Query Answering with Guarded Rules and Expressive Constraints

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Query Answering with Guarded Rules and Expressive Constraints

Antoine Amarilli^1,2, Michael Benedikt², Pierre Bourhis³, Michael Vanden Boom² October 20, 2016

1Télécom ParisTech, Paris, France

2University of Oxford, Oxford, United Kingdom

3CNRS CRIStAL, Lille 1/33

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Open-world query answering (QA)

• We aregiven:

RelationalinstanceI(ground facts)

! LogicalconstraintsΣ

? Boolean conjunctivequeryq

(existentially quantified conjunction of atoms)

• Weask:

• Consider all possiblecompletionsJ⊇I

• Restrict to those that satisfy theconstraintsΣ

→ Isqcertainamong them?

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Open-world query answering (QA)

• We aregiven:

RelationalinstanceI(ground facts)

! LogicalconstraintsΣ

? Boolean conjunctivequeryq

(existentially quantified conjunction of atoms)

• Weask:

• Consider all possiblecompletionsJ⊇I

• Restrict to those that satisfy theconstraintsΣ

→ Isqcertainamong them?

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QA example

Relational instanceI

• CoSup(Jamie)–Jamie is a co-supervised student

! Logical constraintsΣ

• ∀xCoSup(x)→ ∃y zAdv(x,y)∧Adv(x,z)∧Knows(y,z)∧y6=z A co-supervised student has two advisors that know each other

• ∀x yAdv(x,y)→Knows(x,y) Advisors know their students

• ∀x yKnows(x,y)→Knows(y,x)

“Knows” is symmetric

? Boolean conjunctive queryq

• ∃x y zKnows(x,y)∧Knows(y,z)∧Knows(x,z) Is there a clique of 3 people that know each other?

→ The query iscertainunder the instance and constraints

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QA example

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QA example

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QA example

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QA example

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QA example

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QA tasks

Is QAdecidable, and what is thecomplexity?

→ Depends on the language used forlogical constraints

→ E.g., for arbitraryfirst-orderconstraints, QA isundecidable (becausesatisfiabilityis undecidable)

→ Find fragments offirst-order logicwithdecidable QA

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QA tasks

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QA tasks

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QA tasks

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Tuple-generating dependencies (or existential rules)

∀x yφ(x,y)→ ∃zψ(y,z)

where thebodyφandheadψareconjunctions of atoms

→ Guarded TGDs:bodyφhas an atom withallbody variablesx,y

∀x y1y2S(x,y1)∧S(x,y2)∧U(x,y1,y2)→ ∃z S(y2,z)∧T(y1)

→ Frontier-one TGDs:thefrontieryhas onlyonevariable

∀xyS(x,y)→ ∃z S(y,z)

→ Frontier-guarded TGDs (FGTGDs):

bodyφhas an atom withallfrontier variablesy

∀x y1y2S(x,y1)∧S(x,y2)∧R(y1,y2)→ ∃z S(y2,z)∧T(y1)

→ QA for FGTGDs isdecidable[Baget et al., 2011]

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∀x y y S(x,y )∧S(x,y )∧R(y ,y )→ ∃z S(y ,z)∧T(y )

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FGTGDs cannot express everything

• Disjunction:Professors must have a PhDoran equivalent title Negation:Professorswithoutan HDR must have co-supervisors

→ Handled byguarded logicsandguarded negation logics

• Number constraints:Co-supervised students havetwo advisors Functionality:Any person is born inexactly oneplace

→ ∀xy₁y₂ Born(x,y₁)∧Born(x,y₂)→y₁ =y₂ unguarded frontier!

• Transitivity:isLocatedIn, isPartOf, subclass...

→ ∀x y₁y₂Part(y₁,x)∧Part(x,y₂)→Part(y₁,y₂) unguarded frontier!

• Totality:All elephants are bigger than all mice

→ ∀y1y2Elephant(y1)∧Mouse(y2)→y1<y2 unguarded frontier!

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→ ∀xy₁y₂Born(x,y₁)∧Born(x,y₂)→y₁ =y₂

unguarded frontier!

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→ ∀xy₁y₂Born(x,y₁)∧Born(x,y₂)→y₁ =y₂ unguarded frontier!

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→ ∀x y₁y₂Part(y₁,x)∧Part(x,y₂)→Part(y₁,y₂)

unguarded frontier!

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unguarded frontier!

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This talk

Can we extend frontier-guarded TGDs to capture some unguarded constraints while preserving the decidability of QA?

1. Combining Existential Rules and Description Logics A. A., Michael Benedikt, IJCAI’15.

→ Extend FGTGDs withdescription logicconstraints

on arity-two data (includes disjunction, negation, counting) 2. Query Answering with Transitive and Linear-Ordered Data

A. A., Michael Benedikt, P. Bourhis, M. Vanden Boom, IJCAI’16

→ Extend FGTGDs withtransitive relations

andorder relationsthat cannot be used as guards

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This talk

on arity-two data (includes disjunction, negation, counting)

2. Query Answering with Transitive and Linear-Ordered Data A. A., Michael Benedikt, P. Bourhis, M. Vanden Boom, IJCAI’16

→ Extend FGTGDs withtransitive relations

andorder relationsthat cannot be used as guards

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This talk

on arity-two data (includes disjunction, negation, counting) 2. Query Answering with Transitive and Linear-Ordered Data

A. A., Michael Benedikt, P. Bourhis, M. Vanden Boom, IJCAI’16

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Table of contents

Introduction

Combining Existential Rules and Description Logics

Query Answering with Transitive and Linear-Ordered Data

Conclusion

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Rich Description Logics

Rich description logics(DLs) FGTGDs

Emp v CEOt(∃Mgr⁻.Emp) ∀pwvAcpt(p,w,v) → ∃f Trip(p,f,v) Arity-twoonly Arbitraryarity

Rich:disjunction, etc. Conjunctionandimplicationonly Functionality

asserts Funct(Mgr⁻)

! ^n/a

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Emp v CEOt(∃Mgr⁻.Emp) ∀pwvAcpt(p,w,v) → ∃f Trip(p,f,v)

Arity-twoonly Arbitraryarity

! ^n/a

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! ^n/a

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Rich:disjunction, etc. Conjunctionandimplicationonly

Functionality asserts Funct(Mgr⁻)

! ^n/a

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! ^n/a

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Our problem

Can we have the best of both worlds?

• QA is decidable forfrontier-guarded TGDs

• QA is decidable forrich DLs(i.e., expressible inGC², guarded two-variable first-order logic with counting)

→ Is QA decidable forrich DLs+ someclasses of FGTGDs? We show:

• QA isundecidablefor richDLsandFGTGDs

• QA with rich DLs isdecidablefor some newFGTGD classes

• Functional dependenciescan be added under someconditions even tohigher-arity relations

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Our problem

→ Is QA decidable forrich DLs+ someclasses of FGTGDs?

We show:

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Our problem

We show:

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Our problem

We show:

• Functional dependenciescan be added under someconditions

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Undecidability of frontier-guarded plus DLs

Theorem

QA isundecidablefor rich DLs and FGTGDs

Problem:

• DLs can expressFunct(↔functional dependencies, FDs)

• FGTGDs can expressinclusion dependencies(IDs)

∀x yA(x,y)→ ∃zB(y,z)with no variable repetitions

• Implicationof IDs and FDs isundecidable[Mitchell, 1983]

• Implicationreduces toQA [Calì et al., 2003]

→ Restrict tofrontier-oneTGDs: ∀xyφ(x,y)→ ∃zψ(y,z)

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Theorem

QA isundecidablefor rich DLs and FGTGDs Problem:

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Theorem

QA isundecidablefor rich DLs and FGTGDs Problem:

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Undecidability of frontier-one plus DLs

• Restrict tofrontier-oneTGDs: ∀xyφ(x,y)→ ∃zψ(y,z)

• QA for frontier-oneIDsplus FDs isdecidable(separability)

However: Theorem

QA isundecidablefor rich DLs and frontier-one TGDs

Problem:

• Rule heads and bodies may containcycles

• We haveFunctassertions

→ We can build agridand encodetiling problems

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• QA for frontier-oneIDsplus FDs isdecidable(separability) However:

Theorem

Problem:

→ We can build agridand encodetiling problems

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• QA for frontier-oneIDsplus FDs isdecidable(separability) However:

Theorem

Problem:

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Undecidability of frontier-one plus DLs: proof We reduce fromtiling problems:

• finite set ofcolors: , ,

• horizontalandverticalallowed pairs:

Thetiling problemis:

• input:initial configuration:

• output: is there aninfinite tiling?

. . . . . . . . . ... ... ... ... ... ...

→ Undecidablefor some sets of colors and configurations

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. . . . . . . . . ... ... ... ... ... ...

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. . . . . . . . . ... ... ... ... ... ...

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. . . . . . . . . ... ... ... ... ... ...

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• output: is there aninfinite tiling? . . .

. . . . . . ... ... ... ... ... ...

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• output: is there aninfinite tiling? . . .

. . . . . .

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Undecidability of frontier-one plus DLs: proof, cont’d

• FunctionalrelationsDfordownandRforright

• Unary predicateTfortilesandC for eachcolor

Initial instance:C −−−−→^R C −−−−→^R C −−−−→^R C

! Constraints:

• DL Disjunctionto color tiles:TvC tC tC

• Frontier-one TGD:∀yT(y)⇒ ∃tzw

T(y) −−−−→^R T(t)



 y

D



 y

D

T(z) −−−−→^R T(w)

? Query: ∃x y C (x)−−−−→^R C (y)for all forbidden pairs

→ There is anextension of the instanceiff there is atiling

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• Unary predicateTfortilesandC for eachcolor Initial instance:C −−−−→^R C −−−−→^R C −−−−→^R C

! Constraints:

T(y) −−−−→^R T(t)



 y

D



 y

D

T(z) −−−−→^R T(w)

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! Constraints:

T(y) −−−−→^R T(t)



 y

D



 y

D

T(z) −−−−→^R T(w)

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! Constraints:

T(y) −−−−→^R T(t)



 y

D



 y

D

T(z) −−−−→^R T(w)

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! Constraints:

T(y) −−−−→^R T(t)



 y

D



 y

D

T(z) −−−−→^R T(w)

? Query:∃x y C (x)−−−−→^R C (y)for all forbidden pairs

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Decidability of non-looping frontier-one and DLs

Idea:prohibitcyclesin FGTGDs

• R(x,y)S(y,z)T(z,x)is acycle

• R(z,x,y)S(x,y,w)is also acycle

Formally:

• Berge cycle: cycle in the atom–variableincidence graph

• Non-loopingconjunction: no cycle except, e.g.,R(x,y)S(x,y)

• Non-looping frontier-one: non-looping body and head Theorem

QA isdecidablefor non-looping frontier-one TGDs + rich DLs

x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Formally:

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x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Formally:

x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Formally:

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x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Formally:

• Non-looping frontier-one: non-looping body and head

Theorem

x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Formally:

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x y z

R(x,y) T(z,x) S(y,z)

x y

z w

R(z,x,y) S(x,y,w)

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Decidability of non-looping frontier-one and DLs (proof )

• ShredR(a,b,c)toR1(f,a),R2(f,b),R3(f,c)

• AxiomatizetheR_i, e.g.,∀f ∃⁼¹xR1(f,x)

→ QA for theshreddedinstance, TGDs, query, and axioms isequivalentto QA for the original instance, TGDs, query

• Rewriteshredded non-looping frontier-one TGDs toGC²:

• Rewrite∀xyφ(x,y)⇒ ∃zψ(y,z)to∀yφ⁰(y)⇒ψ⁰(y),

withφ⁰(y)andψ⁰(y)the shredding of∃xφ(x,y)and∃zψ(y,z)

• Example:φ(y) :=∃x1x2T(y,x1)∧R(y,y,x2)∧A(x2)

→ ∃x1x2f T(y,x1)∧R1(f,y)∧R2(f,y)∧R3(f,x2)∧A(x2)

→

∃x1T(y,x1)

∧

∃x2f R1(f,y)∧R2(f,y)∧ ∃y2R3(f,y2)∧A(y2)

→ Reduces toQA for GC²: decidable [Pratt-Hartmann, 2009]

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Decidability of non-looping frontier-one and DLs (proof )

• ShredR(a,b,c)toR1(f,a),R2(f,b),R3(f,c)

• AxiomatizetheR_i, e.g.,∀f ∃⁼¹xR1(f,x)

→ QA for theshreddedinstance, TGDs, query, and axioms isequivalentto QA for the original instance, TGDs, query

• Rewriteshredded non-looping frontier-one TGDs toGC²:

• Rewrite∀xyφ(x,y)⇒ ∃zψ(y,z)to∀yφ⁰(y)⇒ψ⁰(y),

withφ⁰(y)andψ⁰(y)the shredding of∃xφ(x,y)and∃zψ(y,z)

• Example:φ(y) :=∃x1x2T(y,x1)∧R(y,y,x2)∧A(x2)

→ ∃x1x2f T(y,x1)∧R1(f,y)∧R2(f,y)∧R3(f,x2)∧A(x2)

→

∃x1T(y,x1)

∧

∃x2f R1(f,y)∧R2(f,y)∧ ∃y2R3(f,y2)∧A(y2)

→ Reduces toQA for GC²: decidable [Pratt-Hartmann, 2009]

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