Reasoning about Disclosure in Data Integration in the Presence of Source Constraints

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Reasoning about Disclosure in Data Integration in the Presence of Source Constraints

DIG Seminar 21/11/19

Michael Benedikt Pierre Bourhis Louis Jachiet Micha¨el Thomazo

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Motivations

Schema+Constraints

Secret

, →

Secret leaked?

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Motivations

Schema+Constraints

Secret

, →

Secret leaked?

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Motivations

Schema+Constraints

Secret

, →

publication

Secret leaked?

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Motivations

Schema+Constraints

Secret

, →

publication

Secret leaked?

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Motivations

Schema+Constraints

Secret

, →

publication

Secret leaked?

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Motivations

Schema+Constraints

Secret

, →

Safe publication?

Secret leaked?

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Example: Hospital setting

Patients Doctors Buildings

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Example: Hospital setting

DocSpec PatSpec

PatDoc DocBldg PatBldg

IsOpen

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Example

Database schema

Predicate Meaning

IsOpen(b,t) Building b is open on Datet PatBdlg(p,b) Patient p is present in Building b

PatSpec(p,s) Patient p was treated for Specialty s PatDoc(p,d) Patient p was treated by Doctor d DocBldg(d,b) Doctor d is associated with Buildingb

DocSpec(d,s) Doctor d is associated with Specialtys

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Example

Views

OpenHours(b,t) = IsOpen(b,t)

VisitingHours(p,t) = PatBdlg(p,b)∧IsOpen(b,t) DocList(d,s,b) = DocSpec(d,s)∧DocBldg(d,b)

Constraints

PatDoc(p,d) → ∃s PatSpec(p,s)∧DocSpec(d,s) PatBdlg(p,b) → ∃d PatDoc(p,d)∧DocBldg(d,b)

Secret

∃p,s PatSpec(p,s)?

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Example

Views

Constraints

Secret

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Example

Views

Constraints

Secret

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Example

OpenHour

B1 Tuesday

B₂ Every day 10-17h VisitingHours Charline Tuesday

DocList

Alice Cancer B1

Alice Cancer B₂ Bob Radiology B₂ Daniel Cancer B1

Constraints

PatBdlg(p,b) → ∃d PatDoc(p,d)∧DocBldg(d,b)

PatDoc(p,d) → ∃s PatSpec(p,s)∧DocSpec(d,s)

Views

VisitingHours(p,t) = PatBdlg(p,b)∧IsOpen(b,t)

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Example

OpenHour B1 Tuesday B₂ Every day 10-17h

VisitingHours Charline Tuesday

DocList

Alice Cancer B1

Constraints

Views

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Example

OpenHour

B1 Tuesday

DocList

Alice Cancer B1

Constraints

Views

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Example

OpenHour

B1 Tuesday

DocList

Alice Cancer B1

Constraints

Views

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Example

OpenHour

B1 Tuesday

DocList

Alice Cancer B1

Constraints

PatBdlg(p,b) → ∃d PatDoc(p,d)∧DocBldg(d,b) PatDoc(p,d) → ∃s PatSpec(p,s)∧DocSpec(d,s) Views

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Formalism

Data represented by databases R(1,17),R(2,42),S(23,45), . . .

, →

^{+ secret} Mappings and secrets are CQ V(x,z) :=R(x,y)∧S(y,z)

Constraints are TGD R(x,y)→ ∃z,S(y,z)

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Formalism

, →

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Formalism

, →

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Safe publication

Secret

, →

Schema Problem

Given (schema, constraintsC, viewsV, secretS) do we have for all an instance such that C( ),V( ) =V( ) and

¬S( )?

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Safe publication

Secret

, → ← -

No secret

Schema Problem

Given (schema, constraintsC, viewsV, secretS) do we have for all an instance such that C( ),V( ) =V( ) and

¬S( )?

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Safe publication

Secret

, → ← -

No secret

View Problem

Given (schema, constraintsC, views V, secretS, visible ) do we have such that C( ), V( ) = and¬S( )?

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Safe publication

Secret

, → ← -

No secret

Schema Problem

Given (schema, constraintsC, viewsV, secretS) do we have for all an instance such that C( ), V( ) =V( ) and

¬S( )?

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Which configurations are decidable/tractable for the schema problem?

Secrets Views Constraints

CQ CQ

ProjMap AtomMap GuardedMap CQMap

Let’s talk about constraints...

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CQ CQ

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CQ

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CQ CQ

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CQ CQ

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CQ CQ

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Ontologies

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Ontologies in a few words

Anontologyrepresents entities and their relationship to each other.

Ontologiescan be seen as a sort of expressive schema.

Ontologiesallows to enrich data by inferring new factsfrom existing ones.

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An example of ontology

With plain words:

Allcatsare mammals.

Allmammalsare animals.

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With plain words:

Allcatsare mammals.

With Tuple Generating Dependencies:

CAT(x)→MAMMAL(x) MAMMAL(x)→ANIMAL(x)

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With plain words:

Allcatsare mammals.

Database: {CAT( )}

Query: Are there animals? (∃X,ANIMAL(X)?)

Answer: Yes: CAT( )⇒MAMMAL( )⇒ANIMAL( )

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With plain words:

Allcatsare mammals.

Database: {CAT( )}

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With plain words:

Allcatsare mammals.

Database: {CAT( )}

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More complex ontological rules

Foreign key constraint:

SEMINAR(team,speaker,room,date)→

∃pers,RESERVED(room,date,pers)

Even more complex constraints:

∃pers,RESERVED(room,date,pers)∧MEMBER(pers,team)

MEMBER(person,team)→

∃date,room, SEMINAR(team,person,room,date)

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∃pers,RESERVED(room,date,pers) Even more complex constraints:

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∃pers,RESERVED(room,date,pers) Even more complex constraints:

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Tuple Generating Dependencies

∀X~, ~Y ϕ(X~, ~Y) ⇒ ∃Z~ ψ(Y~, ~Z)

Often Omitted

Body Head

Frontier

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∀X~, ~Y ϕ(X~, ~Y) ⇒ ∃Z~ ψ(Y~, ~Z)

Often Omitted

Body Head

Frontier

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∀X~, ~Y

ϕ(X~, ~Y) ⇒ ∃Z~ ψ(Y~, ~Z)

Often Omitted

Body Head

Frontier

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∀X~, ~Y

ϕ(X~, ~Y) ⇒ ∃Z~ ψ(Y~, ~Z)

Often Omitted

Body Head

Frontier

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Open World Query Answering

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Open World Query Answering (OWQA)

Open World Query Answering

• A set of facts F

• A set of TGD constraints C

• A conjunctive query Q Do we have, for all :

(F ⊆ ∧ C( ))⇒ Q( )?

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OWQA(F,C,Q) asks ifQ is true in all completions ofF (respectingC).

TheChase algorithms build a universal model.

OWQA(F,C,Q)⇔Chase(F,C)Q

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The Chase

Intuitively the chase simply “applies” the constraints.

With:

• F =CAT( )

• C={CAT(X)→MAMMAL(X),MAMMAL(X)→ ANIMAL(X)}

We obtain:

1. F₁ ={CAT( )}

2. F₂ ={CAT( ),MAMMAL( )}

3. F₃ ={CAT( ),MAMMAL( ),ANIMAL( )}

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The Chase

With:

• F =CAT( )

We obtain:

1. F₁ ={CAT( )}

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The Chase

With:

• F =CAT( )

We obtain:

1. F₁ ={CAT( )}

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The Chase

With:

• F =CAT( )

We obtain:

1. F₁ ={CAT( )}

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The Chase

With:

• F ={PERSON(alice)}

• C={PERSON(X)→ ∃Y,PARENT(X,Y)} We obtain:

1. F₁ ={PERSON(alice)}

2. F₂ ={PERSON(alice),PARENT(alice,Y)}

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The Chase

With:

• C={PERSON(X)→ ∃Y,PARENT(X,Y)}

We obtain:

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The Chase

With:

We obtain:

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The Chase

With:

We obtain:

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The Chase

With:

• C={PERSON(X)→ ∃Y,PARENT(X,Y) PARENT(X,Y)→ ∃PERSON(Y)}

We obtain:

3. F₃ ={PERSON(alice),PARENT(alice,Y),PERSON(Y)} 4. F₄ ={PERSON(alice),PARENT(alice,Y),PERSON(Y),

PARENT(Y,Y⁰)} . . .

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The Chase

With:

We obtain:

3. F₃ ={PERSON(alice),PARENT(alice,Y),PERSON(Y)} 4. F₄ ={PERSON(alice),PARENT(alice,Y),PERSON(Y),

PARENT(Y,Y⁰)} . . .

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The Chase

With:

We obtain:

3. F₃ ={PERSON(alice),PARENT(alice,Y),PERSON(Y)}

4. F₄ ={PERSON(alice),PARENT(alice,Y),PERSON(Y), PARENT(Y,Y⁰)}

. . .

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The Chase

With:

We obtain:

. . .

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The Chase

With:

We obtain:

. . .

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The Chase

The Chase model is not always finite.

When it is not finite it sometimes has a regularity that allows for decidable OWQA.

And in general the OWQA is undecidable. . .

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The Chase

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The Chase

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Decidable Classes of TGD

• UID, the foreign key constraint with one variable

A(x, ~Y)→ ∃Z,B(x, ~Z)

• IncDep, the foreign key constraint

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

• LTGD, the foreign key constraint with repetition of atoms A(x,x,y)→ ∃Z,B(x,y,y,z)

• GTGD, one atom in the body guards all variables

A(x,y,z)∧B(x)∧C(y,z)→ ∃w,D(x,y,w)

• FGTGD, one atom in the body guards all the frontier variables A(w,y,y)∧B(x)∧C(y,z)→ ∃u,D(x,y,u)

• Fr1LTGD, one atom in the body guards the only frontier variable

A(w,y,y)∧B(x)∧C(y,z)→ ∃u,D(x,u)

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A(x, ~Y)→ ∃Z,B(x, ~Z)

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

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A(x, ~Y)→ ∃Z,B(x, ~Z)

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

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A(x, ~Y)→ ∃Z,B(x, ~Z)

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

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A(x, ~Y)→ ∃Z,B(x, ~Z)

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

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A(x, ~Y)→ ∃Z,B(x, ~Z)

A(X~, ~Y)→ ∃Z,B(X~, ~Z)

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Another approach: query rewriting

With

• MAMMAL(X)→ANIMAL(X)

• CAT(X)→MAMMAL(X)

And the query∃X,ANIMAL(X), we obtain:

• ANIMAL(X)

• ANIMAL(X)∨MAMMAL(X)

• ANIMAL(X)∨MAMMAL(X)∨CAT(X)

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With

• ANIMAL(X)

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With

• ANIMAL(X)

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Solving our problem

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Back to our problems

View Problem

Given (schema, constraintsC, views V, secretS, visible ) do we have such that C( ), V( ) = and¬S( )?

Schema Problem

Given (schema, constraintsC, viewsV, secretS) do we have for all an instance such that C( ), V( ) =V( ) and

¬S( )?

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Solving the schema problem

The critical instance The instance

C contains one fact per relation, with one constant: C.

We note

C =V(

C) its view image.

Reduction for schema problem

SchemaProblem(C,V,S) reduces toViewProblem(

C,C,V,S) From Querying Visible and Invisible Information. LICS 2016

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C contains one fact per relation, with one constant: C. We note

C =V(

C) its view image.

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C contains one fact per relation, with one constant: C. We note

C =V(

C) its view image.

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Reduction to Open World Query Answering (OWQA)

• A set of facts F

• A set of TGD constraints C

• A queryQ

Do we have, for all :

(F ⊆ ∧ C( ))⇒ Q( )?

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Reduction to Open World Query Answering

EncodingViewProblem(

C,C,V,S) as OWQA

• The query is S

• The initial facts encode the forward constraints

C⊆ V( )

• The constraints are the original constraints C.

But we also need to encode the backward constraints V( )⊆

C!

For this we use that adom(V( )) ={C}

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C,C,V,S) as OWQA

• The query is S

C⊆ V( )

C!

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C,C,V,S) as OWQA

• The query is S

C⊆ V( )

C!

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C,C,V,S) as OWQA

• The query is S

C⊆ V( )

• The constraints are the original constraintsC.

C!

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C,C,V,S) as OWQA

• The query is S

C⊆ V( )

C!

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C,C,V,S) as OWQA

• The query is S

C⊆ V( )

C!

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Lower bounds

Various reductions from:

• OWQA

• Query evaluation

• Alternating Turing Machines

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Lower bounds

• OWQA

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Lower bounds

• OWQA

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Complexity results

Constraints

Views

ProjMap AtomMap GuardedMap CQMap IncDep PSpace ExpTime 2ExpTime 2ExpTime

LTGD ExpTime ExpTime 2ExpTime 2ExpTime GTGD 2ExpTime 2ExpTime 2ExpTime 2ExpTime FGTGD 2ExpTime 2ExpTime 2ExpTime 2ExpTime

Table 1: Complexity of disclosure

⇒all bounds aretight!

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Complexity results

Constraints

Views

IncDep NP NP ExpTime 2ExpTime

LTGD NP NP ExpTime 2ExpTime

GTGD ExpTime ExpTime ExpTime 2ExpTime FGTGD 2ExpTime 2ExpTime 2ExpTime 2ExpTime

Table 2: Complexity of disclosure in bounded arity

⇒all bounds aretight!

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Complexity results

InPTime:

• CQ secret, foreign keys constraints and projection views

• bounded CQ secret,ProjMap,LTGD

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Complexity results

InPTime:

• CQ secret, foreign keys constraints and projection views

• bounded CQ secret,ProjMap,LTGD

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Future Works

• Implement model checker for publication methods.

• How to synthesize publications automatically?

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Future Works

• Implement model checker for publication methods.

• How to synthesize publications automatically?

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Thank you!

Questions?