Armstrong’s Axioms in Functional Dependency in DBMS

In the world of Database Management Systems (DBMS), ensuring data integrity is paramount. This integrity is maintained through various mechanisms, and one of the most fundamental concepts in this regard is Functional Dependency. Functional Dependency is a critical aspect of designing and maintaining a database, and to understand it better, we turn to Armstrong Axioms in DBMS. These axioms provide a set of rules and principles that help in determining and simplifying functional dependencies within a relational database.

Axioms

Functional Dependency is a crucial concept in Database Management Systems (DBMS). It helps in maintaining data integrity. To understand this concept better, let's explore Armstrong Axioms in DBMS, a set of rules that play a fundamental role in defining and simplifying functional dependencies within a relational database.

Axiom of Reflexivity

The Axiom of Reflexivity is the foundational principle stating that if you have a set of attributes, a functional dependency exists between that set and itself. In simpler terms, it means that any set of attributes functionally determines itself.

Example: In a student database, if we have an attribute 'Student_ID,' it is trivially true that 'Student_ID' determines 'Student_ID.'

Axiom of Augmentation

The Axiom of Augmentation tells us that if a functional dependency exists between two sets of attributes, adding more attributes to both sides of the dependency does not change the dependency.

Example:

If 'Student_ID' determines 'Student_Name,' then it also implies that 'Student_ID, Course_Code' determines 'Student_Name, Course_Code.'

Axiom of Transitivity

The Axiom of Transitivity states that if we have two dependencies, where one attribute set determines another, and the second set determines a third set, then we can infer that the first set determines the third set.

Example:

If 'Student_ID' determines 'Course_Code' and 'Course_Code' determines 'Course_Name,' then 'Student_ID' determines 'Course_Name.'

Secondary Rules

In addition to the primary axioms, Armstrong also introduced several secondary rules:

Union

The Union Rule allows us to combine two or more functional dependencies to derive a new one.

Example:

If 'A' determines 'B' and 'A' determines 'C,' then we can derive 'A' determines 'B, C.'

Composition

The Composition Rule permits us to compose two dependencies to obtain a new one.

Example:

If 'A' determines 'B' and 'C' determines 'D,' we can compose these to derive 'A, C' determines 'B, D.'

Decomposition

The Decomposition Rule allows us to decompose a dependency into two or more dependencies.

Example:

If 'A, B' determines 'C, D' we can decompose it into 'A' determines 'C' and 'B' determines 'D.'

Pseudo Transitivity

The Pseudo Transitivity Rule is an extension of the transitivity axiom, allowing us to infer new dependencies based on existing ones.

Example:

If 'A' determines 'B, C' and 'C' determines 'D,' we can use pseudo transitivity to infer that 'A' determines 'B, D.'

Self Determination

The Self Determination Rule states that if an attribute is part of a set determining another attribute, it also determines itself.

Example:

If 'A' determines 'B,' then 'A' determines 'A.'

Extensivity

The Extensivity Rule suggests that if 'A' determines 'B,' then 'A' also determines 'B, C,' where 'C' is any set of attributes.

Example:

If 'A' determines 'B,' then 'A' determines 'B, C' (for any 'C').

Armstrong Relation

The Armstrong Relation is a set of all functional dependencies that hold in a given relation schema. This relation encapsulates all the functional dependencies that exist in the database.

Why Armstrong Axioms Refer to the Sound and Complete?

Armstrong Axioms in DBMS are referred to as sound and complete because they ensure that no extraneous dependencies are introduced into the set of dependencies. Soundness implies that all the dependencies derived from these axioms are indeed valid in the given schema. Completeness means that every valid dependency can be derived using these axioms.

Advantages of Using Armstrong’s Axioms in Functional Dependency

1. Simplicity:
   Armstrong's Axioms provide a simple and systematic way to derive functional dependencies, making the process more structured and less error-prone.
2. Efficiency:
   By using these axioms, database designers can quickly identify and document the functional dependencies, which is crucial for database normalization.
3. Minimality:
   Armstrong's Axioms help in finding the minimal cover of functional dependencies, reducing redundancy in the database schema.

Disadvantages of Using Armstrong’s Axioms in Functional Dependency

1. Complex Dependencies:
   While the axioms are effective for simple functional dependencies, they may not handle more complex dependencies well, leading to potential inaccuracies.
2. Limited to Deterministic Dependencies:
   Armstrong Axioms in DBMS work best when dealing with deterministic dependencies. In real-world databases, some dependencies might be probabilistic or uncertain, which these axioms do not address.
3. Computational Complexity:
   Deriving all possible dependencies using Armstrong's Axioms can be computationally expensive for large databases.