From Horn-SRIQ to Datalog: A Data-Independent ... fileFrom Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann 3/15 The DL Horn-SRIQ: Syntax C1 ⊓…⊓Cn⊑

From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann

From Horn-SRIQ to Datalog:

/151

A Data-Independent Transformation that Preserves Assertion Entailment

David Carral, Larry González, and Patrick Koopmann

Poster: KRR5901

Introduction

From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /153

The DL Horn-SRIQ: SyntaxC1 ⊓ … ⊓ Cn ⊑ D → EnglishSpeaker ⊓ FrenchSpeaker ⊑ Bilingual,

Vehicle ⊑ Car, Vertebrate ⊓ Invertebrate ⊑ ⊥∃R . C ⊑ D → ∃Attends . Course ⊑ StudentC ⊑ ∃R . D → Director ⊑ ∃Directs . Movie

C ⊑ ≤ 1R . D → PhDStudent ⊑ ≤ 1HasThesisSupervisor . FacultyR1 ∘ … ∘ Rn ⊑ S → HasAncestor ∘ HasAncestor ⊑ HasAncestor, HasMother ⊑ HasParent,

HasParent ∘ HasSister ⊑ HasAuntR− ⊑ S → HasChild− ⊑ HasParent

C(a) → Person(david)R(a, b) → HasFriend(stan, kyle)


The DL Horn-SRIQ: SyntaxC1 ⊓ … ⊓ Cn ⊑ D → EnglishSpeaker ⊓ FrenchSpeaker ⊑ Bilingual,

Vehicle ⊑ Car, Vertebrate ⊓ Invertebrate ⊑ ⊥∃R . C ⊑ D → ∃Attends . Course ⊑ StudentC ⊑ ∃R . D → Director ⊑ ∃Directs . Movie

C ⊑ ≤ 1R . D → PhDStudent ⊑ ≤ 1HasThesisSupervisor . FacultyR1 ∘ … ∘ Rn ⊑ S → HasAncestor ∘ HasAncestor ⊑ HasAncestor, HasMother ⊑ HasParent,

HasParent ∘ HasSister ⊑ HasAuntR− ⊑ S → HasChild− ⊑ HasParent

C(a) → Person(david)R(a, b) → HasFriend(stan, kyle)


C1 ⊓ … ⊓ Cn ⊑ D ↦ ∀x . C1(x) ∧ … ∧ Cn(x) → D(x)∃R . C ⊑ D ↦ ∀x, y . R(x, y) ∧ C(y) → D(x)C ⊑ ∃R . D ↦ ∀x . C(x) → ∃y . R(x, y) ∧ D(y)

C ⊑ ≤ 1R . D ↦ ∀x, y, z . C(x) ∧ R(x, y) ∧ D(y) ∧ R(x, z) ∧ D(z) → y ≈ zR1 ∘ … ∘ Rn ⊑ S ↦ ∀x0, …, xn. R1(x0, x1) ∧ … ∧ Rn(xn− 1, xn) → R(x0, xn)

R− ⊑ S ↦ ∀x, y . R(x, y) → S(y, x)A(a) ↦ A(a)

R(a, b) ↦ R(a, b)

The DL Horn-SRIQ: Semantics


The DL Horn-SRIQ: Semantics

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x)∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x)C ⊑ ∃R . D ↦ C(x) → ∃y . R(x, y) ∧ D(y)

C ⊑ ≤ 1R . D ↦ C(x) ∧ R(x, y) ∧ D(y) ∧ R(x, z) ∧ D(z) → y ≈ zR1 ∘ … ∘ Rn ⊑ S ↦ R1(x0, x1) ∧ … ∧ Rn(xn− 1, xn) → R(x0, xn)

R− ⊑ S ↦ R(x, y) → S(y, x)A(a) ↦ A(a)

R(a, b) ↦ R(a, b)


The DL Horn-SRIQ: SemanticsTerminological axioms



R− ⊑ S ↦ R(x, y) → S(y, x)A(a) ↦ A(a)

R(a, b) ↦ R(a, b)



Assertions / Facts



R− ⊑ S ↦ R(x, y) → S(y, x)A(a) ↦ A(a)

R(a, b) ↦ R(a, b)



Assertions / Facts- = ⟨/, ℱ⟩

Set of facts / ABox

Set of terminological axioms / TBoxOntology



R− ⊑ S ↦ R(x, y) → S(y, x)A(a) ↦ A(a)

R(a, b) ↦ R(a, b)


DatalogFeatures(x, y) → Actor(y)

ActsIn(x, y) → Features(y, x)HasID(x, y) ∧ HasID(x, z) → y ≈ z

Directs(x, y) ∧ Features(y, z) → DirectsActor(x, z)

Reviews(x, y) ∧ IsAuthorOf(z, y) ∧ CollaboratesWith(x, y) → IllegalReviewer(x, y)P(x, y, z) ∧ S(y, w, v) ∧ V(y, v) → D(x)






Person(david)HasFriend(stan, kyle)

R(a, b, c)







R(a, b, c)

Rules







R(a, b, c)

Rules

Assertions / Facts







R(a, b, c)

Rules

Assertions / Facts

2 = ⟨ℛ, ℱ⟩Set of facts

Set of rulesProgram






Remark: Existential quantification is not allowed in Datalog.


R(a, b, c)

Rules

Assertions / Facts


Set of rulesProgram






Remark: Existential quantification is not allowed in Datalog.


R(a, b, c)

Rules

Assertions / Facts


Set of rulesProgramDirector ⊑ ∃Directs . Movie

Director(x) → ∃y . Directs(x, y) ∧ Movie(y)


TBox

Solving Assertion Retrieval with Datalog Rewritings

ABox


DL Reasoner

TBox


ABox


DL Reasoner

TBox


ABox

Entailed ABox Assertions(i.e., facts over named individuals)


Datalog Rule Set

Data-Independent Translation

DL Reasoner

TBox


ABox



Rule Engine

/156

Datalog Rule Set


DL Reasoner

TBox


ABox



Rule Engine

/156

Datalog Rule Set


DL Reasoner

TBox


ABox



Rule Engine

/156

Datalog Rule Set


DL Reasoner

TBox


ABox


Motivation* Research expressivity* Performance improvements


Rule Engine

/156

Datalog Rule Set


DL Reasoner

TBox

Challenges* Correctness and complexity* Implement translation and

evaluate performance


ABox


Motivation* Research expressivity* Performance improvements

Evaluation


Reasoning with Rewritings

Konclude

RDFox

TBox size: 485 Rewriting size: 549

TBox size: 304 Rewriting size: 367


Size of Rewritings

- MOWLCorpus: TBoxes with less 1000 axioms and containing role chain axioms

- 187 TBoxes - 121 computed rewritings w/o

OOM errors


Size of Rewritings

- MOWLCorpus: TBoxes with less 1000 axioms and containing role chain axioms

- 187 TBoxes - 121 computed rewritings w/o

OOM errors

From Horn-ALCHIQ to Datalog

From Horn-ALCHIQ to Datalog

R1 ∘ … ∘ Rn ⊑ S → R ⊑ S


Forest Model Property

C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S



C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D

∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S


“Unnamed-to-Named” Consequences


“Unnamed-to-Named” ConsequencesSuccessor-to-predecessor



C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

Successor-to-predecessor



C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D




C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

S



, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

S



, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S



, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S

Successor-to-predecessor Folding

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R

S

n: E

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R

S, R

n: E

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R

S, R

, Dn: E

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R

S, R

, Dn: E

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R, S

, E

C(x) → G(x)


, F

/1512


C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G

a : C

R

n: D

n′� : E

, G

S


a : C b : D

C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R, S

, E

C(x) → G(x) C(x) ∧ R(x, y) ∧ D(y) → S(x, y)C(x) ∧ R(x, y) ∧ D(y) → E(y)


Computing IQ Rewritings for Horn-ALCHIQConsider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .


Computing IQ Rewritings for Horn-ALCHIQ

For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/

Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .




For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/






For all C1 ⊓ … ⊓ Cn ⊑ D ∈ Ω(/)C1(x) ∧ … ∧ Cn(x) → D(x) ∈ ℛ/








Ω(/) is the set of all axioms of one of the following forms entailed by / .

C1 ⊓ … ⊓ Cn ⊑ DC1 ⊓ … ⊓ Cn ⊑ ∃(R1 ⊓ … ⊓ Rm) . (D1 ⊓ … ⊓ Dk)








C1 ⊓ … ⊓ Cn ⊑ DC1 ⊓ … ⊓ Cn ⊑ ∃(R1 ⊓ … ⊓ Rm) . (D1 ⊓ … ⊓ Dk)

Consequence-based Reasoning Calculi[IJCAI 2017] Yevgeny[AAAI 2012] Eiter et al.



For all C ⊑ ≤ 1R . D ∈ /,C(x) ∧ R(x, y) ∧ D(y) ∧ R(x, z) ∧ D(z) → y ≈ z ∈ ℛ/,C(x) ∧ C1(x) ∧ … ∧ Cn(x) ∧ R(x, y) ∧ D(y) → E(y) ∈ ℛ/if C1 ⊓ … ⊓ Cn ⊑ ∃R . (D ⊓ E) ∈ Ω(/), andC(x) ∧ C1(x) ∧ … ∧ Cn(x) ∧ R(x, y) ∧ D(y) → S(x, y) ∈ ℛ/if C1 ⊓ … ⊓ Cn ⊑ ∃(R ⊓ S) . D ∈ Ω(/)






C1 ⊓ … ⊓ Cn ⊑ DC1 ⊓ … ⊓ Cn ⊑ ∃(R1 ⊓ … ⊓ Rm) . (D1 ⊓ … ⊓ Dk)

Consequence-based Reasoning Calculi[IJCAI 2017] Yevgeny[AAAI 2012] Eiter et al.

From Horn-SRIQ to Datalog

From Horn-SRIQ to Datalog

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From Horn-SRIQ to Datalog:

/1515

A Data-Independent Transformation that Preserves Assertion Entailment

David Carral, Larry González, and Patrick Koopmann

Poster: KRR5901

Documents

From Horn-SRIQ to Datalog: A Data-Independent ... fileFrom Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann 3/15 The DL Horn-SRIQ: Syntax C1 ⊓…⊓Cn⊑