relative information capacity of simple relational database schemata paper by: richard hull...

Relative Information Capacity of Simple Relational Database Schemata

Paper by: Richard HullPresented by: Jose Picado

Outline

• Problem: Data relativism and information capacity– Definition– Examples– Importance

• Hierarchy of dominance measures• Basic results• Discussion

Data relativism

• Represent the same data in different ways

Data relativism

• Represent the same data in different ways• Represent the same data under different

schemas

Data relativism

• Represent the same data in different ways• Represent the same data under different

schemas

Person

name sex spouseSchema 1

Example taken from: Kosky, Anhony. Transforming Databases with Recursive Data Structures, 1996.

Data relativism

• Represent the same data in different ways• Represent the same data under different

schemas

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

Schema 1

Schema 2

Example taken from: Kosky, Anhony. Transforming Databases with Recursive Data Sturctures, 1996.

Relative information capacity

• Expressiveness of a schema• Different schemas representing same data

may have different information capacity

Relative information capacity

• Expressiveness of a schema• Different schemas representing same data

may have different information capacity

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

Schema 1

Schema 2

Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

Relative information capacity

• Expressiveness of a schema• Different schemas representing same data

may have different information capacity

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

Schema 1:• Does not require that the

spouse attribute of a man goes to a woman.

• Does not require that for each spouse attribute in one direction there is a corresponding spouse attribute in another direction.

Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

Relative information capacity

• Expressiveness of a schema• Different schemas representing same data

may have different information capacity

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

Schema 2:• Allows unmarried people to

be represented in the database.

Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

Relative information capacity

• Possible solution: – Transform existing schema to new schema by

structural manipulations

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

transformation

Relative information capacity

• Possible solution: – Transform existing schema to new schema by

structural manipulations– Information capacity preserving?

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

transformation

Importance

• Schema evolution– None of the information stored in the initial

database is lost

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

Importance• Data integration– All information in one of the component

databases is reflected in the integrated database

City

name state

State

name capital

City

name isCapital country

Country

name language currency

City

name place

Country

name language currency capital

State

name capital

Example taken from: Kosky, Anthony. Transforming Databases with Recursive Data Structures, 1996.

Importance

• Database normalization theory• User view construction• Schema simplification• Translation between data models

Hull’s paper

• Introduces theoretical tools for studying measures of relative information capacity– Theoretical frameworks at the time were complex– There was no clear definition about the concept– Hull introduced nice ways of comparing schemata

and their information capacity• Defines a hierarchy of measures to compare

information capacity of schemata

Hull’s paper

• Gives some basic results concerning the previous measures

• Considers only non-keyed relations

Person

id name

Person

id name

123 John

123 Mary

123 John

123 Mary

Non-keyed Keyed

Instances:

Relations:

Definitions

• Schema P is a set of relations• Relations composed of attributes, which may

be of different basic types• Basic types are domain designators (have a

fixed domain of possible values)• I(P) is the instances of P, usually infinite

Person

id name

111 John

222 Mary123 Anne

234 Joe

aaa Jack

bbb Ted

Schema P Instances I(P)

…

Transformation

• P and Q are relational schemata• A transformation from P to Q is a map

Transformation

• P and Q are relational schemata• A transformation from P to Q is a map

PPerson

id nameBirth

id date

Transformation

• P and Q are relational schemata• A transformation from P to Q is a map

P

QPersonInfo

id name bdate

Person

id nameBirth

id date

Transformation

• P and Q are relational schemata• A transformation from P to Q is a map

P

QPersonInfo

id name bdate

Person

id nameBirth

id date

PersonInfo(x,y,z) :- Person(x,y), Birth(x,z).

Dominance

• P and Q are relational schemata

• Q dominates P via if the composition of followed by is the identity on P

Dominance

Person

name sex spouse

Female

name

Male

name

Marriage

husband wife

P

Q

Dominance

1. Take instances of P: I(P)

Person

John male Mary

Mary female John

Anne female Joe

Joe male Anne

Dominance

2. Apply to I(P) Male(x) :- Person(x,y,z), y=“male”.Female(x) :- Person(x,y,z), y=“female”.Marriage(x,y) :- Person(x,u,y), Person(y,v,x), u=“male”, v=“female”

Male

John

Joe

Female

Mary

Anne

Marriage

John Mary

Joe Anne

Person

John male Mary

Mary female John

Anne female Joe

Joe male Anne

Dominance

3. Apply to (I(P))

Person(x,”male”,z) :- Male(x), Marriage(x,z).Person(x,”female”,z) :- Female(x), Marriage(x,z).

Male

John

Joe

Female

Mary

Anne

Marriage

John Mary

Joe Anne

Person

John male Mary

Mary female John

Anne female Joe

Joe male Anne

( (I(P)))

Dominance

4. Compare I(P) and ( (I(P)))

Person

John male Mary

Mary female John

Anne female Joe

Joe male Anne

Person

John male Mary

Mary female John

Anne female Joe

Joe male Anne

I(P)

Dominance

• P and Q are relational schemata

• Q dominates P via if the composition of followed by is the identity on P

Q has at least as much capacity for storing information as P

Information structured according to P can be restructured to “fit” into Q, and restructured again to “fit” into P

Equivalence

• P and Q are equivalent (xxx) if they have equivalent information capacity

• P and Q are equivalent if – Q dominates P (xxx) and – P dominates Q (xxx)

Information dominance measures

1. Calculous dominance2. Generic dominance3. Internal dominance4. Absolute dominance

More restrictive

Less restrictive

Types of equivalency

1. P and Q are equivalent (calc)2. P and Q are equivalent (gen)3. P and Q are equivalent (int)4. P and Q are equivalent (abs)

More restrictive

Less restrictive

Level 1: Calculous dominance

• Only allow transformations to be relational calculus expressions

• Relational calculus:– First order logic or predicate calculus– Predicates: atom,

– Each query Q(x1, …, xn) is a predicate P

Level 1: Calculous dominance

• Only allow transformations to be relational calculus expressions

• are relational calculus expressions

• Q dominates P calculously

Level 2: Generic dominance

• Only allow transformations that treat domain elements as “essentially uninterpreted objects”

• Treat all elements as equals except some set of constants

• Property of all query languages, such as SQL and Datalog

Level 2: Generic dominance

• Only allow transformations that treat domain elements as “essentially uninterpreted objects”

• treat all elements as equals

• Q dominates P generically

Level 3: Internal dominance

• Only allow transformations that do not invent any data

• Invent data: numerical computations or string manipulations

player goals games player performance

performance = goals/games

Level 3: Internal dominance

• Only allow transformations that do not invent any data

• do not invent data• Q dominates P internally

Level 4: Absolute dominance

• Some set of values • : instances of P that contain only values

in Y, where• : cardinality of instances of P containing

only values in Y• If then

Q dominates P absolutely• Easy to compute: based on counting of

instances, instead of transformations

Basic results

• Q dominates P calculously

Q dominates P generically

Q dominates P internally

Q dominates P absolutely

Basic results

• Sometimes absolute and internal dominance hold, but generic and calculous dominance don’t

A A

B B

A B

Q

PQ dominates P (abs, int)• and transformation (int)

does not invent data

Q does not dominate P (gen, calc)• There is no transformation (gen, calc) that

takes instances of P to Q and then back to P

Basic results

• Absolute dominance useful for verifying calculous (not) dominance

A B

A C

A B C

Q

P• Q dominates P calculously

Q dominates P absolutely

• P does not dominate Q absolutelyP does not dominates Q

calculously*under certain constraints

Basic results

• Dominance is preserved by re-namings of basic types (homomorphism)– h(P): homomorphism of P– If Q dominates P then

h(Q) dominates h(P)for any measure of dominance (calc, gen, int, abs)

Basic results

• Calculous dominance does not accurately measure the presence of “semantic correspondence”

Basic results

• Calculous dominance does not accurately measure the presence of “semantic correspondence”

name position goalsname goals minutes S1R1

NAME NUMBER NUMBER NAME NAME NUMBER

title publisher pagestitle pages edition S2R2P

Basic results

• Calculous dominance does not accurately measure the presence of “semantic correspondence”

NAME NAME NUMBER NUMBERT

P

Q

name position goalsname goals minutes S1R1

NAME NUMBER NUMBER NAME NAME NUMBER

title publisher pagestitle pages edition S2R2

Basic results

• Calculous dominance does not accurately measure the presence of “semantic correspondence”

NAME NAME NUMBER NUMBERT

P

Q

Q dominates P (calc), but there is not semantic mapping from P to Q

name position goalsname goals minutes S1R1

NAME NUMBER NUMBER NAME NAME NUMBER

title publisher pagestitle pages edition S2R2

Basic results

• If only non-keyed relational schemata with only one basic type, then all types of dominance are equivalent

Theorem: Let P and Q be non-keyed relational schemata over a single basic type B. Then the following are equivalent:a. Q dominates P (calc)b. Q dominates P (gen)c. Q dominates P (int)d. Q dominates P (abs)

Basic results

• With any reasonable measure of relative information capacity, two non-keyed relational schemata are equivalent iff they are identical

• In the relational model (non-keyed), there is essentially at most one way to represent a given data set

Discussion

• Strong points:– ???

Discussion

• Strong points:1. Provides a theory to study relative information

capacity

Discussion

• Strong points:1. Provides a theory to study relative information

capacity2. Data relativism is important as it arises in many

areas

Discussion

• Strong points:1. Provides a theory to study relative information

capacity2. Data relativism is important as it arises in many

areas3. Defines a hierarchy of dominance measures

Discussion

• Strong points:1. Provides a theory to study relative information

capacity2. Data relativism is important as it arises in many

areas3. Defines a hierarchy of dominance measures4. Gives important results about the relational

model

Discussion

• Weak points:– ???

Discussion

• Weak points:1. Does not support dependencies/constraints• Hierarchy of dominance measures• Basic results

Discussion

• Functional dependency (FD):Given attributes in relation R, the functional dependency means that all tuples in R that agree on attributes must also agree on .

id name address

123 John 21 Kings St.

234 Mary 31 Kings St.

Discussion

• Multivalued dependency (MVD):For MVD , if two tuples of R agree on all the attributes of X, then their components in Y may be swapped, and the result will be two tuples that are also in the relation.

course book lecturer

Machine Learning

Pattern Recognition

John

Artificial Intelligence

AIMA Mary

Discussion

• Inclusion dependency (IND):For , for any tuple t1 in R1, there must exist a tuple t2 in R2, such that

id title

111 Pattern Recognition

222 AIMA

bookid customer

111 John

222 Mary

Book

Order

Discussion

• Weak points:1. Does not support dependencies/constraints• Hierarchy of dominance measures• Basic results

Dependencies change the final result of the paper

Discussion

• Weak points:1. Does not support dependencies/constraints• Hierarchy of dominance measures• Basic results

2. Open questions: • Absolute dominance implies internal dominance?• Generic dominance implies calculous dominance?• Is there a measure for “semantic correspondence”?

Thank you

Quiz

• What are the four formal measures of relative information capacity defined by Hull? Write them in order from most restrictive to less restrictive.