Uncertain Data Management Boolean c-tables

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Uncertain Data Management Boolean c-tables

Antoine Amarilli¹, Silviu Maniu² November 21st, 2017

1Télécom ParisTech

2LRI

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c-tables

Rememberc-tables:

Member./Booking id date teacher class room 2 2017-11-28 NULL1 UDM NULL2

3 2017-12-05 NULL1 UDM NULL2

4 2017-12-12 NULL1 UDM NULL2 ifNULL0is “UDM”

→ Variant:Only allowNULLsin theconditions

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c-tables

Rememberc-tables:

Member./Booking id date teacher class room 2 2017-11-28 NULL1 UDM NULL2

3 2017-12-05 NULL1 UDM NULL2

4 2017-12-12 NULL1 UDM NULL2 ifNULL0is “UDM”

→ Variant:Only allowNULLsin theconditions

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NULLs in conditions

•

^Thepossible tuplesare exactly therows

•

Each row may either bekeptordeleted

→ Depends on thecondition

→ Finitenumber of possible worlds

→ at most2^Nif 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

2 2017-11-28 Antoine UDM C42 ifNULL1is “Antoine”

2 2017-11-28 Silviu UDM C42 ifNULL1is “Silviu”

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Table of contents

Definitions

Boolean c-tables

Expressiveness

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Boolean c-tables

WithBoolean c-tables, we impose:

•

the possible values of eachNULL_i areTrueandFalse

We cansimplify notation

→ We write theNULLsasBoolean variablesx_i

→ We replacex_i=Trueby justx_i

→ We replacex_i=Falseby¬x_i

→ We canrewrite:

• x_i=x_jto(x_i∧x_j)∨(¬x_i∧ ¬x_j)

• x_i6=x_jto(x_i∧ ¬x_j)∨(¬x_i∧x_j)

→ The conditions becomeBoolean expressions

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•

the possible values of eachNULL_i areTrueandFalse We cansimplify notation

→ We canrewrite:

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•

→ We canrewrite:

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•

→ We canrewrite:

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Expressiveness

Theorem

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

Proof: Two steps:

1. We can pick theNULLsin afinite domain

2. We can rewrite anyfinite domaintoTrueandFalse

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Expressiveness

Theorem

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

Proof:Two steps:

1. We can pick theNULLsin afinite domain

2. We can rewrite anyfinite domaintoTrueandFalse

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Step 1: Reducing to a finite domain

•

We can choose amonginfinitely manyvalues for theNULLs

•

However, the values only appear in theconditions:

• NULL_i=NULL_j

• NULL_i= “c”

• Boolean combinations

•

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

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Step 1: Reducing to a finite domain

•

We can choose amonginfinitely manyvalues for theNULLs

•

However, the values only appear in theconditions:

• NULL_i=NULL_j

• NULL_i= “c”

• Boolean combinations

•

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

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Step 1: Reducing to a finite domain (example)

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

Consider the following:

•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

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

→ What are the possibleassignments?

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

→ true, false

• (y,x)withx6=candy6=x

→ false, false

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•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

→ true, false

→ false, false

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•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

→ true, false

→ false, false

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•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

→ true, false

→ false, false

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•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

→ true, false

→ false, false

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•

^NULL¹ ⁼^NULL²

•

^NULL² ^{= “c”}

• (c,c)

→ true,true

• (x,c)withx6=c

→ false,true

• (x,x)withx6=c

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Step 1: Reducing to a finite domain (concluding)

•

Consider allconstantsthat appear:C

•

^Consider^Ndifferent valuesV, whereNis the number ofNULLs

→ Gives ourdomainD ··=C t V Lemma

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

•

under the standard semantics

•

^when^NULLsrange over the finiteD.

→ Forsimplicity, let’spadDto have exactly2^kvalues for somek

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•

→ Gives ourdomainD ··=C t V

Lemma

•

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•

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•

Forsimplicity, let’s to have exactly values for some

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Step 2: Rewriting to Boolean variables

•

^The^domain^{has size}²^k

•

^{Write its}^valuesⁱⁿ^binary

•

Encode eachNULL_i to variablesx_i¹, . . . ,x_i^k

→ Can wetranslatethe conditions?

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•

Encode eachNULL_i to variablesx¹_i, . . . ,x_i^k

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•

Encode eachNULL_i to variablesx¹_i, . . . ,x_i^k

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Step 2: Rewriting to Boolean variables (example)

•

^NULL⁷ ⁼⁰⁰¹

→ x¹₇=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷⁼^NULL⁸

→ x¹₇=x¹₈andx²₇ =x₈²andx³₇=x³₈

•

^NULL⁷⁶⁼^NULL⁸

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷⁼^NULL⁸

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷⁼^NULL⁸

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷⁼^NULL⁸

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷⁼^NULL⁸

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷ ⁼^NULL⁸

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

→ x₇¹=x¹₈andx²₇ =x₈²andx³₇=x³₈

•

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷ ⁶⁼^NULL⁸

→ negatethe above

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•

^NULL⁷ ⁼⁰⁰¹

→ x₇¹=0andx²₇=0andx³₇=1

→ ¬x₇¹∧ ¬x²₇∧x³₇

•

^NULL⁷ ⁶⁼⁰⁰¹

→ negatethe above

•

^NULL⁷ ⁶⁼^NULL⁸

→ negatethe above

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Concluding

1. We have moved to afinite domain (without changing the relation)

2. We haverewrittento Boolean variables (we changed the relation)

→ Itsufficesto study Boolean c-tables

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Concluding

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Concluding

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Table of contents

Definitions

Boolean c-tables

Expressiveness

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Strong representation system

•

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

→ Yes!

→ c-tablesare a strong representation system

→ NULLswill neverappearby themselves outside of conditions

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•

→ Yes!

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•

→ Yes!

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Capturing all uncertain relations

•

^{Fix a}^finite^{set of}possible tuples

•

^Apossible world: a subset of thepossible tuples

•

^Anfinite uncertain relation:(finite) set ofpossible worlds

Booking date teacher room 21 Antoine Saphir 21 Silviu Saphir 21 Silviu C47 28 Antoine Saphir 28 Antoine C47 28 Silviu Saphir

→ Can we capture allfinite uncertain relations?

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•

^Anfinite uncertain relation:(finite) set ofpossible worlds Booking

date teacher room 21 Antoine Saphir 21 Silviu Saphir 21 Silviu C47 28 Antoine Saphir 28 Antoine C47 28 Silviu Saphir

→ Can we capture allfinite uncertain relations?

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•

^Anfinite uncertain relation:(finite) set ofpossible worlds Booking

date teacher room 21 Antoine Saphir 21 Silviu Saphir 21 Silviu C47 28 Antoine Saphir 28 Antoine C47 28 Silviu Saphir

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Capturing finite uncertain relations

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Make multiplecopiesof possible worlds so there are2^k possible worlds

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^{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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•

^{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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•

^{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 thepossible worldswhere 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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For eachtuple, write thepossible worldswhere 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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For eachtuple, write thepossible worldswhere 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 one}non-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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•

^{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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•

^{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 trickto 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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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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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 ¬x ∧x ∨x ∧ ¬x ∨x ∧x

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Conclusion We have studied:

•

^First:_•

Codd tableswithNULLs

• v-tableswithnamedNULLs

• c-tableswithnamedNULLsandconditions

•

^Then:• c-tables withNULLsonly in conditions

• Booleanc-tables: Boolean variables We have shown:

→ Anyc-tablewithNULLsonly in conditions rewritesto a Boolean c-table

→ Boolean c-tablescaptureall finiteuncertain relations

→ Boolean c-tables are astrong representation system

→ c-tables are astrong representation system

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•

^First:_•

•

• Booleanc-tables: Boolean variables

We have shown:

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•

^First:_•

•

• Booleanc-tables: Boolean variables We have shown:

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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.