August12,2014 AntoineAmarilli Open-WorldFiniteQueryAnsweringUnderNumberRestrictions

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Introduction Existing approaches Result Proof ideas Conclusion

Open-World Finite Query Answering Under Number Restrictions

Antoine Amarilli^1,2

1Télécom ParisTech, Paris, France

2University of Oxford, Oxford, United Kingdom

August 12, 2014

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Open-world query answering

Evaluatequery q overinstance I,open-world assumption:

The instanceIiscorrectbutincomplete

Consider all possiblecompletionsJsatisfyingconstraintsΣ Certain answersto queryqamong those completions

⇒ Formally: I,Σ|=qifJ|=qfor allJ⊇I s.t.J|=Σ

Constraints:

TGDs, especiallyinclusion dependencies (ID)

⇒ Unaryinclusion dependencies (UID):R[A]⊆S[B]

Number restrictions, especiallyfunctional dependencies(FD) Finitevs unrestricted QA

Instance: List of employees

Constraint 1: Each employee reviews some employee (UID) Constraint 2: At most one reviewer per employee (FD)

Query: Are all employees reviewed?

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Open-world query answering

⇒ Formally: I,Σ|=qifJ|=qfor allJ⊇I s.t.J|=Σ Constraints:

Number restrictions, especiallyfunctional dependencies(FD)

Finitevs unrestricted QA Instance: List of employees

Query: Are all employees reviewed?

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Open-world query answering

⇒ Formally: I,Σ|=qifJ|=qfor allJ⊇I s.t.J|=Σ Constraints:

Number restrictions, especiallyfunctional dependencies(FD) Finitevs unrestricted QA

Instance: List of employees

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Undecidability barrier

Entailment of IDsandFDs is undecidable[Mitchell, 1983]

Already for binary IDs andunary FDs:

R[A,B]⊆S[C,D],R[A]→R[B]

⇒ QA (finite or not) is also undecidable[Calì et al., 2003]

(Remark: this proof requires constants in the query)

⇒ We can’t have everything

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Undecidability barrier

Entailment of IDsandFDs is undecidable[Mitchell, 1983]

Already for binary IDs andunary FDs:

R[A,B]⊆S[C,D],R[A]→R[B]

⇒ QA (finite or not) is also undecidable[Calì et al., 2003]

(Remark: this proof requires constants in the query)

⇒ We can’t have everything

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Idea 1: Separability

The chasefor IDs: universal model

Intuition: apply all IDs with fresh elements

FDs are separablefrom IDs if they do not impact the chase Sufficient conditions for separability, e.g., non-conflicting:

⇒ exported positions must not be astrict supersetof a key When separable, we canignore FDs (just check them onI)

⇒ The chase is infinitein general so it doesn’t work in the finite

⇒ Finite QA undecidablefor separable IDs/FDs [Rosati, 2006] (intuition: their finite consequencesmay not be separable)

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Idea 1: Separability

The chasefor IDs: universal model

Intuition: apply all IDs with fresh elements

FDs are separablefrom IDs if they do not impact the chase Sufficient conditions for separability, e.g., non-conflicting:

⇒ exported positions must not be astrict supersetof a key When separable, we canignore FDs (just check them onI)

⇒ The chase is infinitein general so it doesn’t work in the finite

⇒ Finite QA undecidablefor separable IDs/FDs [Rosati, 2006]

(intuition: their finite consequencesmay not be separable)

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Idea 2: Finite controllability

Finite controllability means that finite and infinite QA coincide IDs are finitely controllable [Rosati, 2006]

⇒ Construction: finite chase(chase with distant reuses) Generalizes to the guarded fragment[Barany et al., 2010]

⇒ (Guardedmeans that∀/∃must be covered by an atom)

⇒ Intuition: query acyclificationand cycleblowup Generalises to IDs/FDs with foreign keyscondition [Rosati, 2006]

⇒ FDs are not expressiblein the guarded fragment.

⇒ IDs/FDs are notfinitely controllable!

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Idea 2: Finite controllability

Finite controllability means that finite and infinite QA coincide IDs are finitely controllable [Rosati, 2006]

⇒ Construction: finite chase(chase with distant reuses) Generalizes to the guarded fragment[Barany et al., 2010]

⇒ (Guardedmeans that∀/∃must be covered by an atom)

⇒ Intuition: query acyclificationand cycleblowup Generalises to IDs/FDs with foreign keyscondition [Rosati, 2006]

⇒ FDs are not expressiblein the guarded fragment.

⇒ IDs/FDs are notfinitely controllable!

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Idea 3: Arity-two

Finite and unrestricted QA decidablein arity-two for the two-variable guarded fragment and counting constraints [Pratt-Hartmann, 2009]

Intuition: again, encode the acyclic partof the query Satisfiabilitydecidableby reduction to an inequation system Explicit construction for DLs [Ibáñez-García et al., 2014]

⇒ Only forarity-two signatures

⇒ No clear way to generalizeto higher arity

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Idea 3: Arity-two

Finite and unrestricted QA decidablein arity-two for the two-variable guarded fragment and counting constraints [Pratt-Hartmann, 2009]

Intuition: again, encode the acyclic partof the query Satisfiabilitydecidableby reduction to an inequation system Explicit construction for DLs [Ibáñez-García et al., 2014]

⇒ Only forarity-two signatures

⇒ No clear way to generalizeto higher arity

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Our setting

So:

FiniteQA

TGDs and EGDs withinteraction (not FC) High-aritysignatures

⇒ Can we haveall three?

⇒ What if we restrict the language to UIDsandFDs? Nodirect encodingto arity-two (unlike UIDs/UKDs...) UIDsare important IDs in practice

UIDsmatch the DL intuition UIDsare less expressive than BIDs and...

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Our setting

So:

FiniteQA

⇒ What if we restrict the language to UIDsandFDs?

Nodirect encodingto arity-two (unlike UIDs/UKDs...) UIDsare important IDs in practice

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Our setting

So:

FiniteQA

⇒ What if we restrict the language to UIDsandFDs?

Nodirect encodingto arity-two (unlike UIDs/UKDs...) UIDsare important IDs in practice

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Finite closure for UIDs and FDs

Implication of UIDs/FDs isdecidable and PTIME [Cosmadakis et al., 1990]

Unrestricted and finite do not coincide

For unrestricted: implication of FDs and UIDs in isolation For finite: addcycle reversal:

Consider onlyunary FDs: R[i]→S[j] WhenR[i]⊆S[j]we have|R[i]| ≤ |S[j]| WhenR[i]→S[j] we have|R[i]| ≥ |S[j]| Inequalitycycleswith this encoding

⇒ In the finite, such cycles must bereversed

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Finite closure for UIDs and FDs

For unrestricted: implication of FDs and UIDs in isolation

For finite: addcycle reversal:

Consider onlyunary FDs: R[i]→S[j] WhenR[i]⊆S[j]we have|R[i]| ≤ |S[j]| WhenR[i]→S[j] we have|R[i]| ≥ |S[j]| Inequalitycycleswith this encoding

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Finite closure for UIDs and FDs

Consider onlyunary FDs: R[i]→S[j]

WhenR[i]⊆S[j]we have|R[i]| ≤ |S[j]| WhenR[i]→S[j]we have|R[i]| ≥ |S[j]| Inequalitycycleswith this encoding

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Finite closure for UIDs and FDs

Consider onlyunary FDs: R[i]→S[j]

WhenR[i]⊆S[j]we have|R[i]| ≤ |S[j]| WhenR[i]→S[j]we have|R[i]| ≥ |S[j]| Inequalitycycleswith this encoding

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Finite closure example

I

R( )

^R[2]_R[2]^⊆_→^R[1]_R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2] Add

⇒ No finite model!

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Finite closure example

I

R( ) R( )

R( )

^R[2]_R[2]^⊆_→^R[1]_R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2] Add

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Finite closure example

I

R( ) R( ) R( )

R( )

^R[2]_R[2]^⊆_→^R[1]_R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2] Add

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Finite closure example

I

R( ) R( ) R( ) R( )

R( )

R[2]⊆R[1]

R[2]→R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2] Add

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Finite closure example

I

R( ) R( ) R( ) R( )

R( )

R[2]⊆R[1]

R[2]→R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2] Add

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Finite closure example

I

R( ) R( ) R( ) R( )

R( )

R[2]⊆R[1]

R[2]→R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]|

Add R[1]⊆R[2] Add

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Finite closure example

I

R( ) R( ) R( ) R( )

R( )

R[2]⊆R[1]

R[2]→R[1]

⇒ |R[2]| ≤ |R[1]|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2]

Add R[1]→R[2]

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Finite closure example

I

R( ) R( ) R( ) R( )

R( )

!

_⇒ ^R[2]^R[2]_|_R[2]^⊆^→_{| ≤ |}^R[1]^R[1]_R[1]_|

⇒ |R[1]| ≤ |R[2]|

⇒ |R[2]|=|R[1]| Add R[1]⊆R[2]

Add R[1]→R[2]

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Finite controllability up to closure

In arity-two, UIDs/UFDs finitely controllableup to finite closure [Rosati, 2008, Ibáñez-García et al., 2014]

⇒ To perform finite QAon instance I, UIDs/UFDsΣ:

ComputeΣ^∗ thefinite closureofΣ Check ifIsatisfies the UFDsofΣ^∗ Performunrestricted QA withIandΣ^∗

Easy because UIDs/UFDs arenon-conflictingsoseparable

⇒ Does this also hold with higher-arityrelations and FDs?

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Finite controllability up to closure

In arity-two, UIDs/UFDs finitely controllableup to finite closure [Rosati, 2008, Ibáñez-García et al., 2014]

⇒ To perform finite QAon instance I, UIDs/UFDsΣ:

ComputeΣ^∗ thefinite closureofΣ Check ifIsatisfies the UFDsofΣ^∗ Performunrestricted QA withIandΣ^∗

Easy because UIDs/UFDs arenon-conflictingsoseparable

⇒ Does this also hold with higher-arityrelations and FDs?

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The result

Theorem

UIDs and FDs, though not finitely controllable, are finitely controllable up to finite closure, on arbitrary arity signatures.

⇒ It suffices to show that for any k,I, and Σ^∗, there is afinite completionofIbyΣ^∗ which isuniversalfor queries of size≤k

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The result

Theorem

UIDs and FDs, though not finitely controllable, are finitely controllable up to finite closure, on arbitrary arity signatures.

⇒ It suffices to show that for any k,I, and Σ^∗, there is a finite completionofIbyΣ^∗ which isuniversalfor queries of size≤k

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase

R( ) R( ) R( )

Considerk-neighborhood equivalence

Quotient the chase by this relation May violate FDs

Notuniversal (cycles) even for ≤k Yet universal for ≤k acyclic queries Keep ahomomorphismto this quotient

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase

R( ) R( ) R( )

Considerk-neighborhood equivalence Quotient the chase by this relation

May violate FDs

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Considerk-neighborhood equivalence Quotient the chase by this relation

May violate FDs

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Considerk-neighborhood equivalence Quotient the chase by this relation May violate FDs

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( ) !

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Notuniversal (cycles) even for≤k

Yet universal for ≤k acyclic queries Keep ahomomorphismto this quotient

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( ) !

Notuniversal (cycles) even for≤k

Yet universal for ≤k acyclic queries Keep ahomomorphismto this quotient

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Notuniversal (cycles) even for≤k Yet universal for ≤k acyclic queries

Keep ahomomorphismto this quotient

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Quotienting the chase

R[2] ⊆ R[1]

R( )

chase chase/≡2

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Notuniversal (cycles) even for≤k Yet universal for ≤k acyclic queries Keep ahomomorphismto this quotient

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Frugal chase steps

R( )

Follow thechase

Partition positions:

Exportedposition Dangerouspositions (determiners for an UFD) Non-dangerouspositions (the rest)

Createfresh elementsfor dangerous Reuseclusters for non-dangerous created byinitial chasing

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Frugal chase steps

R( )

R[3] ⊆ S[1]

Follow thechase

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( ) * * * *

Follow thechase

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( ) * * * *

Follow thechase Partition positions:

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( )

^danger

non-danger

* * * *

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( )

^danger

non-danger

* * * *

Createfresh elementsfor dangerous

Reuseclusters for non-dangerous created byinitial chasing

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( )

^danger

non-danger

* *

Createfresh elementsfor dangerous

Reuseclusters for non-dangerous created byinitial chasing

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( )

^danger

non-danger

* *

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Frugal chase steps

R( )

R[3] ⊆ S[1]

S( )

^danger

non-danger initial segment

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Blueprint

Infinite functional paths... ... but only within acycle

Connect it back(match elements) Morecomplex if many positions of many relations are involved... Uses cardinalityalong cycles... ... but also initial chasing

to force “generic neighborhoods”

I R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

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Blueprint

Infinite functional paths...

... but only within acycle

I R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

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Blueprint

I R( )

R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

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Blueprint

I R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

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Blueprint

I R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

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Blueprint

I R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

⊆ ←

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Blueprint

I R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

⊆ ←

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Blueprint

I R( )

R( ) R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

⊆ ←

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Blueprint

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

⊆ ←

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Blueprint

Connect it back(match elements)

Morecomplex if many positions of many relations are involved... Uses cardinalityalong cycles... ... but also initial chasing

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

⊆ ←

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Blueprint

Connect it back(match elements)

Morecomplex if many positions of many relations are involved... Uses cardinalityalong cycles... ... but also initial chasing

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

R( )

⊆ ←

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Blueprint

Connect it back(match elements) Morecomplexif many positions of many relations are involved...

Uses cardinalityalong cycles... ... but also initial chasing

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

R( )

⊆ ←

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Blueprint

Uses cardinalityalong cycles...

... but also initial chasing

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

R( )

⊆ ←

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Blueprint

Uses cardinalityalong cycles...

... but also initial chasing

I R( ) R( ) R( )

R( ) R( ) A( )

R[2] ⊆ R[1] R[2] → R[1]

R( )

⊆ ←

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Dependency graph

R( ) * * * * ( ) * *

S

Build a DAGon the dependencycycles

We never create fresh elements for ahigher dependencyin the DAG Satisfy cycles along a topological sort

⇒ Finite extension that satisfies UIDs/UFDs with ahomomorphism to the quotient

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Dependency graph

R( ) * * * * ( ) * *

S

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Dependency graph

R( ) * * * * ( ) * *

S

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Dependency graph

R( ) * * * * ( ) * *

S

Build a DAGon the dependencycycles We never create fresh elements

for ahigher dependencyin the DAG

Satisfy cycles along a topological sort

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Dependency graph

R( ) * * * * ( ) * *

S

for ahigher dependencyin the DAG Satisfy cycles along a topological sort

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Dependency graph

R( ) * * * * ( ) * *

S

for ahigher dependencyin the DAG Satisfy cycles along a topological sort

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Higher-arity FDs

Ignored so far

May only be triggered at non-dangerous reuses

Idea: if non-dangerousbut dangerous for higher-arity FD then no unary key

R( )

export non-danger

Idea:

create manyreuse candidates combine them in differentpatterns

⇒ Lemma: if no UKD then O(n^>1) patterns forO(n) elements

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Higher-arity FDs

Ignored so far

R( )

export non-danger

!

Idea:

create manyreuse candidates combine them in differentpatterns

⇒ Lemma: if no UKD then O(n^>1) patterns forO(n) elements

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Higher-arity FDs

Ignored so far

R( )

export non-danger

!

Idea:

create manyreuse candidates

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Higher-arity FDs

I

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Higher-arity FDs

R( ) I

reuse candidates

R( )

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Higher-arity FDs

R( ) I

reuse candidates O(n)

elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

R( )

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Higher-arity FDs

R( )

R( ) R( ) I

reuse candidates

...

O(n) elements

O(n^>1)

patterns

O(n) reuses

R( )

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Blowing up cycles

Usually: product with a group ofhigh girth[Otto, 2002]

Ensures all non-instancecyclesare large

We cannot do it directlyas it would violate theFDs

However, intuition:

FD violations can only appear onnon-dangerous reuses... ... and those facts are mapped to thesame quotient fact ... so cycles on them areself-homomorphicin the quotient

R( ) R( )

model M

non-danger reuses

⇒ Blow up cycles, but not those cycles

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Blowing up cycles

We cannot do it directlyas it would violate theFDs However, intuition:

FD violations can only appear onnon-dangerous reuses...

... and those facts are mapped to thesame quotient fact ... so cycles on them areself-homomorphicin the quotient

R( ) R( )

model M

non-danger reuses

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Blowing up cycles

R( ) R( )

model M chase/≡_k

non-danger reuses

R( )

h

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Blowing up cycles

R( ) R( )

model M chase/≡_k

non-danger reuses

R( )

h

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Blowing up cycles via the product

chase/≡

_k

k-acyclic-

universal

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Blowing up cycles via the product

chase/≡

_k

hom

k-acyclic- universal

M

satisﬁes Σ*

safe overlaps

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Blowing up cycles via the product

chase/≡

_k

hom

k-acyclic- universal

M

satisﬁes Σ*

safe overlaps

k-universal

chase/≡ ×G

_k

M ×G

prod prod

satisﬁes Σ*

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Blowing up cycles via the product

chase/≡

_k

hom

k-acyclic- universal

M

satisﬁes Σ*

safe overlaps

k-universal

chase/≡ ×G

_k

M ×G

prod prod

hom

satisﬁes Σ*

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Summary

UIDs/FDs finitely controllable up to closure Maindifferences with arity-two:

Clustersfor non-dangerous reuses Combinationsfor higher-arity FDs More complexreconnexionsalong cycles

More elaboratecycle elimination(via the quotient)

⇒ Generalize to richer unary languagesfor high arity?

Need construction forfinite implication Does thefinite model constructionadapt?

Thanks for your attention!

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Summary

UIDs/FDs finitely controllable up to closure Maindifferences with arity-two:

Clustersfor non-dangerous reuses Combinationsfor higher-arity FDs More complexreconnexionsalong cycles

More elaboratecycle elimination(via the quotient)

⇒ Generalize to richer unary languagesfor high arity?

Need construction forfinite implication Does thefinite model constructionadapt?

Thanks for your attention!

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References I

Barany, V., Gottlob, G., and Otto, M. (2010).

Querying the guarded fragment.

InLICS.

Calì, A., Lembo, D., and Rosati, R. (2003).

On the decidability and complexity of query answering over inconsistent and incomplete databases.

InPODS.

Cosmadakis, S. S., Kanellakis, P. C., and Vardi, M. Y. (1990).

Polynomial-time implication problems for unary inclusion dependencies.

JACM, 37(1).

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References II

Ibáñez-García, Y., Lutz, C., and Schneider, T. (2014).

Finite model reasoning in horn description logics.

InKR.

Mitchell, J. C. (1983).

The implication problem for functional and inclusion dependencies.

Information and Control, 56(3).

Otto, M. (2002).

Modal and guarded characterisation theorems over finite transition systems.

InLICS.

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References III

Pratt-Hartmann, I. (2009).

Data-complexity of the two-variable fragment with counting quantifiers.

Inf. Comput., 207(8).

Rosati, R. (2006).

On the decidability and finite controllability of query processing in databases with incomplete information.

InPODS.

Rosati, R. (2008).

Finite model reasoning in DL-Lite.

InESWC.

August12,2014 AntoineAmarilli Open-WorldFiniteQueryAnsweringUnderNumberRestrictions

Open-World Finite Query Answering Under Number Restrictions

Open-world query answering

Open-world query answering

Open-world query answering

Table of Contents

Undecidability barrier

Undecidability barrier

Idea 1: Separability

Idea 1: Separability

Idea 2: Finite controllability

Idea 2: Finite controllability

Idea 3: Arity-two

Idea 3: Arity-two

Table of Contents

Our setting

Our setting

Our setting

Finite closure for UIDs and FDs

Finite closure for UIDs and FDs

Finite closure for UIDs and FDs

Finite closure for UIDs and FDs

Finite closure example

I

R( )

R( )

Finite closure example

I

R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( ) R( )

R( )

Finite closure example

I

R( ) R( ) R( ) R( )

R( )

!

Finite controllability up to closure

Finite controllability up to closure

The result

The result

Table of Contents

Quotienting the chase

Quotienting the chase

Quotienting the chase

R[2] ⊆ R[1]

R( )

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Quotienting the chase

R[2] ⊆ R[1]

R( )

R( ) R( ) R( )

R( ) R( ) R( ) R( )

Quotienting the chase

R[2] ⊆ R[1]

R( )

R( ) R( ) R( )

R( ) R( ) R( ) R( ) !

Quotienting the chase

R[2] ⊆ R[1]

R( )