Well-designed XML Data

1
Well-designed XML Data
Marcelo Arenas and Leonid Libkin University of
Toronto
2
Outline

• Part 1 - Database Normalization from the 1970s
  and 1980s.
• Part 2 - Classical theory revisited normalizing
  XML documents.
• Part 3 - Classical theory re-done new
  justifications for normalization.

2
3
Part 1 Classical Normalization

• 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
4
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
BAD alternative
GOOD alternative S(number, title), number ? title
T(number, section, room), ?
4
5
Problems with BAD Update Anomaly
number title section room
CSC258 Computer Organization 1 LP266
CSC258 Computer Organization 2 GB258
CSC258 Computer Organization 3 GB248
CSC434 Database Systems 1 GB248
Title of CSC258 is changed to Computer
Organization I.
5
6
Problems with BAD Update Anomaly
number title section room
CSC258 Computer Organization 1 LP266
CSC258 Computer Organization 2 GB258
CSC258 Computer Organization 3 GB248
CSC434 Database Systems 1 GB248
Title of CSC258 is changed to Computer
Organization I.
5
7
Problems with BAD Update Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
CSC434 Database Systems 1 GB248
Title of CSC258 is changed to Computer
Organization I.
The instance stores redundant information.
5
8
Deletion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
CSC434 Database Systems 1 GB248
CSC434 is not given in this term.
6
9
Deletion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
CSC434 Database Systems 1 GB248
CSC434 is not given in this term.
6
10
Deletion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
CSC434 is not given in this term.
Additional effect all the information about
CSC434 was deleted.
6
11
Insertion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
A new course is created (CSC336, Numerical
Methods)
7
12
Insertion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248

A new course is created (CSC336, Numerical
Methods)
7
13
Insertion Anomaly
number title section room
CSC258 Computer Organization I 1 LP266
CSC258 Computer Organization I 2 GB258
CSC258 Computer Organization I 3 GB248
CSC336 Numerical Methods ? ?
A new course is created (CSC336, Numerical
Methods)
The instance stores attributes that are not
directly related.
7
14
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
Title of CSC258 is changed to Computer
Organization I.
8
15
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
Title of CSC258 is changed to Computer
Organization I.
8
16
Avoiding Update Anomalies
number title
CSC258 Computer Organization I
CSC434 Database Systems
number section room
CSC258 1 LP266
CSC258 2 GB258
CSC258 3 GB248
CSC434 1 GB248
Title of CSC258 is changed to Computer
Organization I.
CSC434 is not given in this term.
The instance does not store redundant information.
8
17
Avoiding Update Anomalies
number title
CSC258 Computer Organization I
CSC434 Database Systems
number section room
CSC258 1 LP266
CSC258 2 GB258
CSC258 3 GB248
CSC434 1 GB248
CSC434 is not given in this term.
8
18
Avoiding Update Anomalies
number title
CSC258 Computer Organization I
CSC434 Database Systems
number section room
CSC258 1 LP266
CSC258 2 GB258
CSC258 3 GB248
CSC434 is not given in this term.
A new course is created (CSC336, Numerical
Methods)
The title of CSC434 is not removed from the
instance.
8
19
Avoiding Update Anomalies
number title
CSC258 Computer Organization I
CSC434 Database Systems

number section room
CSC258 1 LP266
CSC258 2 GB258
CSC258 3 GB248
A new course is created (CSC336, Numerical
Methods)
8
20
Avoiding Update Anomalies
number title
CSC258 Computer Organization I
CSC434 Database Systems
CSC336 Numerical Methods
number section room
CSC258 1 LP266
CSC258 2 GB258
CSC258 3 GB248
A new course is created (CSC336, Numerical
Methods)
No information about sections has to be provided.
Each relation stores attributes that are directly
related.
8
21
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.

9
22
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.

10
23
Part 2 XML and Normalization
XML Document
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
11
24
Part 2 XML and Normalization
XML Document
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
11
25
Part 2 XML and Normalization
XML Document
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
11
26
Part 2 XML and Normalization
XML Document
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
11
27
Part 2 XML and Normalization
XML Document
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
11
28
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
29
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
30
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt

lt/coursegt
ltcourse cnoCSC434gt

lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
31
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
32
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
33
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
34
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
35
Part 2 XML and Normalization
XML Document
DTD
ltcoursesgt
ltcourse cnoCSC258gt
lttaken_bygt
ltstudent snost1gt
ltnamegt Fox lt/namegt
ltgradegt B lt/gradegt
lt/studentgt
lt/taken_bygt
lt/coursegt
lt/coursesgt
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
name ? PCDATA
grade ? PCDATA
11
36
XML Databases
XML Schema (D, ?)
D
?
Two students with the same _at_sno value must have
the same name.
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
12
37
Redundancy in XML
courses
course
course
info
_at_cno
_at_cno
taken_by
taken_by
name
_at_sno
CSC258
CSC434
st1
Fox
student
student
student
. . .
_at_sno
name
grade
grade
name
_at_sno
st1
st1
A
B
Fox
Fox
13
38
XML Database Normalization
DTD
Data dependency
Two students with the same _at_sno value must have
the same name.
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? name, grade
14
39
XML Database Normalization
DTD
Data dependency
, info
_at_sno is the identifier of info elements.
courses ? course
course ? _at_cno
course ? taken_by
taken_by ? student
student ? _at_sno
student ? grade
Two students with the same _at_sno value must have
the same name.
info ? _at_sno
info ? name
14
40
A Non-relational Example
DBLP
conf
conf
title
issue
issue
ICDT
_at_year
article
article
article
_at_year
1999
2001

_at_year
title
title
_at_year
author
_at_year
title
author
1999
1999
Dong
2001
Jarke
. . .
. . .
. . .
15
41
XNF XML Normal Form

• It eliminates two types of anomalies.
• It was defined for XML functional dependencies
• DBLP.conf._at_title ? DBLP.conf
• DBLP.conf.issue ? DBLP.conf.issue.article._at_year
  

16
42
Problems to Address

• Functional dependencies for XML.
• Normal form for XML documents (XNF).
• Generalizes BCNF.
• Algorithm for normalizing XML documents.
• Implication problem for functional dependencies.

17
43
Framework Paths in DTDs

• Paths(D) all paths in a DTD D
• courses.course
• courses.course._at_cno
• courses.course.student.name
• courses.course.student.name.S
• We distinguish three kinds of elements
  attributes (_at_), strings (S) and element types.
• FDs are defined by means of a relational
  representation of XML documents.

18
44
Framework XML Trees
courses
v0
course
course
v1
. . .
_at_cno
student
student
v2
v5
cs100
_at_sno
grade
_at_sno
grade
name
name
v3
v4
v6
v7
123
456
S
S
S
S
Fox
B
Smith
A-
19
45
Tree Tuples

• Relational representation tree tuples - mappings
• t Paths(D) ? Vertices ? Strings ? ?
• A tree tuple represents an XML tree

courses
t(courses) v0 t(courses.course)
v1 t(courses.course._at_cno) cs100 t(courses.cour
se.student) v2 t(p) ?, for the remaining paths
v0
course
v1
_at_cno
student
v2
cs100
20
46
XML Tree set of Tree Tuples
courses
courses
v0
v0
course
course
course
course
v1
. . .
v1
. . .
_at_cno
student
_at_cno
student
student
student
v2
v5
v2
v5
cs100
cs100
_at_sno
grade
_at_sno
grade
_at_sno
grade
_at_sno
grade
name
name
name
name
v3
v4
v6
v7
v3
v4
v6
v7
123
456
123
456
S
S
S
S
S
S
S
S
Fox
B
Smith
A-
Fox
B
Smith
A-
21
47
Functional Dependencies for XML

• Expressions of the form
• X ? Y
• defined over a DTD D, where X, Y are finite
• non-empty subsets of Paths(D).
• XML tree T can be tested for satisfaction of X ?
  Y if
• X ? Y ? Paths(T) ? Paths(D)
• T ? X ? Y if for every pair u, v of tree tuples
  in T
• u.X v.X and u.X ? ? implies u.Y v.Y

22
48
FD Examples

• University DTD courses ? course
• course ? _at_cno, student
• student ? _at_sno, name, grade
• Two students with the same _at_sno value must have
  the same name
• courses.course.student._at_sno ? courses.course.stud
  ent.name.S
• Every student can have at most one grade in every
  course
• courses.course,
• courses.course.student._at_sno ?
  courses.course.student.grade.S

23
49
Implication Problem for FD

• Given a DTD D and a set of functional
  dependencies ? ? ?
• (D, ?) ? ? if for any XML tree T conforming to D
  and satisfying ? , it is the case that T ? ?
• (D, ?) ? (D, ?) ? ?
• Functional dependency ? is trivial if it is
  implied by the DTD alone (D, ?) ? ?

24
50
XNF XML Normal Form

• XML specification a DTD D and a set of
  functional dependencies ?.
• A Relational DB is in BCNF if for every
  non-trivial functional dependency X ? Y in the
  specification, X is a key.
• (D, ?) is in XNF if
• For each non-trivial FD X ? p._at_l or X ? p.S in
  (D, ?), X ? p is in (D, ?).

25
51
Back to DBLP

• DBLP is not in XNF
• DBLP.conf.issue ? DBLP.conf.issue.article._at_year
  ? (D,?)
• DBLP.conf.issue ?
• DBLP.conf.issue.article ? (D,?)
• Proposed solution is in XNF.

26
52
Normalization Algorithm

• The algorithm applies two transformations until
  the
• schema is in XNF.
• If there is an anomalous FD of the form
• DBLP.conf.issue ? DBLP.conf.issue.article._at_year
• then apply the DBLP example rule.
• Otherwise choose a minimal anomalous FD and
  apply the University example rule.

27
53
Normalizing XML Documents

• Theorem The decomposition algorithm terminates
  and outputs a specification in XNF.
• Furthermore, it does not lose information
• Unnormalized Normalized
• XML document XML Document
• Q1, Q2 are XQuery core queries.

Q1
Q2
28
54
Part 3 What was Missing? Justification!

• What is a good database design?
• Well-known solutions BCNF, 4NF,
• 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.
• Previous work was specific for the relational
  model.
• Classical problems have to be revisited in the
  XML context.

29
55
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.

30
56
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
31
57
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
32
58
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 gt
  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 ...

33
59
Problems with the Measure

• The measure cannot distinguish between different
  types of data dependencies.
• It cannot distinguish between different instances
  of the same schema

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
entropy 0
entropy 0
34
60
A General Measure
A B C
1 2 3
1 2 4
Instance I of schema R(A,B,C), A ? B
35
61
A General Measure
A B C
1 2 3
1 2 4
Instance I of schema R(A,B,C), A ? B
Initial setting pick a position p ? Pos(I) and
pick k such that adom(I) ? 1, , k. For
example, k 7.
35
62
A General Measure
A B C
1 2 3
1 2 4
Instance I of schema R(A,B,C), A ? B
Initial setting pick a position p ? Pos(I) and
pick k such that adom(I) ? 1, , k. For
example, k 7.
35
63
A General Measure
A B C
1 3
1 2 4
Instance I of schema R(A,B,C), A ? B
Initial setting pick a position p ? Pos(I) and
pick k such that adom(I) ? 1, , k. For
example, k 7.
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
35
64
A General Measure
A B C
1 3
1 2 4
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
35
65
A General Measure
A B C
3
1 2
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
35
66
A General Measure
A B C
3
1 2
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
35
67
A General Measure
A B C
2 3
1 2
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
35
68
A General Measure
A B C
1 2 3
1 2 1
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
35
69
A General Measure
A B C
4 2 3
1 2 7
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
35
70
A General Measure
A B C
1 2 3
1 2 3
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
48/
35
71
A General Measure
A B C
3
1 2
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
48/
For a ? 2, P(a X)
35
72
A General Measure
A B C
a 3
1 2
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
48/
For a ? 2, P(a X)
35
73
A General Measure
A B C
2 a 3
1 2 7
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
48/
For a ? 2, P(a X)
35
74
A General Measure
A B C
1 a 3
1 2 6
Instance I of schema R(A,B,C), A ? B
Computation for every X ? Pos(I) p, compute
probability distribution P(a X), a ? 1, , k.
P(2 X)
48/
(48 6 ? 42) 0.16
For a ? 2, P(a X)
42/
(48 6 ? 42) 0.14
Entropy 2.8057
(log 7 2.8073)
35
75
A General Measure
A B C
1 3
1 2 4
Instance I of schema R(A,B,C), A ? B

• Value we consider the average over all sets
  X ? Pos(I) p.
• Average 2.4558 lt log 7 (maximal entropy)
• It corresponds to conditional entropy.
• It depends on the value of k ...

35
76
A General Measure

• Previous value
• For each k, we consider the ratio
• How close the given position p is to having the
  maximum possible information content.
• General measure

36
77
Basic Properties

• The measure is well defined
• For every set of firstorder constraints ?
  defined over a schema S, every I ? inst(S,?), and
  every p ? Pos(I) exists.
• Bounds

37
78
Basic Properties

• The measure does not depend on a particular
  representation of constraints. If ?1 and ?2 are
  equivalent
• 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
38
79
Well-Designed Databases

• Definition A database specification (S,?) is
  well-designed if for every I ? inst(S,?) and
  every p ? Pos(I), 1.
• In other words, every position in every instance
  carries the maximum possible amount of
  information.
• We would like to test this definition in the
  relational world ...

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80
Relational Databases

• ? is a set of data dependencies over a schema S
• ? ? (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.

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Relational Databases

• The problem of verifying whether a relational
  schema is well-designed is undecidable.
• If the schema contains only universal constraints
  (FDs, MVDs, JDs, ), then the problem becomes
  decidable.
• Now we would like to apply our definition in the
  XML world ...

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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).

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Positions in an XML Tree
DBLP
conf
conf
title
issue
issue
ICDT
ICDT
article
article
article

_at_year
title
title
_at_year
author
_at_year
title
author
1999
1999
Dong
2001
Jarke
. . .
. . .
. . .
1999
1999
Dong
2001
Jarke
. . .
. . .
. . .
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84
Well-Designed XML Data

• We consider k such that adom(T) ? 1, ,k.
• For each k
• We consider the ratio
• General measure

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85
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),
  1.
• For functional dependencies
• Theorem An XML specification (D,?) is in XNF if
  and only if (D,?) is well-designed.

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86
Normalization Algorithms

• The information-theoretic measure can also be
  used for reasoning about normalization
  algorithms.
• For BCNF and XNF decomposition algorithms
• Theorem After each step of these decomposition
  algorithms, the amount of information in each
  position does not decrease.

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87
Future Work

• We would like to consider more complex XML
  constraints and characterize good designs they
  give rise to.
• We would like to characterize 3NF by using the
  measure developed in this paper.
• In general, we would like to characterize
  non-perfect normal forms.
• 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?

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