Uncertain Data Management Boolean c-tables

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Definitions Boolean c-tables Expressiveness

Uncertain Data Management Boolean c-tables

Antoine Amarilli¹, Silviu Maniu²

1Télécom ParisTech

2LRI

November 30th, 2015

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c-tables

Rememberc-tables:

Member ▷◁Booking id date teacher class room 1 2015-12-07 NULL₁ UDM NULL₂ 2 2015-12-07 NULL₁ UDM NULL₂

3 2015-12-07 NULL₁ UDM NULL₂ ifNULL₀ is “UDM”

→ Variant: Only allowNULLs in theconditions

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c-tables

Rememberc-tables:

Member ▷◁Booking id date teacher class room 1 2015-12-07 NULL₁ UDM NULL₂ 2 2015-12-07 NULL₁ UDM NULL₂

3 2015-12-07 NULL₁ UDM NULL₂ ifNULL₀ is “UDM”

→ Variant: Only allowNULLs in the conditions

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NULLs in conditions

The possible tuplesare exactly therows Each row may either be kept ordeleted

→ Depends on thecondition

→ Finitenumber of possible worlds

→ at most2^N if we haveNrows

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NULLs in conditions

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NULLs in conditions

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Example

Member ▷◁Booking id date teacher class room

1 2015-12-07 Antoine UDM C42 if NULL₁ is “Antoine”

2 2015-12-07 Antoine UDM C42 if NULL1 is “Antoine”

3 2015-12-07 Antoine UDM C42 if NULL₁ is “Antoine”

1 2015-12-07 Silviu UDM C42 if NULL₁ is “Silviu”

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Boolean c-tables

WithBoolean c-tables, we impose:

the possible values of each NULL_i are True andFalse

We cansimplify notation

→ We write theNULLs asBoolean variablesx_i

→ We replace x_i=True by justx_i

→ We replace x_i=False by¬x_i

→ We can rewrite:

xi=xj to(xi∧xj)∨(¬xi∧ ¬xj) xi̸=xj to(xi∧ ¬xj)∨(¬xi∧xj)

→ The conditions becomeBoolean expressions

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Boolean c-tables

the possible values of each NULL_i are True andFalse We cansimplify notation

→ We can rewrite:

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Boolean c-tables

→ We can rewrite:

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Boolean c-tables

→ We can rewrite:

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Expressiveness

Theorem

We can always rewrite a c-table havingNULLs only in conditions to a Boolean c-table.

Twosteps:

1 We can pick the NULLs in afinite domain

2 We can rewrite any finite domaintoTrue andFalse

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Expressiveness

Theorem

We can always rewrite a c-table havingNULLs only in conditions to a Boolean c-table.

Twosteps:

1 We can pick the NULLs in afinite domain

2 We can rewrite anyfinite domain toTrue andFalse

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Reducing to a finite domain

We can choose among infinitely many values for theNULLs However, the values only appear in theconditions:

NULLi =NULLj

NULLi = “c”

Boolean combinations

We call two assignments of values toNULLs equivalent if all conditions evaluate to thesame

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Reducing to a finite domain

We can choose among infinitely many values for theNULLs However, the values only appear in theconditions:

NULLi =NULLj

NULLi = “c”

Boolean combinations

We call two assignments of values toNULLsequivalent if all conditions evaluate to thesame

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Reducing to a finite domain (example)

→ Call two assignments of values to NULLs equivalent if all conditions evaluate to thesame.

Consider the following:

NULL₁ =NULL₂ NULL₂ = “c”

→ E.g.: The assignment(a,d) is equivalentto (b,d)

→ What are the possible assignments? (c,c)

→ true,true (x,c)withx̸=c

→ false,true (x,x)withx̸=c

→ true, false

(y,x)withx̸=candy̸=x

→ false, false

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Reducing to a finite domain (example)

→ What are the possible assignments?

(c,c)

→ true, false

→ false, false

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Reducing to a finite domain (example)

(c,c)

→ true,true

(x,c)withx̸=c

→ true, false

→ false, false

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Reducing to a finite domain (example)

(c,c)

→ false,true

(x,x)withx̸=c

→ true, false

→ false, false

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Reducing to a finite domain (example)

(c,c)

→ true, false

→ false, false

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Reducing to a finite domain (example)

(c,c)

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Reducing to a finite domain (concluding)

Consider allconstantsthat appear: C ConsiderNdifferent values V, whereNis the number of NULLs

→ Gives ourdomain D ··=C ⊔ V Lemma

For any c-table withNULLs only in conditions, its set of possible worlds is the same:

under the standard semantics when NULLs range over the finite D.

→ For simplicity, let’spadD

to have exactly 2^k values for somek

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Reducing to a finite domain (concluding)

→ Gives ourdomain D ··=C ⊔ V

Lemma

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Reducing to a finite domain (concluding)

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Reducing to a finite domain (concluding)

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Rewriting to Boolean variables

The domainhas size2^k.

Write its values in binary

Encode each NULLi to variables x¹_i, . . . ,x^k_i

→ Can we translatethe conditions?

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Rewriting to Boolean variables

The domainhas size2^k. Write its values inbinary

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Rewriting to Boolean variables

The domainhas size2^k. Write its values inbinary

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above NULL₇=NULL₈

→ x¹₇=x¹₈ andx²₇=x²₈ andx³₇=x³₈ NULL₇̸=NULL₈

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇

NULL₇̸= 001

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

NULL₇=NULL₈

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ x¹₇=x¹₈ andx²₇=x²₈ andx³₇=x³₈

NULL₇̸=NULL₈

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

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Rewriting to Boolean variables (example)

NULL₇= 001

→ x¹₇= 0andx²₇= 0 andx³₇= 1

→ ¬x¹₇∧ ¬x²₇∧x³₇ NULL₇̸= 001

→ negatethe above

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Concluding

→ We have moved to afinite domain (without changing the table)

→ We have rewrittento Boolean variables (we changed the table)

→ It sufficesto study Boolean c-tables

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Concluding

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Concluding

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Strong representation system

Are Boolean c-tables astrong representation system for relational algebra? ...

→ Yes!

→ c-tablesare

→ NULLs will neverappearby themselves

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Strong representation system

→ Yes!

→ c-tablesare

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Strong representation system

→ Yes!

→ c-tablesare

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Capturing all uncertain relations

Fix a set ofpossible tuples

A possible world: a subset of the possible tuples An uncertain relation: set ofpossible worlds

Booking

date teacher room 23 Antoine C017 23 Silviu C017 23 Silviu C47 30 Antoine C017 30 Antoine C47 30 Silviu C017

Booking

date teacher room 23 Antoine C017

23 Silviu C017

23 Silviu C47 30 Antoine C017 30 Antoine C47 30 Silviu C017

→ Can we capture alluncertain relations?

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Capturing all uncertain relations

Booking

23 Silviu C017

→ Can we capture alluncertain relations?

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Capturing all uncertain relations

Booking

23 Silviu C017

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Capturing uncertain relations

Make multiple copies of possible worlds so there are 2^k possible worlds

Write each possible worldin binary

00 v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f

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Capturing uncertain relations

Write each possible worldin binary 00

v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f

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Capturing uncertain relations

Write each possible worldin binary 00

v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f

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Numbering tuples

For eachtuple, write the possible worlds where it appears

00 v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f v w

a d 00 01 10 11

b e 01

c f 01 10 11

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Numbering tuples

For eachtuple, write the possible worlds where it appears 00

v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f

v w

a d 00 01 10 11

b e 01

c f 01 10 11

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Numbering tuples

For eachtuple, write the possible worlds where it appears 00

v w a d b e c f

01 v w a d b e c f

10 v w a d b e c f

11 v w a d b e c f v w

a d 00 01 10 11

b e 01

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Making a condition

Create onenon-Boolean variable

→ chooses the world

Obtain a non-Booleanc-table

v w

a d 00 01 10 11

b e 01

c f 01 10 11

v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11

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Making a condition

Obtain a non-Booleanc-table v w

a d 00 01 10 11

b e 01

c f 01 10 11

v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11

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Making a condition

Obtain a non-Booleanc-table v w

a d 00 01 10 11

b e 01

c f 01 10 11

v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11

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Making a Boolean c-table

Use theprevious trick to rewrite to aBoolean c-table

v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11 v w

a d ¬x₁∧ ¬x₂∨ ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

b e ¬x₁∧x₂

c f ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

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Making a Boolean c-table

Use theprevious trick to rewrite to aBoolean c-table v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11

v w

a d ¬x₁∧ ¬x₂∨ ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

b e ¬x₁∧x₂

c f ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

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Making a Boolean c-table

Use theprevious trick to rewrite to aBoolean c-table v w

a d x= 00∨x= 01∨x= 10∨x= 11

b e x= 01

c f x= 01∨x= 10∨x= 11 v w

a d ¬x₁∧ ¬x₂∨ ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

b e ¬x₁∧x₂

c f ¬x₁∧x₂∨x₁∧ ¬x₂∨x₁∧x₂

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Conclusion

We have studied:

First:

Codd tableswithNULLs v-tableswithnamedNULLs

c-tableswithnamedNULLs andconditions

Then:

c-tables withNULLsonly in conditions Booleanc-tables: Boolean variables We have shown:

→ Any c-tablewith NULLsonly in conditions rewrites to a Boolean c-table

→ Boolean c-tables captureall finite uncertain tables

→ Boolean c-tables are a strong representation system

→ c-tables are a strong representation system

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Conclusion

We have studied:

First:

c-tableswithnamedNULLs andconditions Then:

c-tables withNULLsonly in conditions Booleanc-tables: Boolean variables

We have shown:

→ Any c-tablewith NULLsonly in conditions rewrites to a Boolean c-table

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Conclusion

We have studied:

First:

c-tableswithnamedNULLs andconditions Then:

c-tables withNULLsonly in conditions Booleanc-tables: Boolean variables We have shown:

→ Any c-tablewith NULLs only in conditions rewrites to a Boolean c-table

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References I

Abiteboul, S., Hull, R., and Vianu, V. (1995).

Foundations of Databases.

Addison-Wesley.

http://webdam.inria.fr/Alice/pdfs/all.pdf.

Suciu, D., Olteanu, D., Ré, C., and Koch, C. (2011).

Probabilistic Databases.

Morgan & Claypool.

Unavailable online.

Uncertain Data Management Boolean c-tables