Armstrong Axioms
The term Armstrong Axioms refers to the sound and complete set of inference rules or axioms, introduced by William W. Armstrong, that is used to test the logical implication of functional dependencies. If F is a set of functional dependencies then the closure of F, denoted as F+, is the set of all functional dependencies logically implied by F. Armstrong’s Axioms are a set of rules, that when applied repeatedly, generates a closure of functional dependencies.
Axioms
- Axiom of Reflexivity: If A is a set of attributes and B is a subset of A, then A holds B. If B⊆A then A→B. This property is trivial property.
- Axiom of Augmentation: If A→B holds and Y is the attribute set, then AY→BY also holds. That is adding attributes to dependencies, does not change the basic dependencies. If A→B, then AC→BC for any C.
- Axiom of Transitivity: Same as the transitive rule in algebra, if A→B holds and B→C holds, then A→C also holds. A→B is called A functionally which determines B. If X→Y and Y→Z, then X→Z.
Secondary Rules
These rules can be derived from the above axioms.
- Union: If A→B holds and A→C holds, then A→BC holds. If X→Y and X→Z then X→YZ.
- Composition: If A→B and X→Y hold, then AX→BY holds.
- Decomposition: If A→BC holds then A→B and A→C hold. If X→YZ then X→Y and X→Z.
- Pseudo Transitivity: If A→B holds and BC→D holds, then AC→D holds. If X→Y and YZ→W then XZ→W.
- Self Determination: It is similar to the Axiom of Reflexivity, i.e. A→A for any A.
- Extensivity: Extensivity is a case of augmentation. If AC→A, and A→B, then AC→B. Similarly, AC→ABC and ABC→BC. This leads to AC→BC.
Armstrong Relation
Armstrong Relation can be stated as a relation that is able to satisfy all functional dependencies in the F+ Closure. In the given set of dependencies, the size of the minimum Armstrong Relation is an exponential function of the number of attributes present in the dependency under consideration.
Armstrong’s Axioms in Functional Dependency in DBMS
Prerequisite – Functional Dependencies
This article contains Armstrong’s Axioms and how Armstrong’s Axioms are used to decide about the functional dependency on the database. We will be also learning about the Secondary Rules and Armstrong Relations. We will learn each thing in detail. Before moving ahead, you must have a knowledge of Functional Dependency.