This document discusses schema refinement in database design, focusing on functional dependencies and their role in identifying redundancies and anomalies in designs. It contrasts poor design examples with more efficient relational schemas and introduces concepts such as candidate keys and Armstrong's axioms. The lecture sets the stage for understanding normalization and how to ensure relations are in 'good form' while avoiding loss of information.

1. 1
Lecture 6:
Schema refinement: Functional
dependencies
www.cl.cam.ac.uk/Teaching/current/Databases/

2. 2
Recall: Database design
lifecycle
• Requirements analysis
– User needs; what must database do?
• Conceptual design
– High-level description; often using E/R model
• Logical design
– Translate E/R model into relational schema
• Schema refinement
– Check schema for redundancies and anomalies
• Physical design/tuning
– Consider typical workloads, and further optimise
Next
two
lectures

3. 3
Today’s lecture
• Why are some designs bad?
• What’s a functional dependency?
• What’s the theory of functional
dependencies?
• (Next lecture: How can we use this theory
to classify redundancy in relation design?)

4. 4
Not all designs are
equally good
• Why is this design bad?
Data(sid,sname,address,cid,cname,grade)
• Why is this one preferable?
Student(sid,sname,address)
Course(cid,cname)
Enrolled(sid,cid,grade)

5. 5
An instance of our bad
design
sid sname addres
s
ci
d
cname grade
124 Britney USA 206 Database A++
204 Victoria Essex 202 Semantics C
124 Britney USA 201 S/Eng I A+
206 Emma London 206 Database B-
124 Britney USA 202 Semantics B+

6. 6
Evils of redundancy
• Redundancy is the root of many problems
associated with relational schemas
– Redundant storage
– Update anomalies
– Insertion anomalies
– Deletion anomalies
– LOW TRANSACTION THROUGHPUT
• In general, with higher redundancy, if
transactions are correct (no anomalies),
then they have to lock more objects thus
causing greater contention and lower
throughput

7. 7
Decomposition
• We remove anomalies by replacing the schema
Data(sid,sname,address,cid,cname,grade)
with
Student(sid,sname,address)
Course(cid,cname)
Enrolled(sid,cid,grade)
• Note the implicit extra cost here
• Two immediate questions:
1. Do we need to decompose a relation?
2. What problems might result from a decomposition?

8. 8
Functional dependencies
• Recall:
– A key is a set of fields where if a pair of tuples
agree on a key, they agree everywhere
• In our bad design, if two tuples agree on
sid, then they also agree on address,
even though the rest of the tuples may not
agree

9. 9
Functional dependencies
cont.
• We can say that sid determines address
– We’ll write this
sid  address
• This is called a functional dependency
(FD)
• (Note: An FD is just another integrity
constraint)

10. 10
Functional dependencies
cont.
• We’d expect the following functional
dependencies to hold in our Student
database
– sid  sname,address
– cid  cname
– sid,cid  grade
• A functional dependency X  Y is simply a
pair of sets (of field names)
– Note: the sloppy notation A,B  C,D rather
than {A,B}  {C,D}

11. 11
Formalities
• Given a relation R=R(A1:1, …, An:n), and
X, Y ({A1, …, An}), an instance r of R
satisfies XY, if
– For any two tuples t1, t2 in R, if t1.X=t2.X then
t1.Y=t2.Y
• Note: This is a semantic assertion. We
can not look at an instance to determine
which FDs hold (although we can tell if the
instance does not satisfy an FD!)

12. 12
Properties of FDs
• Assume that X  Y and Y  Z are known to
hold in R. It’s clear that X  Z holds too.
• We shall say that an FD set F logically implies
X  Y, and write F [X  Y
– e.g. {X  Y, Y  Z} [ X  Z
• The closure of F is the set of all FDs logically
implied by F, i.e.
F+ @ {XY | F [ XY}
• The set F+ can be big, even if F is small 

13. 13
Closure of a set of FDs
• Which of the following are in the closure of
our Student FDs?
– addressaddress
– cidcname
– cidcname,sname
– cid,sidcname,sname

14. 14
Candidate keys and FDs
• If R=R(A1:1, …, An:n) with FDs F and
X{A1, …, An}, then X is a candidate key
for R if
– X  A1, …,An  F+
– For no proper subset YX is
Y  A1, …,An  F+

15. 15
Armstrong’s axioms
• Reflexivity: If YX then F XY
– (This is called a trivial dependency)
– Example: sname,addressaddress
• Augmentation: If F XY then
F X,WY,W
– Example: As cidcname then
cid,sidcname,sid
• Transitivity: If F XY and F YZ then
F XZ
– Example: As sid,cidcid and
cidcname, then sid,cidcname

16. 16
Consequences of
Armstrong’s axioms
• Union: If F XY and F XZ then
F XY,Z
• Pseudo-transitivity: If F XY and
F W,YZ then F X,WZ
• Decomposition: If F XY and ZY then
F XZ
Exercise: Prove that these are
consequences of Armstrong’s axioms

17. 17
Proof of Union Rule
Suppose that F XY and F XZ.
By augmentation we have
F XX,Y
since X U X = X. Also by augmentation
F X,YZ,Y
Therefore, by transitivity we have
F XZ,Y
QED

18. 18
Functional Dependencies Can be
useful in Algebraic Reasoning
Suppose R(A,B,C) is a relation schema
with dependency AB, then
)
(
, R
R B
A

 )
(
, R
C
A

A
(This is called Heath’s rule.)

19. 19
Proof of Heath’s Rule
)
(
, R
C
A

A
First show that
Suppose
then
and
Since
we have )
(
, R
C
A

A

20. 20
Proof of Heath’s Rule (cont.)
A
In the other direction, we must show that
Suppose Then there must exist records
and
There must also exist
Therefore, we have
so that
QED
But the functional dependency tells us that

21. 21
Equivalence
• Two sets of FDs, F and G, are said to be
equivalent if F+=G+
• For example:
{(A,BC), (AB)} and
{(AC), (AB)}
are equivalent
• F+ can be huge – we’d prefer to look for
small equivalent FD sets

22. 22
Minimal cover
• An FD set, F, is said to be minimal if
1. Every FD in F is of the form XA, where A is
a single attribute
2. For no XA in F is F-{XA} equivalent to F
3. For no XA in F and ZX is
(F-{XA}){ZA} equivalent to F
• For example, {(AC), (AB)} is a
minimal cover for {(A,BC), (AB)}

23. 23
More on closures
• FACT: If F is an FD set, and XYF+ then
there exists an attribute AY such that
XAF+

24. 24
Why Armstrong’s axioms?
• Soundness
– If F XY is deduced using the rules, then
XY is true in any relation in which the
dependencies of F are true
• Completeness
– If XY is is true in any relation in which the
dependencies of F are true, then F XY can
be deduced using the rules

25. 25
Soundness
• Consider the Augmentation rule:
– We have XY, i.e. if t1.X=t2.X then t1.Y=t2.Y
– If in addition t1.W=t2.W then it is clear that
t1.(Y,W)=t2.(Y,W)

26. 26
Soundness cont.
Consider the Transitivity rule:
– We have XY, i.e. if t1.X=t2.X then t1.Y=t2.Y
(*)
– We have YZ, i.e. if t1.Y=t2.Y then t1.Z=t2.Z
(**)
– Take two tuples s1 and s2 such that s1.X=s2.X
then from (*) s1.Y=s2.Y and then from (**)
s1.Z=s2.Z

27. 27
Completeness
• Exercise
– (You may need the fact from slide 23)

28. 28
Attribute closure
• If we want to check whether XY is in a closure of the
set F, could compute F+ and check – but expensive 
• Cheaper: We can instead compute the attribute
closure, X+, using the following algorithm:
• Then F XY iff Y is a subset of X+
Try this with sid,snamecname,grade
closure:= X;
repeat until no change{
if UVF, where Uclosure
then closure:=closureV
};

29. 29
Preview of next lecture:
Goals of normalisation
• Decide whether a relation is in “good form”
• If it is not, then we will “decompose” it into
a set of relations such that
– Each relation is in “good form”
– The decomposition has not lost any
information that was present in the original
relation
• The theory of this process and the notion
of “good form” is based on FDs

30. 30
Summary
You should now understand:
• Redundancy and various forms of
anomalies
• Functional dependencies
• Armstrong’s axioms
Next lecture: Schema refinement:
Normalisation