Encyclopedia of Database Systems

Living Edition
| Editors: Ling Liu, M. Tamer Özsu

Armstrong Axioms

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4899-7993-3_1554-2


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]...

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Recommended Reading

  1. 1.
    Abiteboul S, Hull R, Vianu V. Foundations of databases. Reading: Addison-Wesley; 1995.Google Scholar
  2. 2.
    Armstrong W. Dependency structures of data base relationships. In: Proceedings of the IFIP Congress; 1974.Google Scholar

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© Springer Science+Business Media LLC 2016

Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada