Multi-valued Dependencies
Salman Azhar
Multi-valued Dependencies
Fourth Normal Form
These slides use some figures, definitions, and explanations from Elmasri-Navathe’s
Fundamentals of Database Systems
and Molina-Ullman-Widom’s Database Systems
1
A New Form of Redundancy
• Multi-valued dependencies (MVDs) express a
condition among tuples of a relation that exists
when the table (relation) is trying to represent
– more than one many-many relationship.
– then certain columns (attributes) become
independent of one another
– and their values must appear in all combinations.
2
Example
Customers(name, addr, phones, sodasLiked)
• A customer’s phones are independent of the sodas
they like.
• Many-many relations:
– Customers  Phones
Customers  Sodas
• Each phone appear with each soda in all
combinations.
– E.g., For 3 phones (Home, Work, Cell) and 10 sodas, we
need 30 tuples
• Repetition is unlike FD redundancy.
• There is only one FD: name  addr
3
Tuples Implied by Independence
If we have pink tuples, then
Then the blue tuples must also be in the relation
name addr
sue a
sue a
phones sodasLiked
p1
s1
p2
s2
sue
sue
p2
p1
a
a
s1
s2
name-phone and name-soda relations are independent:
The green relationship implies existence of blue relations
4
Definition of MVD
• A multi-valued dependency (MVD) X  Y :
– if an assertion that if two rows of a table agree on
all the attributes of X,
– then their components may be swapped in the set of
attributes Y,
– and the result will be two tuples that are also in the
relation.
5
Example
• The name-addr-phones-sodasLiked example
illustrated two MVDs
– name  phones
– name  sodasLiked.
6
Picture of MVD X  Y
X
Y
others
equal
exchange
We must exchange
all components of
Ys (not just some)
If there is an FD X  Y,
then swapping Ys
components doesn’t
change anything.
Other attributes
get copied
remain the same
Every FD is an MVD.
7
MVD Rules
• Every FD is an MVD.
– If X Y, then swapping Y ’s between two tuples that
agree on X doesn’t change the tuples.
– Therefore, the “new” tuples are surely in the relation,
and we know X Y.
• Complementation:
– If X Y, and Z is all the other attributes
(complement of X and Y) then X Z.
– Reason: Swapping Ys that agree on X is the same
effect as swapping Zs. 
8
Splitting Doesn’t Hold
• Recall in FDs we could split right side, but could NOT
split left side
– Recall FD A, B  C does not imply AC and BC
– Recall FD A B, C does imply AB and AC
• MVDs, we cannot generally split the left side
• BUT, we cannot split the right side EITHER!!!
– E.g., name  areaCode phone
• in (408) 555-9999 and (812)-555-1111, you can’t swap area
codes
– sometimes you have to deal with several attributes on the
right side.
9
Example
• Consider phone has been split into area code and
number and adding manf giving the following relation:
Customers(name, areaCode, phone, sodasLiked, manf)
• We can claim:
– A customer can have several phones,
• the number divided between areaCode and phone
– A customer can like several sodas,
• each with its own manufacturer.
– Phones are sodasLiked are independent, so we get
• all possible combinations of areaCode-phone pairs
& sodasLiked-manf pairs
10
Example, Continued
• Since the areaCode-phone pairs for a customer are
independent of the sodasLiked-manf pairs, we expect
the following MVDs:
name  areaCode phone
name  sodasLiked manf
• Observe that we can’t split the right hand sides:
– areaCode and phone are linked
– sodasLiked and manf are linked
• in (408) 555-9999 and (812)-555-1111, you can’t swap area codes!!!
11
Example Data
Here is possible data satisfying these MVDs:
name areaCode
phone
sodasLiked
Sue 812
555-1111
Pepsi
Sue 812
555-1111
Sprite
Sue 408
555-9999
Pepsi
Sue 408
555-9999
Sprite
Now all possible
combinations are
represented.
Swapping any pairs does
not yield new rows.
manf
PepsiCo
CocaCola
PepsiCo
CocaCola
We cannot swap area codes or phones by themselves.
Swapping (812) And (408) gives us incorrect phone numbers
Note: neither name  areaCode
nor name  phone holds for this relation.
12
Fourth Normal Form
• The redundancy caused by MVDs can’t be
removed by transforming the database schema
to BCNF.
• There is a stronger normal form, called 4th
Normal for (4NF), that (intuitively):
– treats MVDs as FDs when it comes to
decomposition
– but not when determining keys of the relation.
13
4NF Definition
•
A relation R is in 4NF if whenever X Y
is a nontrivial MVD, then X is a superkey.
–
Nontrivial means that:
1. Y is not a subset of X
(swapping components does not change tuples)
2. X and Y are not, together, all the attributes.
(swapping components yields the same tuples)
–
Note that the definition of “superkey” still
depends on FDs only.
14
BCNF Versus 4NF
• Remember that
– Every FD X  Y is also an MVD, X  Y.
• Thus, if R is in 4NF, it is certainly in BCNF.
– Because any BCNF violation is a 4NF violation.
• However, R could be in BCNF and not 4NF,
because MVDs are “invisible” to BCNF.
15
Decomposition and 4NF
•
If X  Y is a 4NF violation for relation R,
we can decompose R using the same
technique as for BCNF.
1. XY is one of the decomposed relations.
2. All but (Y – X) is the other.
16
Example
Customers(name, addr, phones, sodasLiked)
FD:
MVDs:
name  addr
name  phones
name  sodasLiked
• Key is {name, phones, sodasLiked}.
• All dependencies violate 4NF because name is not a
superkey.
17
Example, Continued
• Decompose using name  addr:
1. Customers1(name, addr)
 In 4NF, only dependency is name  addr.
2. Customers2(name, phones, sodasLiked)
 Not in 4NF because MVDs
 name  phones and name  sodasLiked apply.
 No FD’s, so all three attributes form the key.
18
Example: Decompose Customers2
• Recall
– Customers2(name, phones, sodasLiked)
• Either MVD
– name  phones
– name  sodasLiked
• tells us to decompose to:
– Customers3(name, phones)
– Customers4(name, sodasLiked)
19
Example: Decompose Customers2
• 4th Normal Form Consists of:
– Customers1(name, addr)
• Contact
– Customers3(name, phones)
• Phone
– Customers4(name, sodasLiked)
• Likes
• The following FD and MVDs are satisfied:
FD:
MVDs:
name  addr
name  phones
name  sodasLiked
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Continue Normal Forms
• 5th Normal Form
• 6th Normal Form
• 7th Normal Form
21
Salman Teaching 4th Normal Form
• How would I look teaching 937th Normal Form?
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Salman Teaching 957th Normal Form
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