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Armstrong's axioms

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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 1974 paper.[1] 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.

Contents

  • Axioms 1
    • Axiom of reflexivity 1.1
    • Axiom of augmentation 1.2
    • Axiom of transitivity 1.3
  • Additional rules 2
    • Union 2.1
    • Decomposition 2.2
    • Pseudo transitivity 2.3
  • Armstrong relation 3
  • External links 4
  • References 5

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; this notation is rather standard in database theory when dealing with sets of attributes.

Axiom of reflexivity

If Y \subseteq X then X \to Y

Axiom of augmentation

If X \to Y, then X Z \to Y Z for any Z

Axiom of transitivity

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

Additional rules

These rules can be derived from above axioms.

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 X \to Y and YZ \to W then XZ\to W

Armstrong relation

Given a set of functional dependencies F, the 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.[2]

External links

  • UMBC CMSC 461 Spring '99
  • CS345 Lecture Notes from Stanford University

References

  1. ^ William Ward Armstrong: Dependency Structures of Data Base Relationships, page 580-583. IFIP Congress, 1974.
  2. ^ Beeri, C.; Dowd, M.; Fagin, R.; Statman, R. (1984). "On the Structure of Armstrong Relations for Functional Dependencies" (PDF). Journal of the ACM 31: 30–46.  
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