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 in his 1974 paper.[1] The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as ) when applied to that set (denoted as ). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure .

More formally, let denote a relational scheme over the set of attributes with a set of functional dependencies . We say that a functional dependency is logically implied by ,and denote it with if and only if for every instance of that satisfies the functional dependencies in , r also satisfies . We denote by the set of all functional dependencies that are logically implied by .

Furthermore, with respect to a set of inference rules , we say that a functional dependency is derivable from the functional dependencies in by the set of inference rules , and we denote it by if and only if is obtainable by means of repeatedly applying the inference rules in to functional dependencies in . We denote by the set of all functional dependencies that are derivable from by inference rules in .

Then, a set of inference rules is sound if and only if the following holds:

that is to say, we cannot derive by means of functional dependencies that are not logically implied by . The set of inference rules is said to be complete if the following holds:

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

Axioms (Primary Rules)

Let be a relation scheme over the set of attributes . Henceforth we will denote by letters , , any subset of and, for short, the union of two sets of attributes and by instead of the usual ; this notation is rather standard in database theory when dealing with sets of attributes.

Axiom of reflexivity

If is a set of attributes and is a subset of , then holds . Hereby, holds [ ] means that functionally determines .

If then .

Axiom of augmentation

If holds and is a set of attributes, then holds . It means that attribute in dependencies does not change the basic dependencies.

If , then for any .

Axiom of transitivity

If holds and holds , then holds .

If and , then .

Additional rules (Secondary Rules)

These rules can be derived from the above axioms. Useful acronym DCUPS.

Decomposition

If then and .

Proof

1. (Given)
2. (Reflexivity)
3. (Transitivity of 1 & 2)

Composition

If and then .

Proof

1. (Given)
2. (Given)
3. (Augmentation of 1 & A)
4. (Decomposition of 3)
5. (Augmentation 2 & X)
6. (Decomposition of 5)
7. (Union 4 & 6)

Union (Notation)

If and then .

Proof

1. (Given)
2. (Given)
3. (Initial Rule)
4. (Augmentation of 1 & X)
5. (Augmentation of 2 & Y)
6. (Transitivity of 3 & 4)
7. (Transitivity of 4 & 5)

Pseudo transitivity

If and then .

Proof

1. (Given)
2. (Given)
3. (Augmentation of 1 & Z)
4. (Transitivity of 3 and 2)

Self determination

for any . This follows directly from the axiom of reflexivity.

Armstrong relation

Given a set of functional dependencies , an Armstrong relation is a relation which satisfies all the functional dependencies in the closure 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]

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. doi:10.1145/2422.322414.
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