Functional Dependency and Attribute Closure

Fundamental ideas in database management and design are functional dependency and attribute closure. They are essential to maintaining data integrity and building effective, organized, and normalized databases.

Functional Dependency

A functional dependency A->B in a relation holds if two tuples having the same value of attribute A also have the same value for attribute B. For Example, in relation to STUDENT shown in Table 1, Functional Dependencies

STUD_NO->STUD_NAME, STUD_NO->STUD_PHONE hold

but

STUD_NAME->STUD_STATE do not hold

Student Table

Advantages of Functional Dependencies

Through the identification and removal of redundant or unneeded data, they aid in the reduction of data redundancy in databases.
By guaranteeing that data is correct and consistent throughout the database, they enhance data integrity.
They make it simpler to add, edit, and remove data, which helps with database management.

Disadvantages of Functional Dependencies

The process of identifying functional dependencies can be time-consuming and complex, especially in large databases with many tables and relationships.
Overly restrictive functional dependencies can result in slow query performance or data inconsistencies, as data that should be related may not be properly linked.
Functional dependencies do not take into account the semantic meaning of data, and may not always reflect the true relationships between data elements.

How to find Functional Dependencies for a Relation?

Functional Dependencies in a relation are dependent on the domain of the relation. Consider the STUDENT relation given in Table 1.

We know that STUD_NO is unique for each student. So STUD_NO->STUD_NAME, STUD_NO->STUD_PHONE, STUD_NO->STUD_STATE, STUD_NO->STUD_COUNTRY and STUD_NO -> STUD_AGE all will be true.
Similarly, STUD_STATE->STUD_COUNTRY will be true as if two records have same STUD_STATE, they will have same STUD_COUNTRY as well.
For relation STUDENT_COURSE, COURSE_NO->COURSE_NAME will be true as two records with same COURSE_NO will have same COURSE_NAME.

Functional Dependency Set

Functional Dependency set or FD set of a relation is the set of all FDs present in the relation. For Example, FD set for relation STUDENT shown in table 1 is:

 { STUD_NO->STUD_NAME, STUD_NO->STUD_PHONE, STUD_NO->STUD_STATE, STUD_NO->STUD_COUNTRY, 
  STUD_NO -> STUD_AGE, STUD_STATE->STUD_COUNTRY }

Attribute Closure

Attribute closure of an attribute set can be defined as set of attributes which can be functionally determined from it.

How to find attribute closure of an attribute set?

To find attribute closure of an attribute set:

Add elements of attribute set to the result set.
Recursively add elements to the result set which can be functionally determined from the elements of the result set.

Using FD set of table 1, attribute closure can be determined as:

(STUD_NO)+ = {STUD_NO, STUD_NAME, STUD_PHONE, STUD_STATE, STUD_COUNTRY, STUD_AGE}
(STUD_STATE)+ = {STUD_STATE, STUD_COUNTRY}

Advantages of Attribute Closure

Attribute closures help to identify all possible attributes that can be derived from a set of given attributes.
They facilitate database design by identifying relationships between attributes and tables, which can help to optimize query performance.
They ensure data consistency by identifying all possible combinations of attributes that can exist in the database.

Disadvantages of Attribute Closure

The process of calculating attribute closures can be computationally expensive, especially for large datasets.
Attribute closures can become too complex to manage, especially as the number of attributes and tables in a database grows.
Attribute closures do not take into account the semantic meaning of data, and may not always accurately reflect the relationships between data elements.

How to Find Candidate Keys and Super Keys Using Attribute Closure?

If attribute closure of an attribute set contains all attributes of relation, the attribute set will be super key of the relation.
If no subset of this attribute set can functionally determine all attributes of the relation, the set will be candidate key as well. For Example, using FD set of table 1,

(STUD_NO, STUD_NAME)+ = {STUD_NO, STUD_NAME, STUD_PHONE, STUD_STATE, STUD_COUNTRY, STUD_AGE}

(STUD_NO)+ = {STUD_NO, STUD_NAME, STUD_PHONE, STUD_STATE, STUD_COUNTRY, STUD_AGE}

(STUD_NO, STUD_NAME) will be super key but not candidate key because its subset (STUD_NO)+ is equal to all attributes of the relation. So, STUD_NO will be a candidate key.

Prime and Non-Prime Attributes

Attributes which are parts of any candidate key of relation are called as prime attribute, others are non-prime attributes. For Example, STUD_NO in STUDENT relation is prime attribute, others are non-prime attribute.

Conclusion

Tools like functional dependency and attribute closure are helpful when designing and optimizing databases. They are useful for:

Determine the connections between the tables and the attributes.
Boost query efficiency
Ascertain data coherence.

GATE Questions

Q.1: Consider the relation scheme R = {E, F, G, H, I, J, K, L, M, N} and the set of functional dependencies {{E, F} -> {G}, {F} -> {I, J}, {E, H} -> {K, L}, K -> {M}, L -> {N} on R. What is the key for R? (GATE-CS-2014)

A. {E, F}
B. {E, F, H}
C. {E, F, H, K, L}
D. {E}

Solution:

Finding attribute closure of all given options, we get:
{E,F}+ = {EFGIJ}
{E,F,H}+ = {EFHGIJKLMN}
{E,F,H,K,L}+ = {{EFHGIJKLMN}
{E}+ = {E}
{EFH}+ and {EFHKL}+ results in set of all attributes, but EFH is minimal. So it will be candidate key. So correct option is (B).

Q.2: How to check whether an FD can be derived from a given FD set?

Solution:

To check whether an FD A->B can be derived from an FD set F,

Find (A)+ using FD set F.

If B is subset of (A)+, then A->B is true else not true.

Q.3: In a schema with attributes A, B, C, D and E following set of functional dependencies are given
{A -> B, A -> C, CD -> E, B -> D, E -> A}
Which of the following functional dependencies is NOT implied by the above set? (GATE IT 2005)

A. CD -> AC
B. BD -> CD
C. BC -> CD
D. AC -> BC

Solution:

Using FD set given in question,
(CD)+ = {CDEAB} which means CD -> AC also holds true.
(BD)+ = {BD} which means BD -> CD can’t hold true. So this FD is no implied in FD set. So (B) is the required option.
Others can be checked in the same way.

Q.4: Consider a relation scheme R = (A, B, C, D, E, H) on which the following functional dependencies hold: {A–>B, BC–> D, E–>C, D–>A}. What are the candidate keys of R? [GATE 2005]

(a) AE, BE
(b) AE, BE, DE
(c) AEH, BEH, BCH
(d) AEH, BEH, DEH

Solution:

(AE)+ = {ABECD} which is not set of all attributes. So AE is not a candidate key. Hence option A and B are wrong.
(AEH)+ = {ABCDEH}
(BEH)+ = {BEHCDA}
(BCH)+ = {BCHDA} which is not set of all attributes. So BCH is not a candidate key. Hence option C is wrong.
So correct answer is D.

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