Functional Dependencies
Definition:
If two tuples agree on the attributes
A1, A2, … A n
then they must also agree on the attributes
B1, B2, … B m
Formally:
A1, A2, … A n
B1, B2, … B m
Motivating example for the study of functional dependencies:
Name
Social Security Number
Phone Number
Examples
Product: name
Person: ssn
Company: name
price, manufacturer
name, age
stock price, president
Key of a relation is a set of attributes that:
- functionally determines all the attributes of the relation
- none of its subsets determines all the attributes.
Superkey: a set of attributes that contains a key.
Finding the Attributes of a
Relation
Given a relation constructed from an E/R diagram, what is its key?
Rules:
1. If the relation comes from an entity set,
the key of the relation is the set of attributes which is the
key of the entity set.
Person
address
name
ssn
Rules for Binary Relationships
name
Product
price
Person
buys
name
ssn
Several cases are possible for a binary relationship E1 - E2:
1. Many-many: the key includes the key of E1 together with the
key of E2.
What happens for:
2. Many-one:
3. One-one:
Rules for Multiway Relationships
None, really.
Except: if there is an arrow from the relationship to E, then
we don’t need the key of E as part of the relation key.
Product
Purchase
Payment Method
Person
Store
Some Properties of FD’s
A1, A2, … A n
B1, B2, … B m
Is equivalent to
A1, A2, … A n
B1
A1, A2, … A n
B2
…
A1, A2, … A n
A1, A2, … A n
A
i
Bm
Always holds.
Splitting rule
and
Combing rule
Comparing Functional
Dependencies
Functional dependencies: a statement about the set of allowable
databases.
Entailment and equivalence: comparing sets of functional dependencies
Entailment: a set of functional dependencies S1 entails a set S2 if:
any database that satisfies S1 much also satisfy S2.
Example:
{A B,
entails
A C
B
C}
Equivalence: two sets of FD’s are equivalent if each entails the
other.
{A B, B C }
is equivalent to {A B, A C, B
C}
Closure of a set of Attributes
Given a set of attributes A and a set of dependencies C,
we want to find all the other attributes that are functionally
determined by A.
In other words, we want to find the maximal set of attributes B,
such that for every B in B,
C
entails A
B.
Closure Algorithm
Start with Closure=A.
Until closure doesn’t change do:
if A1, A2, … A n
A1, A2, … A n
B is in C, and
are all in the closure, and
B is not in Closure
then
add B to closure.
Example
A B
A D
B
A F
C
E
D
B
Closure of {A,B}:
Closure of {A, F}:
Problems in Designing Schema
Name
Fred
Fred
Joe
Joe
Problems:
- redundancy
- update anomalies
- deletion anomalies
SSN
123-321-99
123-321-99
909-438-44
909-438-44
Phone Number
(201)
(206)
(908)
(212)
555-1234
572-4312
464-0028
555-4000
Relation Decomposition
Break the relation into two relations:
Name
Fred
Joe
SSN
123-321-99
909-438-44
Name
Phone Number
Fred
Fred
Joe
Joe
(201)
(206)
(908)
(212)
555-1234
572-4312
464-0028
555-4000
Decompositions in General
Let R be a relation with attributes A , A , … A
1
2
n
Create two relations R1 and R2 with attributes
B1, B2, … B m
Such that:
B1, B2, … B m
C1, C2, … C l

C1, C2, … C l =
And
-- R1 is the projection of R on
-- R2 is the projection of R on
A1, A2, … A n
B1, B2, … B m
C1, C2, … C l
Boyce-Codd Normal Form
A simple condition for removing anomalies from relations:
A relation R is in BCNF if and only if:
Whenever there is a nontrivial dependency A1, A2, … A n
for R , it is the case that { A , A , … A }
1
2
n
a super-key for R.
B
In English (though a bit vague):
Whenever a set of attributes of R is determining another attribute,
should determine all the attributes of R.
Example
Name
Fred
Fred
Joe
Joe
SSN
123-321-99
123-321-99
909-438-44
909-438-44
What are the dependencies?
What are the keys?
Is it in BCNF?
Phone Number
(201)
(206)
(908)
(212)
555-1234
572-4312
464-0028
555-4000
And Now?
SSN
123-321-99
909-438-44
SSN
123-321-99
123-321-99
909-438-44
909-438-44
Name
Fred
Joe
Phone Number
(201)
(206)
(908)
(212)
555-1234
572-4312
464-0028
555-4000
What About This?
Name
Gizmo
Price
Category
$19.99
gadgets
Question:
Find an example of a 2-attribute relation that is not in BCNF.
More Decompositions
Name
Address
Name
Address
Name
Move-Date
What’s wrong?
Move-Date
More Careful Strategy
Find a dependency that violates the BCNF condition:
A1, A2, … A n
Others
R1
B1, B2, … B m
A’s
B’s
R2
Example Decomposition
Name Social-security-number Age Eye Color Phone Number
Functional dependencies:
Name + Social-security-number
Age, Eye Color
What if we also had an attribute Draft-worthy, and the FD:
Age
Draft-worthy
Decomposition Based on BCNF
is Necessarily Correct
Attributes A, B, C.
FD: A
Relations R1[A,B]
R2[A,C]
C
Tuples in R1: (a,b)
Tuples in R2: (a,c),
(a,d)
Tuples in the join of R1 and R2: (a,b,c), (a,b,d)
Can (a,b,d) be a bogus tuple?
Multivalued Dependencies
Name
Fred
Fred
Fred
Fred
SSN
Phone Number
123-321-99
123-321-99
123-321-99
123-321-99
(206)
(206)
(206)
(206)
572-4312
572-4312
432-8954
432-8954
The multivalued dependencies are:
Name, SSN
Name, SSN
Phone Number
Course
4th Normal form: replace FD by MVD.
Course
CSE-444
CSE-341
CSE-444
CSE-341