Post on 30-Aug-2019
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
From Horn-SRIQ to Datalog:
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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)
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
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
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
The DL Horn-SRIQ: SemanticsTerminological axioms
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
The DL Horn-SRIQ: SemanticsTerminological axioms
Assertions / Facts
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
The DL Horn-SRIQ: SemanticsTerminological axioms
Assertions / Facts- = ⟨/, ℱ⟩
Set of facts / ABox
Set of terminological axioms / TBoxOntology
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
Rules
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
Rules
Assertions / Facts
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
Rules
Assertions / Facts
2 = ⟨ℛ, ℱ⟩Set of facts
Set of rulesProgram
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
Remark: Existential quantification is not allowed in Datalog.
Person(david)HasFriend(stan, kyle)
R(a, b, c)
Rules
Assertions / Facts
2 = ⟨ℛ, ℱ⟩Set of facts
Set of rulesProgram
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
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)
Remark: Existential quantification is not allowed in Datalog.
Person(david)HasFriend(stan, kyle)
R(a, b, c)
Rules
Assertions / Facts
2 = ⟨ℛ, ℱ⟩Set of facts
Set of rulesProgramDirector ⊑ ∃Directs . Movie
Director(x) → ∃y . Directs(x, y) ∧ Movie(y)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /156
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /156
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /156
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /156
Datalog Rule Set
Data-Independent Translation
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
/156
Datalog Rule Set
Data-Independent Translation
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
/156
Datalog Rule Set
Data-Independent Translation
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
/156
Datalog Rule Set
Data-Independent Translation
DL Reasoner
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
Motivation* Research expressivity* Performance improvements
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
/156
Datalog Rule Set
Data-Independent Translation
DL Reasoner
TBox
Challenges* Correctness and complexity* Implement translation and
evaluate performance
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox Assertions(i.e., facts over named individuals)
Motivation* Research expressivity* Performance improvements
Evaluation
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /158
Reasoning with Rewritings
Konclude
RDFox
TBox size: 485 Rewriting size: 549
TBox size: 304 Rewriting size: 367
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /159
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-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /159
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
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃R . DC1 ⊓ … ⊓ Cn ⊑ D
∃R . C ⊑ DC ⊑ ≤ 1R . DR ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” ConsequencesSuccessor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
S
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
S
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R
S
n: E
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R
S, R
n: E
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R
S, R
, Dn: E
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R
S, R
, Dn: E
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
a : C b : D
C ⊑ ∃S. ES ⊑ RE ⊑ DC ⊑ ≤ 1R . D R, S
, E
C(x) → G(x)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/1512
“Unnamed-to-Named” Consequences
C ⊑ ∃R . DD ⊑ ∃S. E∃S. E ⊑ F∃R . F ⊑ G
a : C
R
n: D
n′� : E
, G
S
Successor-to-predecessor Folding
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)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQConsider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
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 / .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQ
For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/
For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/
Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQ
For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/
For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/
For all C1 ⊓ … ⊓ Cn ⊑ D ∈ Ω(/)C1(x) ∧ … ∧ Cn(x) → D(x) ∈ ℛ/
Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQ
For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/
For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/
For all C1 ⊓ … ⊓ Cn ⊑ D ∈ Ω(/)C1(x) ∧ … ∧ Cn(x) → D(x) ∈ ℛ/
Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
Ω(/) is the set of all axioms of one of the following forms entailed by / .
C1 ⊓ … ⊓ Cn ⊑ DC1 ⊓ … ⊓ Cn ⊑ ∃(R1 ⊓ … ⊓ Rm) . (D1 ⊓ … ⊓ Dk)
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQ
For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/
For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/
For all C1 ⊓ … ⊓ Cn ⊑ D ∈ Ω(/)C1(x) ∧ … ∧ Cn(x) → D(x) ∈ ℛ/
Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
Ω(/) is the set of all axioms of one of the following forms entailed by / .
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 David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn-ALCHIQ
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(/)
For all C ⊑ ∀R . D ∈ /,C(x) ∧ R(x, y) → D(y) ∈ ℛ/
For all R ⊑ S ∈ /,R(x, y) → S(x, y) ∈ ℛ/
For all C1 ⊓ … ⊓ Cn ⊑ D ∈ Ω(/)C1(x) ∧ … ∧ Cn(x) → D(x) ∈ ℛ/
Consider some Horn-ALCHIQ TBox / .Then, the rule set ℛ/ defined as follows is an IQ-preserving rewriting for / .
Ω(/) is the set of all axioms of one of the following forms entailed by / .
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 David Carral, Larry González, and Patrick Koopmann
From Horn-SRIQ to Datalog:
/1515
A Data-Independent Transformation that Preserves Assertion Entailment
David Carral, Larry González, and Patrick Koopmann
Poster: KRR5901