July7,2015 AntoineAmarilli ,MichaelBenedikt Open-WorldFiniteQueryAnsweringwithNumberRestrictions

Introduction Context and result Proof ideas Conclusion

Open-World Finite Query Answering with Number Restrictions

Antoine Amarilli^1,2, Michael Benedikt²

1Télécom ParisTech, Paris, France

2University of Oxford, Oxford, United Kingdom

July 7, 2015

1/15

Introduction Context and result Proof ideas Conclusion

Open-world query answering (QA)

Open-worldquery answering:

RelationalinstanceI(ground facts),correctbutincomplete Boolean conjunctivequeryq

Consider all possiblecompletionsJ⊇I

Isqcertain, i.e., holds on all completions? If yes,I|=unrq

Introduction Context and result Proof ideas Conclusion

Constraints

Impose constraintson the possible completions Inclusion dependencies: some facts imply more facts:

∀xdEmployee(x,d)⇒ ∃yAdvises(x,y,d) Employee[1,2]⊆Advises[1,3]

→ Unaryinclusion dependencies (UIDs): one shared position Functional dependencies(FDs): uniqueness constraints:

∀xx^′ydAdvises(x,y,d)∧Advises(x^′,y,d)→x=x^′ Advises[2,3]→Advises[1]

→ Consider allJ⊇Ithat satisfy the constraints

→ Is qcertain on all such completions? If yes, I,Σ|=_unr q

3/15

Introduction Context and result Proof ideas Conclusion

Constraints

Impose constraintson the possible completions Inclusion dependencies: some facts imply more facts:

∀xdEmployee(x,d)⇒ ∃yAdvises(x,y,d) Employee[1,2]⊆Advises[1,3]

→ Unaryinclusion dependencies (UIDs): one shared position

Functional dependencies(FDs): uniqueness constraints:

∀xx^′ydAdvises(x,y,d)∧Advises(x^′,y,d)→x=x^′ Advises[2,3]→Advises[1]

→ Consider allJ⊇Ithat satisfy the constraints

→ Is qcertain on all such completions? If yes, I,Σ|=_unr q

Introduction Context and result Proof ideas Conclusion

Constraints

Impose constraintson the possible completions Inclusion dependencies: some facts imply more facts:

∀xdEmployee(x,d)⇒ ∃yAdvises(x,y,d) Employee[1,2]⊆Advises[1,3]

→ Unaryinclusion dependencies (UIDs): one shared position Functional dependencies(FDs): uniqueness constraints:

∀xx^′ydAdvises(x,y,d)∧Advises(x^′,y,d)→x=x^′ Advises[2,3]→Advises[1]

→ Consider allJ⊇Ithat satisfy the constraints

→ Is qcertain on all such completions? If yes, I,Σ|=_unr q

3/15

Introduction Context and result Proof ideas Conclusion

Constraints

Impose constraintson the possible completions Inclusion dependencies: some facts imply more facts:

∀xdEmployee(x,d)⇒ ∃yAdvises(x,y,d) Employee[1,2]⊆Advises[1,3]

→ Unaryinclusion dependencies (UIDs): one shared position Functional dependencies(FDs): uniqueness constraints:

∀xx^′ydAdvises(x,y,d)∧Advises(x^′,y,d)→x=x^′ Advises[2,3]→Advises[1]

→ Consider allJ⊇Ithat satisfy the constraints

→ |

Introduction Context and result Proof ideas Conclusion

Finite vs infinite

Instance: List of adviser–advisee pairs: Advises(a,b), . . . UID: Each advisee advises someone

Advises[2]⊆Advises[1]

FD: Each advisee has only one adviser Advises[2]→Advises[1]

Query: Are all advisers themselves advised by someone?

q is not certain:

Advises(a,b),Advises(b,b₂),Advises(b₂,b₃), . . . q is certain on all finite completions

→ FiniteQA: only finite completions. If yes,I,Σ|=_fin q

→ Σ finitely controllable: ∀I∀q,I,Σ|=_fin q iffI,Σ|=_unr q

4/15

Introduction Context and result Proof ideas Conclusion

Finite vs infinite

Instance: List of adviser–advisee pairs: Advises(a,b), . . . UID: Each advisee advises someone

Advises[2]⊆Advises[1]

FD: Each advisee has only one adviser Advises[2]→Advises[1]

Query: Are all advisers themselves advised by someone?

q is not certain:

Advises(a,b),Advises(b,b₂),Advises(b₂,b₃), . . . q is certain on all finite completions

→ FiniteQA: only finite completions. If yes,I,Σ|=_fin q

→ Σ finitely controllable: ∀I∀q,I,Σ|=_fin q iffI,Σ|=_unr q

Introduction Context and result Proof ideas Conclusion

Finite vs infinite

Instance: List of adviser–advisee pairs: Advises(a,b), . . . UID: Each advisee advises someone

Advises[2]⊆Advises[1]

FD: Each advisee has only one adviser Advises[2]→Advises[1]

Query: Are all advisers themselves advised by someone?

q is not certain:

Advises(a,b),Advises(b,b₂),Advises(b₂,b₃), . . . q is certain on all finite completions

→ FiniteQA: only finite completions. If yes,I,Σ|=_finq

→ Σ finitely controllable: ∀I∀q,I,Σ|=_fin q iffI,Σ|=_unr q

4/15

Introduction Context and result Proof ideas Conclusion

Finite vs infinite

Instance: List of adviser–advisee pairs: Advises(a,b), . . . UID: Each advisee advises someone

Advises[2]⊆Advises[1]

FD: Each advisee has only one adviser Advises[2]→Advises[1]

Query: Are all advisers themselves advised by someone?

q is not certain:

Advises(a,b),Advises(b,b₂),Advises(b₂,b₃), . . . q is certain on all finite completions

Introduction Context and result Proof ideas Conclusion

Table of contents

1 Introduction

2 Context and result

3 Proof ideas

4 Conclusion

5/15

Introduction Context and result Proof ideas Conclusion

Existing results

(Finite) open-world query answering is undecidable for IDs and FDs [Calì et al., 2003]

IDs alone arefinitely controllable [Rosati, 2006] Infinite open-world query answering is decidable for UIDsand FDs [Calì et al., 2012]

but not finitely controllable(see above; [Rosati, 2006]) (Finite) open-world query answering is decidable for IDs and FDs with relations of arity≤2

[Pratt-Hartmann, 2009, Ibáñez-García et al., 2014]

Introduction Context and result Proof ideas Conclusion

Existing results

(Finite) open-world query answering is undecidable for IDs and FDs [Calì et al., 2003]

IDs alone arefinitely controllable [Rosati, 2006]

Infinite open-world query answering is decidable for UIDsand FDs [Calì et al., 2012]

but not finitely controllable(see above; [Rosati, 2006]) (Finite) open-world query answering is decidable for IDs and FDs with relations of arity≤2

[Pratt-Hartmann, 2009, Ibáñez-García et al., 2014]

6/15

Introduction Context and result Proof ideas Conclusion

Existing results

(Finite) open-world query answering is undecidable for IDs and FDs [Calì et al., 2003]

IDs alone arefinitely controllable [Rosati, 2006]

Infinite open-world query answering is decidable for UIDsand FDs [Calì et al., 2012]

but not finitely controllable(see above; [Rosati, 2006])

(Finite) open-world query answering is decidable for IDs and FDs with relations of arity≤2

[Pratt-Hartmann, 2009, Ibáñez-García et al., 2014]

Introduction Context and result Proof ideas Conclusion

Existing results

(Finite) open-world query answering is undecidable for IDs and FDs [Calì et al., 2003]

IDs alone arefinitely controllable [Rosati, 2006]

Infinite open-world query answering is decidable for UIDsand FDs [Calì et al., 2012]

but not finitely controllable(see above; [Rosati, 2006]) (Finite) open-world query answering is decidable for IDs and FDs with relations of arity≤2

[Pratt-Hartmann, 2009, Ibáñez-García et al., 2014]

6/15

Introduction Context and result Proof ideas Conclusion

Problem and main result

Study finite QAfor UIDs and FDs onarbitrary signatures Is it decidable? What is thecomplexity?

Theorem

Finite open-world query answering for UIDs and FDs has

PTIMEdata complexity and NP-completecombined complexity.

Introduction Context and result Proof ideas Conclusion

Problem and main result

Study finite QAfor UIDs and FDs onarbitrary signatures Is it decidable? What is thecomplexity?

Theorem

Finite open-world query answering for UIDs and FDs has

PTIMEdata complexity and NP-completecombined complexity.

7/15

Introduction Context and result Proof ideas Conclusion

Main idea

Find which UIDs and FDs are implied over finite instances

→ This isPTIME[Cosmadakis et al., 1990]

→ Itdiffersfrom unrestricted implication

Finite closure: add all finitely implied dependencies

→ Taking the finite closureensure finite controllability?

Theorem

For any instanceI, conjunctive queryq, UIDs and FDsΣ, lettingΣ^∗ be the finite closureofΣ,

we have I,Σ|=_fin q iff I,Σ^∗|=_unrq.

Introduction Context and result Proof ideas Conclusion

Main idea

Find which UIDs and FDs are implied over finite instances

→ This isPTIME[Cosmadakis et al., 1990]

→ Itdiffersfrom unrestricted implication

Finite closure: add all finitely implied dependencies

→ Taking the finite closureensure finite controllability?

Theorem

For any instanceI, conjunctive queryq, UIDs and FDsΣ, lettingΣ^∗ be the finite closureofΣ,

we have I,Σ|=_fin q iff I,Σ^∗|=_unrq.

8/15

Introduction Context and result Proof ideas Conclusion

Main idea

Find which UIDs and FDs are implied over finite instances

→ This isPTIME[Cosmadakis et al., 1990]

→ Itdiffersfrom unrestricted implication

Finite closure: add all finitely implied dependencies

→ Taking the finite closureensure finite controllability?

Theorem

For anyinstance I, conjunctive queryq, UIDs and FDsΣ, lettingΣ^∗ be the finite closureofΣ,

Introduction Context and result Proof ideas Conclusion

Table of contents

1 Introduction

2 Context and result

3 Proof ideas

4 Conclusion

9/15

Introduction Context and result Proof ideas Conclusion

Rephrasing the result

Assume that the UIDs and FDs Σare finitely closed

Assume that aninfinite completionJ satisfiesΣbut notq

→ Construct a finite completionJ^′ satisfying Σbut not q: StartwithI

FollowJandadd factsto satisfy theUIDsofΣ... ... usingfinitely many elements(reuse them) ... but respecting theFDsofΣ

... and without makingqinadvertantly true

Introduction Context and result Proof ideas Conclusion

Rephrasing the result

Assume that the UIDs and FDs Σare finitely closed Assume that aninfinite completionJ satisfiesΣbut notq

→ Construct a finite completionJ^′ satisfying Σbut not q: StartwithI

FollowJandadd factsto satisfy theUIDsofΣ... ... usingfinitely many elements(reuse them) ... but respecting theFDsofΣ

... and without makingqinadvertantly true

10/15

Introduction Context and result Proof ideas Conclusion

Rephrasing the result

Assume that the UIDs and FDs Σare finitely closed Assume that aninfinite completionJ satisfiesΣbut notq

→ Construct a finite completionJ^′ satisfying Σbut not q:

StartwithI

FollowJandadd factsto satisfy theUIDsofΣ...

... usingfinitely many elements(reuse them) ... but respecting theFDsofΣ

... and without makingqinadvertantly true

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

Introduction Context and result Proof ideas Conclusion

Constructing the finite completion

Makeassumptions, we lift them from bottom to top:

5. Assume that the queryq is acyclic

e.g., ∃xy R(x)S(x,y) but not∃xyz R(x,y)S(y,z)T(z,x)

→ Product withgroup of high girthinspired by [Otto, 2002] –skipped

4. Assume that there are only unary FDs e.g., R[1]→R[2]but not R[1,3]→R[2]

→ Recombinationof elements inpatternswhen reusing –skipped

3. Assume a certain reversibility propertyon the UIDs and FDs

→ DecompositionofUIDsbased on [Cosmadakis et al., 1990] –skipped

2. Assume we have the trivial query ⊥

→ Exemplified soon

1. Assume that the relations all have arity two

→ Reuseelements orextendprevious case –skipped

0. Solve the problem!

→ Exemplified soon

11/15

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

R S

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

e

R S

a

b c

d

12/15

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

a

b c

d

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

12/15

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

a

b c

d

UFDs:

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

12/15

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

a

b c

d

UFDs:

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

12/15

Introduction Context and result Proof ideas Conclusion

0. Satisfying UIDs and UFDs in arity-two

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

a

b c

d

UFDs:

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

and UFDs...

a b

c d

f e

13/15

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

a b

c d

f e

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

and UFDs...

a b

c d

f e

query:

!

13/15

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

a b

c d

f e

query:

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

and UFDs...

a b

c d

f e

query:

13/15

Introduction Context and result Proof ideas Conclusion

2. Adding acyclic queries

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆ UIDs:

a b

c d

f e

query:

Introduction Context and result Proof ideas Conclusion

Table of contents

1 Introduction

2 Context and result

3 Proof ideas

4 Conclusion

14/15

Introduction Context and result Proof ideas Conclusion

Summary

Finiteopen-world query answering with:

relational instance arbitrary aritysignature UIDsandFDs

conjunctive query

→ We have finite controllabilityup to finite closure Future work:

Simplifythe proof

More expressivefrontier-one logics [Ibáñez-García et al., 2014] Finite controllability and closure forpath FDsin arity-two?

Thanks for your attention!

Introduction Context and result Proof ideas Conclusion

Summary

Finiteopen-world query answering with:

relational instance arbitrary aritysignature UIDsandFDs

conjunctive query

→ We have finite controllabilityup to finite closure

Future work:

Simplifythe proof

More expressivefrontier-one logics [Ibáñez-García et al., 2014] Finite controllability and closure forpath FDsin arity-two?

Thanks for your attention!

15/15

Introduction Context and result Proof ideas Conclusion

Summary

Finiteopen-world query answering with:

relational instance arbitrary aritysignature UIDsandFDs

conjunctive query

→ We have finite controllabilityup to finite closure Future work:

Simplifythe proof

More expressivefrontier-one logics [Ibáñez-García et al., 2014]

Finite controllability and closure forpath FDsin arity-two?

Thanks for your attention!

Introduction Context and result Proof ideas Conclusion

Summary

Finiteopen-world query answering with:

relational instance arbitrary aritysignature UIDsandFDs

conjunctive query

→ We have finite controllabilityup to finite closure Future work:

Simplifythe proof

More expressivefrontier-one logics [Ibáñez-García et al., 2014]

Finite controllability and closure forpath FDsin arity-two?

Thanks for your attention!

15/15

References I

Calì, A., Gottlob, G., and Pieris, A. (2012).

Towards more expressive ontology languages: The query answering problem.

Artif. Intel., 193.

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

References II

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

Finite model reasoning in Horn description logics.

InKR.

Otto, M. (2002).

Modal and guarded characterisation theorems over finite transition systems.

InLICS.

Pratt-Hartmann, I. (2009).

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

Inf. Comput., 207(8).

2/7

References III

Rosati, R. (2006).

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

InPODS.

1. Supporting arbitrary arity

R S

1 2

3

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

4/7

1. Supporting arbitrary arity

e

R S

a

b c

d

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

4/7

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

4/7

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist

With UFDs: bijectivemaps within ⇄-classes

4/7

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

With UFDs: bijectivemaps within ⇄-classes

1. Supporting arbitrary arity

e

R S

R[2] S[1]⊆

⊆

S[2] R[1]⊆

⊆

R[1] R[2]→

→

S[1] S[2]→→

a

b c

d

UFDs:

1 2

3

f

g

When no UFDs: reuseelements from suitable positions

→ Mustsaturateinitially to make sure such elements exist With UFDs: bijectivemaps within ⇄-classes

4/7

3. From reversible to arbitrary UIDs and UFDs

⊆ ⊆

⊆

⊆

⊆

⊆ ⊆

⊆

⊆ R

S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

3. From reversible to arbitrary UIDs and UFDs

⊆ ⊆

⊆

⊆

⊆

⊆ ⊆

⊆

⊆ R

S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

5/7

3. From reversible to arbitrary UIDs and UFDs

⊆

⊆ ⊆

⊆

⊆ R

S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

3. From reversible to arbitrary UIDs and UFDs

⊆ ⊆

⊆

⊆ R

S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

5/7

3. From reversible to arbitrary UIDs and UFDs

R S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

3. From reversible to arbitrary UIDs and UFDs

R S

T U

UIDs can bedecomposed in reversiblesubsets The subsets can be solved independently Relies on the finite implicationconstruction [Cosmadakis et al., 1990]

5/7

4. From UFDs to FDs

a

b c

d e

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

6/7

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

6/7

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

a'

b' c'

d' e'

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

a'

b' c'

d' e'

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

6/7

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

a'

b' c'

d' e'

Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

a'

b' c'

d'

e' Create initially many reusable elements Reuse them innew combinations

Dense interpretations lemma:

linearly many elements givesuper-linearly many combinations

6/7

4. From UFDs to FDs

a

b c

d e

S[2] ⊆ R[2]

UID:

R[1,3] → R[2]

FD:

a'

b' c'

d' e' Create initially many reusable elements

5. From acyclic queries to arbitrary queries

e

d f

b

a c

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

7/7

5. From acyclic queries to arbitrary queries

e

d f

b

a c

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

5. From acyclic queries to arbitrary queries

e

d f

b

a c

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

7/7

5. From acyclic queries to arbitrary queries

e

d f

b

a c

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

query:

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

5. From acyclic queries to arbitrary queries

e

d f

e'

d' f'

b

a c

b'

a' c'

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

7/7

5. From acyclic queries to arbitrary queries

e

d f

e'

d' f'

b

a c

b'

a' c'

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

5. From acyclic queries to arbitrary queries

e

d f

e'

d' f'

b

a c

b'

a' c'

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]

Product with a group of high girth[Otto, 2002] Must betweakedto avoid violating FDs

7/7

5. From acyclic queries to arbitrary queries

e

d f

e'

d' f'

b

a c

b'

a' c'

S[2] ⊆ R[1]

UID:

S[2] ⊆ R[2]