y ⊆ X, then X → Y.

• Augmentation: If X → Y , then XZ → YZ.

• Transitivity: If X → Y and Y → Z, then X → Z.

Note that in the above rules XZ refers to the union of two attribute sets X and Z. Armstrong axioms are sound and complete: a functional dependency f is derivable from a set of functional dependencies Σ by applying the axioms if and only if Σ⊧f (refer to [1]...