Database Normalization Revisited: An information-theoretic approach Leonid Libkin Joint work with Marcelo Arenas and Solmaz Kolahi

Database Normalization Revisited:

An information-theoretic approach

Leonid Libkin

Joint work with Marcelo Arenas and Solmaz Kolahi

Sources

M. Arenas, L. An information-theoretic approach to normal forms for relational and XML data,

PODS’03; J.ACM, 2005. S. Kolahi, L. Redundancy vs dependency-

preservation in normalisation: an information-theoretic analysis of 3NF,

PODS’06.

Outline

Part 1 - Database Normalization from the 1970s and 1980s

Part 2: Classical theory re-done: new justification for normal forms:• BCNF and relatives (academic, eliminate

redundancies)

• 3NF (practical, may leave some redundancies)

Part 3: An XML application

2

If you haven’t taught “Intro to DB” lately…

Design: decide how to represent the information in a particular data model.

• Even for simple application domains there is a large number of ways of representing the data of interest.

We have to design the schema of the database.

• Set of relations.

• Set of attributes for each relation.

• Set of data dependencies.

3

Normalization Theory Today

Normalization theory for relational databases was developed in the 70s and 80s.

Why do we need normalization theory today?• New data models have emerged: XML.

• XML documents can contain redundant information.

Redundant information in XML documents:• Can be discovered if the user provides semantic

information.

• Can be eliminated.

15

Designing a Database: An Example

Attributes: number, title, section, room.

Data dependency: every course number is associated with only one title.

Relational Schema:

R(number, title, section, room),

number title

GOOD alternative:

S(number, title), number title

T(number, section, room), Ø

4

BAD alternative:

Problems with BAD:Redundancies and Update Anomalies

number title section room

CSC258 Computer Organization 1 LP266

CSC258 Computer Organization 2 GB258


CSC434 Database Systems 1 GB248

5

Deletion Anomaly


CSC258 Computer Organization I

1 LP266


2 GB258


3 GB248

CSC434 Database Systems 1 GB248CSC434 is not given in this term.

6

Deletion Anomaly



1 LP266


2 GB258


3 GB248

CSC434 Database Systems 1 GB248CSC434 is not given in this term.

6

Deletion Anomaly



1 LP266


2 GB258


3 GB248

CSC434 is not given in this term.

Additional effect: all the information about CSC434 was deleted.

6

Avoiding Update Anomalies

number

title

CSC258

Computer Organization

CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC258

3 GB248

CSC434

1 GB248

8


number

title

CSC258

Computer Organization I

CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC258

3 GB248

CSC434

1 GB248

The instance does not store redundant information.

8


number

title

CSC258


CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC258

3 GB248

CSC434

1 GB248CSC434 is not given in this term.

8


number

title

CSC258


CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC258

3 GB248

CSC434 is not given in this term.

The title of CSC434 is not removed from the instance.

8

Normalization Theory

Main idea: a normal form defines a condition that a well designed database should satisfy.

Normal form: syntactic condition on the database schema.• Defined for a class of data dependencies.

Main problems:

• How to test whether a database schema is in a particular normal form.

• How to transform a database schema into an equivalent one satisfying a particular normal form.

10

BCNF: a Normal Form for FDs

Functional dependency (FD) over R(A1, …, An) : X Y , X, Y {A1, …, An}.

X Y : two rows with the same X-values must have the same Y-values.• Number Title in our example

Key dependency : X A1 …. An

• X is a key: two distinct rows must have distinct X-values.

11

BCNF: a Normal Form for FDs

is a set of FD over R(A1, …, An).

Relation schema R(A1, …, An), is in BCNF if for every nontrivial X Y in , X is a key.

A relational schema is in BCNF if every relation schema is in BCNF.

In BCNF: S(number, title), number title

T(number, section, room),

Not in BCNF:


number title


T(number, section, room),



12

BCNF Decomposition

Relation schema: R(X,Y,Z), • Not in BCNF: implies X Y and but not X A, for

every A Z.

Basic decomposition: replace R(X,Y,Z) by S(X,Y) and T(X,Z).

Example:

13


number title

S(number, title), number title


Lossless Decomposition





number

title

CSC258


CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC434

1 GB248

∏number, title (R) ∏number, section, room (R)

14

Lossless Decomposition





number

title

CSC258


CSC434

Database Systems

number

section room

CSC258

1 LP266

CSC258

2 GB258

CSC434

1 GB248

S JoinT

14

How to justify good designs?

What is a good database design?

• Well-known solutions: BCNF, 4NF, 3NF…

But what is it that makes a database design good?

• Elimination of update anomalies.

• Existence of algorithms that produce good designs: lossless decomposition, dependency preservation.

34

Problems with traditional approaches

Many papers tried to justify normal forms. Problem: tied very closely to the relational

model. Relied on well-defined notions of

queries/updates.

These days we want to deal with other data models, in particular XML.

We need an approach that extends to other models, in particular, XML.

Justification of Normal Forms

Problematic to evaluate XML normal forms.

• No XML update language has been standardized.

• No XML query language yet has the same “yardstick” status as relational algebra.

• We do not even know if implication of XML FDs is decidable!

We need a different approach.

• It must be based on some intrinsic characteristics of the data.

• It must be applicable to new data models.

• It must be independent of query/update/constraint issues.

Our approach is based on information theory.

35

Information Theory

Entropy measures the amount of information provided by a certain event.

Assume that an event can have n different outcomes with probabilities p1, …, pn.

Amount of information gained by knowing that event i occurred :Average amount of information gained (entropy) :

Entropy is maximal if each pi = 1/n :

36

ip

1log

∑=

n

i ii p

p1

1log

nlog

Entropy and Redundancies

Database schema: R(A,B,C), A B

Instance I:

Pick a domain properly containing adom(I) :

• Probability distribution: P(4) = 0 and P(a) = 1/5, a ≠ 4

• Entropy: log 5 ≈ 2.322

A B C

1 2 3

1 2 4

A B C

1 2 3

1 2 4

A B C

1 2

1 2 4

A B C

1 2 3

1 2 4

A B C

1 3

1 2 4

Pick a domain properly containing adom(I) : {1, …, 6}

• Probability distribution: P(2) = 1 and P(a) = 0, a ≠ 2

• Entropy: log 1 = 0

{1, …, 6}

37

Entropy and Normal Forms

Let be a set of FDs over a schema S.

Theorem (S,) is in BCNF if and only if for every instance of (S,) and for every domain properly containing adom(I), each position carries non-zero amount of information (entropy > 0).

A similar result holds for 4NF and MVDs.

This is a clean characterization of BCNF and 4NF, but the measure is not accurate enough ...

38

Problems with the Measure

The measure cannot distinguish between different types of data dependencies.

It cannot distinguish between different instances of the same schema:

A B C

1 2 3

1 2 4

1 5

A B C

1 2 3

1 4

entropy = 0

R(A,B,C), A B

entropy = 0

39

A General Measure

Instance I of schema R(A,B,C), A B :

A B C

1 2 3

1 2 4

40

A General Measure


Initial setting: pick a position p Pos(I) and pick k such that adom(I) {1, …, k}. For example, k = 7.

A B C

1 2 3

1 2 4

40

A General Measure



A B C

1 2 3

1 2 4

40

A General Measure



A B C

1 3

1 2 4

Computation: for every X Pos(I) – {p}, compute probability distribution P(a | X), a {1, …, k}.

40

A General Measure


A B C

1 3

1 2 4


40

A General Measure


A B C

3

1 2


40

A General Measure


A B C

3

1 2


P(2 | X) =

40

A General Measure


A B C

2 3

1 2


P(2 | X) =

40

A General Measure


A B C

1 2 3

1 2 1


P(2 | X) =

40

A General Measure


A B C

4 2 3

1 2 7


P(2 | X) =

40

A General Measure


A B C

1 2 3

1 2 3


P(2 | X) = 48/

40

A General Measure


A B C

3

1 2


P(2 | X) = 48/

For a ≠ 2, P(a | X) =

40

A General Measure


A B C

a 3

1 2


P(2 | X) = 48/

For a ≠ 2, P(a | X) =

40

A General Measure


A B C

2 a 3

1 2 7


P(2 | X) = 48/

For a ≠ 2, P(a | X) =

40

A General Measure


A B C

1 a 3

1 2 6


P(2 | X) = 48/

For a ≠ 2, P(a | X) = 42

(48 + 6 * 42) = 0.16

/ (48 + 6 * 42) = 0.14

Entropy ≈ 2.8057 (log 7 ≈ 2.8073)

40

A General Measure


A B C

1 3

1 2 4

Value : we consider the average over all sets X Pos(I) – {p}.

•Average: 2.4558 < log 7 (maximal entropy)

•It corresponds to conditional entropy.

•It depends on the value of k ...40

A General Measure: Relative Information Content (RIC)

Previous value: RICIk(Σ|p)

For each k, we consider the ratio: RICIk(Σ|p) / log k

• How close the given position p is to having the maximum possible information content.

General measure (Arenas, L. 2003):

RICI(Σ|p) = limk ∞ RICIk(Σ|p) / log k

41

Basic Properties

The measure is well defined:

For every set of first order constraints Σ, every instance I of Σ, and every position p in I, RICI(Σ|p) exists.

Bounds: 0 ≤ RICI(Σ|p) ≤ 1

Closer to 1 = Less redundancy

42

Basic Properties

The measure does not depend on a particular representation of constraints.

It overcomes the limitations of the simple measure: R(A,B,C), A B

A B C

1 2 3

1 2 4

1 5

A B C

1 2 3

1 4

0.875 0.781

43

Well-Designed Databases

Definition A database specification (S,) is well-designed if for every I inst(S,) and every p Pos(I), RICI(Σ|p) = 1.

In other words, every position in every instance carries the maximum possible amount of information.

44

Relational Databases (Arenas, L.’03)

is a set of data dependencies over a schema S:

= Ø: (S,) is well-designed.

is a set of FDs: (S,) is well-designed if and only if (S,) is in BCNF.

is a set of FDs and MVDs: (S,) is well-designed if and only if (S,) is in 4NF.

is a set of FDs and JDs:

• If (S,) is in PJ/NF or in 5NFR, then (S,) is well-designed. The converse is not true.

• A syntactic characterization of being well-designed is given in [AL03].

45

Decidability Issues

If Σ is a set of First-Order integrity constraints, then the problem of verifying whether a relational schema is well-designed is undecidable.

If Σ contains only universal constraints (FDs, MVDs, JDs, …), then the problem becomes decidable.

High complexity (coNEXPTIME) by reduction to the (complement) of Bernays-Schönfinkel satisfiability.

46

3NF

BCNF is the most popular textbook normal form.

In practice 3NF is much more common.

From Oracle's “General Database Design FAQ”:

after defining 1NF, 2NF, and 3NF, it says: that there are other normal forms but “their

definitions are of academic concern only, and are rarely required for practical purposes”

Reminder: 3NF

A candidate key: a minimal (wrt subset) key A prime attribute: an attribute that belongs to

a candidate key.

BCNF: For a nontrivial FD X A, where A is an attribute, X must be a key.

3NF (Bernstein/Zaniolo): For a nontrivial FD X A, X must be a key OR A must be prime.

Why 3NF?

Because some relational schemas do not have decompositions that are both:

• In BCNF, and

• Preserve all functional dependencies

Example: ABC, AB C, C A On the other hand, there always exists a

lossless dependency preserving 3NF decomposition.

Redundancies vs Dependency-Preservation

To achieve complete elimination of redundancies (BCNF), one has to pay in terms of dependency preservation.

Losing constraints is often undesirable (database integrity must be enforced).

What is we only consider normal form that guarantee lossless dependency-preserving decomposition?

Which is best? Is it 3NF?

3NF: how low can one go?

Is there a lower bound for RICI(Σ|p) if Σ is in 3NF?

PROPOSITION (Kolahi DBPL 2005) For every ε > 0, there exists a 3NF schema Σ, an

instance I and a position p so that RICI(Σ|p) < ε.

BUT: I has many attributes (increasing with 1/ε) Σ can be further decomposed into better 3NF

designs using the standard synthesis algorithm.

How good is 3NF?

Let NF be a dependency-preserving normal form (guaranteeing lossless dependency-preserving decompositions) based on functional dependencies.

The guaranteed information content of NF is inf { c in [0,1] | for all schemas Σ, exists an NF-decomposition Σ1,…, Σm such that RI(Σi|p) ≥ c

for positions p in all instances I of Σi}

PRICE(NF) = 1 – Guaranteed Information Content(NF)

Why 3NF? -- Answer

PRICE(NF): the smallest amount of information content loss one needs to tolerate to achieve dependency-preservation.

PRICE(NF) > 0 (BCNF isn’t dependency-preserving)

PRICE(NF) is lower ==> NF is better.

THEOREM (Kolahi, L.)

• PRICE(3NF) = ½.

• PRICE(NF) ≥ ½ for every other dependency- preserving NF.

Why is PRICE(3NF)=1/2?

We said earlier that RICI(Σ|p) could be below any given ε > 0.

But those schemas can are “bad” 3NFs that can be further decomposed into “good” 3NFs, and for “good” 3NFs we guarantee PRICE=1/2.

“Good” 3NF = 3NF schemas produced by the standard synthesis algorithm.

So the result justifies not only 3NF but also the algorithm that is most commonly used to produce 3NF designs.

Comparing normal forms

We can use the information-theoretic measure to compare normal forms.

Define, for a condition P, the set of possible values of the information-theoretic measure:

POSSP(m) = { RICI(Σ|p) | I has m attributes,

Σ satisfies P } Define the GAIN function: inf POSSP1(m)

GAINP1/P2 (m) = ------------------

inf POSSP2(m)

Comparison of normal forms

THEOREM (Kolahi, L.) For all m > 2:

•GAIN3NF/All (m) = 2

•GAIN”good” 3NF/All = 2m-2

•GAIN”good” 3NF/3NF = 2m-3

The measure extends beyond relations

It can be used to reason about designs in other datamodels:• Nested relational

• XML

In particular it can be used to justify a normal form proposed recently for XML:• Called XNF (Arenas, L., 2002)

• Generalizes BCNF to XML documents

XML Databases

XML schema: (D,).

• D is a DTD.

• is a set of data dependencies over D.

We would like to evaluate XML normal forms.

The notion of being well-designed extends from relations to XML.

• The measure is robust; we just need to define the set of positions in an XML tree T: Pos(T).

47

Positions in an XML Tree

DBLP

conf conf

title issueissue

article articlearticle

@yeartitle title @year

“ICDT”

author @yeartitleauthor“1999” “1999”“Dong” “2001”“Jarke”“. . .” “. . .” “. . .”

“ICDT”

“1999” “1999”“Dong” “2001”“Jarke”“. . .” “. . .” “. . .”

48

XML normalization

DBLP

conf conf

title issueissue

article articlearticle

@yeartitle title @year

@year

“ICDT”

@year

author @yeartitleauthor“1999”

“1999”

“1999”“Dong” “2001”“Jarke”

“2001”

“. . .” “. . .” “. . .”

20

XNF: an XML normal form

XNF is achieved by repeated transformations of two kinds:

•As above in the DBLP example, and

•Splitting multiple attributes of the same element type in the same manner as in the case of BCNF normalization for relations.

There is also a formal definition which is a natural analog of BCNF in the XML context.

Well-Designed XML Data

We consider k such that adom(T) {1, …,k}.

For each k : RICTk(Σ|p)

We consider the ratio: RICTk(Σ|p) / log k

General measure: RICT(Σ|p) = limk ∞ RICT

k(Σ|p) / log k

49

XNF: XML Normal Form

For arbitrary XML data dependencies:

Definition An XML specification (D,) is well-designed if for every T inst(D,) and every p Pos(T), RICT(Σ|p) = 1.

For functional dependencies:

Theorem An XML specification (D,) is in XNF if and only if (D,) is well-designed.

50

Future Work

What is an analog of 3NF for XML? We would like to develop better

characterizations of normalization algorithms using our measure.

• Why is the “usual” BCNF decomposition algorithm good?

• Why does it always stop?

What else can this measure be used for? What about nonuniform distributions?

• Are they meaningful here?

• If so, how do the results change?

52

Documents

Database Normalization Revisited: An information-theoretic approach Leonid Libkin Joint work with Marcelo Arenas and Solmaz Kolahi