Functional dependency and normalization

Functional
Dependencies
prepared by Visakh V,Assistant
Professor,LBSITW

There are two levels at which we discuss the goodness of
relation schema
•Logical (or Conceptual)Level
•Implementation (or storage ) Level
Professor,LBSITW

Professor,LBSITW

4 Informal measures of quality for
relation schemas are
I. Semantics of the attribute
• interpretation of attribute values
from tuples
• The attribute belonging to one relation
have certain real-world meaning and a
proper interpretation associated with
them
Professor,LBSITW

ii. Reducing redundant information in tuples
• minimize the storage space used by the base
relation
• Problem of update anomalies
• Insertion of anomalies
• Deletion of anomalies
• Modification of anomalies
iii. Reducing the NULL values in the tuples
• Waste space at storage levels
• Also lead to problems with understanding
the meaning of attributes (ex:count,join )
iv. Disallowing the possibility of generating
spurious tuples
• Seems to be genuine but it is falseprepared by Visakh V,Assistant
Professor,LBSITW

• Most important concept in relational schema design
theory.
• A functional dependency is a constraint between
two set of attributes from the database.
• it is denoted by X  Y(functional dependency
from x to y or y is functionally dependent on X.)
• i.e. Y component of a tuple in r depend on , or are
determined by , the values of the Y component.
•A functional dependency is a property of the of the
semantics or meaning .prepared by Visakh V,Assistant
Professor,LBSITW

Inference ???
•Dept_no  Mgr_SSN , each department has one manager
• Mgr_SSN  Mgr_phone : Each manager has a unique phone number
• then we can infer Dept_no  Mgr_phone
Professor,LBSITW

Closure : includes all dependencies that can
be inferred from the given set F, it is denoted
by F
closure ???
+
Let us see some examples on board
Professor,LBSITW

The notation F XY means the functional
dependency XY is inferred from functional
dependencies F.
To determine a systematic way to infer dependencies,
we use inference rules that can be used to infer new
dependencies
Professor,LBSITW

Inference rules for functional dependencies are :
• IR1 (reflexive rule) : If Y subset-of X,
then X  Y
•IR2 (augmentation rule) : If X -> Y, then XZ -> YZ
•IR3 (transitive rule) :If X Y & Y Z, then
X  Z
•IR4 (decomposition ,or projective rule) : If X YZ then X Y
•IR5 (Union or additive rule) : If XY,XZ then XYZ
•IR6 (pseudo transitive rule) : If XY,WYZ then
WX  Z
Trivial
otherwise
Non-Trivial
Professor,LBSITW

Inference rules IR1 through IR3
are sound and complete
Dependency that we
can infer from F by
using IR1 to IR3 holds
every relation state r
of R that satisfies the
dependencies in F.
Using IR1to IR3
repeatedly to infer
dependencies until no
more dependencies
can be inferred from F
The closure of F , can be determined from F by using
inference rules IR1 to IR3 are known as Armstrong ‘s
inference rules.
Professor,LBSITW

Cover : A set of functional dependencies F is said
to cover another set of dependencies E if every
FD in E is also in F .
Or we can say that E is covered by F.
+
Two sets of functional dependencies E and F are
equivalent if E = F .+ +
Professor,LBSITW

Minimal cover : minimal cover of a functional dependencies E
is a set functional dependencies F that satisfies the property
that every dependency in E is in the closure of F of F.+
•Every set of FDs has an equivalent minimal set
•There can be several equivalent minimal sets
•There is no simple algorithm for computing a minimal
set of FDs that is equivalent to a set F of FDs
•To synthesize a set of relations, we assume that we
start with a set of dependencies that is a minimal set
Professor,LBSITW

• Set F:= E
• Replace each functional dependencies X{A1,A2,. . . .,An}
in F by the n functional dependencies X A1,XA2,. ..
.,XAn.
• For each functional dependencies X A in
For each attribute B that is an element of X
• if {F – {XA}} U {(x-{B}) a} }
Algorithm
Finding a Minimal cover F for a set of functional dependencies E
CANONICAL formprepared by Visakh V,Assistant
Professor,LBSITW

Find minimum cover for E
{B A, DA,ABD}
{AD,BCA,BCD,CB,EA,ED}
{ AB -> C, C -> A, BC -> D, ACD -> B, D -> E, D ->
G, BE -> C, CG -> B, CG -> D, CE -> A, CE -> G}
Professor,LBSITW

{noname
noage
no,nameage
noage,name}
Professor,LBSITW

NORMAL
FORMS
Professor,LBSITW

•First proposed by Codd (1972)
•Initially proposed first ,second and
3NF
• later Boyce and Codd proposed
BCNF
• later 4th and 5th NF are proposed
based on the concept of multi-
valued dependency and join
dependencies
Professor,LBSITW

Advantages
•Minimizing redundancy
•Minimizing insertion , deletion, and
modification of anomalies
Normal form a relation refers to the highest
normal form condition that it meets, and hence
indicate the degree to which it has been
normalized prepared by Visakh V,Assistant
Professor,LBSITW

Non-prime attribute
A non-prime attribute is an attribute that does not occur in any
candidate key.
Prime attribute
A prime attribute, conversely, is an attribute that does occur in
some candidate key.
Professor,LBSITW

•Disallow multi valued attributes and composite attributes
• it states that domain of an attributes must include only
atomic (simple, indivisible)values.
•Ex: address
Professor,LBSITW

Partial key
Professor,LBSITW

Must be in 1NF
Professor,LBSITW

Full functional dependency : A functional dependency X Y is a
FULL FUNCTIONAL dependency if removal of any attribute A from X
means that dependency does not hold any more;
Partial dependency : A functional dependency X Y is a partial
dependency if some attribute A in X can be removed from X and
dependency still holds
Def : A relation schema R is in 2NF if every non
prime attribute A in R is fully functionally
dependent on the primary key of R
Professor,LBSITW

•it must be in 2NF
Def : A Relation Schema R is in 3nF ,
Whenever a non-trivial functional
dependency holds in R ,either
(a)X is a super key of R or (b) A is a
prime attribute of R
X A
Professor,LBSITW

X A If A is non prime attribute
then X must be super key
Professor,LBSITW

Def : A Relation Schema R is in BCNF ,
Whenever a non-trvial functional
dependency holds in R , X is a
superkey
X A
Professor,LBSITW

More stricter than 3NF
X A Always the left hand side
must be super key
whether A is prime or non
prime
Professor,LBSITW

Here {student, course  instructor}
{instructor course}
It is 3NF but not BCNF
Professor,LBSITW

In BCNF must check TWO conditions
• X Y allowed ,if it is trivial functional dependency
OR
• X is a super key for schema R
BCNF is
More stricter than 3NF
Professor,LBSITW

A 3NF table which does not have
multiple overlapping candidate keys
is guaranteed to be in BCNF
Ex: 3NF but not BCNF
Court Start Time End Time Rate Type
1 09:30 10:00 SAVER
1 11:00 12:00 SAVER
1 14:00 15:30 STANDARD
2 10:00 11:30 PREMIUM-B
2 11:30 13:30 PREMIUM-B
2 15:00 16:30 PREMIUM-A
2 9:30 10:00 PREMIUM-A
Professor,LBSITW

Here the candidate keys are
S1: {COURT,START TIME}
S2:{COURT,END TIME}
S3:{RATE TYPE,START TIME}
S4:{RATE TYPE, END TIME}
Here no non prime attributes , all are prime attribute
belongs to some candidate key.
So the table is 2NF and 3NF.but not in BCNF.prepared by Visakh V,Assistant
Professor,LBSITW

Here no non prime attributes , all are prime attribute
belongs to some candidate key.
So the table is 2NF and 3NF,but not in BCNF because of
rate type  court.
Professor,LBSITW

Today's Bookings
Rate Type Start Time End Time
SAVER 09:30 10:30
SAVER 11:00 12:00
STANDARD 14:00 15:30
PREMIUM-B 10:00 11:30
PREMIUM-B 11:30 13:30
PREMIUM-A 15:00 16:30
Rate Types
Rate Type Court Member Flag
SAVER 1 Yes
STANDARD 1 No
PREMIUM-A 2 Yes
PREMIUM-B 2 No
Now it is in BCNF
Professor,LBSITW

Problems when using BCNF
Person Shop Type Nearest Shop
Teena Optician Eagle Eye
Meenu Hairdresser Snippets
Priya Bookshop Merlin Books
Sreedevi Bakery Sree bakers
Sreedevi Hairdresser Sweeney
Sreedevi Optician Eagle Eye
Dependency:
A, B  C
C  B
Not BCNFprepared by Visakh V,Assistant
Professor,LBSITW

Shop Near Person
Person Shop
Teena Eagle Eye
Meenu Snippets
Priya Merlin Books
Sreedevi Sree Bakers
Sreedevi Sweeney
Sreedevi Eagle Eye
Shop
Shop Shop Type
Eagle Eye Optician
Snippets Hairdresser
Merlin Books Bookshop
Sree Bakers Bakery
Sweeney Hairdresser
Professor,LBSITW

Problem : It allow us to record data such as,a
person’s multiple shops with same type ,It
violates the dependency
{person,shoptype}{shop}
So BCNF is not always possible
Professor,LBSITW

(Lets do some problems)
Professor,LBSITW

MULTIVALUED DEPENDENCY
: A consequence of first normal form, Which disallows an attribute
have a set of values
: A multivalued dependency is a special case of a join dependency,
Course Book Lecturer
AHA Silberschatz John D
AHA Nederpelt William M
AHA Silberschatz William M
AHA Nederpelt John D
AHA Silberschatz Christian G
AHA Nederpelt Christian G
OSO Silberschatz John D
OSO Silberschatz William M
{course} {book}
and equivalently
{course}  {lecturer}.
Professor,LBSITW

Definition for 4th Normal Form
A relation schema R is in 4NF with respect to a set
of dependencies F (that include functional
dependencies and multivalued dependencies ),if for
every non trivial multivalued dependency X Y
in F closure,X is a super key of R.
A trivial multivalued dependency X  Y is one
where either Y is a subset of X, or X and Y together
form the whole set of attributes of the relation.
Professor,LBSITW

•In dependency theory, a join dependency is a
constraint on the set of legal relations over a
database scheme.
• A table T is subject to a join dependency if T can
always be recreated by joining multiple tables
each having a subset of the attributes of T.
• If one of the tables in the join has all the
attributes of the table T, the join dependency is
called trivial
Professor,LBSITW

•If JOIN dependency is present carry out a
multiway decomposition in to 5th Normal
form
•Such dependency is very peculiar semantic
constraint, that is very difficult to detect I
practice. So 5NF very rarely done in practice.
Professor,LBSITW

NON ADDITIVE (LOSELESS) JOIN DEPENDENCY
•Which ensures that no spurious tuples are generated
when a NATURAL JOIN operation is applied to the
relations in the decomposition
• lossless refers to loss information ,not to loss of
tuples.
Professor,LBSITW

Example of lossy decomposition
A B C
1 1 1
1 1 2
1 2 1
A B C
1 1 1
1 1 2
1 2 1
1 2 2
A B
1 1
1 2
A C
1 1
1 2
Original table Decomposition
Reconstruction
Professor,LBSITW

Functional dependency and normalization

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