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 $F^{+}$ ) 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 $F$ .

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 the above axioms. Useful acronym DCUPS.

Decomposition

If $X \to YZ$ then $X \to Y$ and $X \to Z$

Composition

If $X\to Y$ and $A\to B$ then $XA\to YB$

Union (Notation)

If $X\to Y$ and $X \to Z$ then $X \to YZ$

Pseudo transitivity

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

Self determination

If $I\to I$

Armstrong relation

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.^[2]

External links

References

↑ William Ward Armstrong: Dependency Structures of Data Base Relationships, page 580-583. IFIP Congress, 1974.
↑ 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.

Database management systems
Types	Object-oriented comparison Relational comparison Document-oriented Graph NoSQL NewSQL
Concepts	Database ACID Armstrong's axioms CAP theorem CRUD Null Candidate key Foreign key Superkey Surrogate key Unique key
Objects	Relation table column row View Transaction Transaction log Trigger Index Stored procedure Cursor Partition
Components	Concurrency control Data dictionary JDBC XQJ ODBC Query language Query optimizer Query plan
Functions	Administration and automation Query optimization Replication
Related topics	Database models Database normalization Database storage Distributed database Federated database system Referential integrity Relational algebra Relational calculus Relational database Relational DBMS Relational model Object-relational database Transaction processing

Database normalization
Main topics	Normalization Denormalization
Normal forms	First normal form Second normal form Third normal form Boyce–Codd normal form Fourth normal form Fifth normal form Domain-key normal form Sixth normal form

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