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 : Syntax
C 1 ⊓ … ⊓ C n ⊑ D → EnglishSpeaker ⊓ FrenchSpeaker ⊑ Bilingual , Vehicle ⊑ Car , Vertebrate ⊓ Invertebrate ⊑ ⊥
∃ R . C ⊑ D → ∃ Attends . Course ⊑ Student C ⊑ ∃ R . D → Director ⊑ ∃ Directs . Movie
C ⊑ ≤ 1 R . D → PhDStudent ⊑ ≤ 1 HasThesisSupervisor . Faculty
R 1 ∘ … ∘ R n ⊑ S → HasAncestor ∘ HasAncestor ⊑ HasAncestor , HasMother ⊑ HasParent , HasParent ∘ HasSister ⊑ HasAunt
R − ⊑ S → HasChild − ⊑ HasParent C ( a ) → Person ( david )
R ( a , b ) → HasFriend ( stan , kyle )
The DL Horn- SRIQ : Syntax
C 1 ⊓ … ⊓ C n ⊑ D → EnglishSpeaker ⊓ FrenchSpeaker ⊑ Bilingual , Vehicle ⊑ Car , Vertebrate ⊓ Invertebrate ⊑ ⊥
∃ R . C ⊑ D → ∃ Attends . Course ⊑ Student C ⊑ ∃ R . D → Director ⊑ ∃ Directs . Movie
C ⊑ ≤ 1 R . D → PhDStudent ⊑ ≤ 1 HasThesisSupervisor . Faculty
R 1 ∘ … ∘ R n ⊑ S → HasAncestor ∘ HasAncestor ⊑ HasAncestor , HasMother ⊑ HasParent , HasParent ∘ HasSister ⊑ HasAunt
R − ⊑ S → HasChild − ⊑ HasParent
C ( a ) → Person ( david )
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /154
C 1 ⊓ … ⊓ C n ⊑ D ↦ ∀ x . C 1 ( x ) ∧ … ∧ C n ( 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 ⊑ ≤ 1 R . D ↦ ∀ x , y , z . C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z R 1 ∘ … ∘ R n ⊑ S ↦ ∀ x 0 , …, x n . R 1 ( x 0 , x 1 ) ∧ … ∧ R n ( x n − 1 , x n ) → R ( x 0 , x n )
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
C 1 ⊓ … ⊓ C n ⊑ D ↦ C 1 ( x ) ∧ … ∧ C n ( 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 ⊑ ≤ 1 R . D ↦ C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z R 1 ∘ … ∘ R n ⊑ S ↦ R 1 ( x 0 , x 1 ) ∧ … ∧ R n ( x n − 1 , x n ) → R ( x 0 , x n )
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 : Semantics
Terminological axioms
C 1 ⊓ … ⊓ C n ⊑ D ↦ C 1 ( x ) ∧ … ∧ C n ( 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 ⊑ ≤ 1 R . D ↦ C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z R 1 ∘ … ∘ R n ⊑ S ↦ R 1 ( x 0 , x 1 ) ∧ … ∧ R n ( x n − 1 , x n ) → R ( x 0 , x n )
R − ⊑ S ↦ R ( x , y ) → S ( y , x ) A ( a ) ↦ A ( a )
R ( a , b ) ↦ R ( a , b )
The DL Horn- SRIQ : Semantics
Terminological axioms
Assertions / Facts
C 1 ⊓ … ⊓ C n ⊑ D ↦ C 1 ( x ) ∧ … ∧ C n ( 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 ⊑ ≤ 1 R . D ↦ C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z R 1 ∘ … ∘ R n ⊑ S ↦ R 1 ( x 0 , x 1 ) ∧ … ∧ R n ( x n − 1 , x n ) → R ( x 0 , x n )
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 : Semantics
Terminological axioms
Assertions / Facts
𝒪 = ⟨𝒯, ℱ⟩
Set of facts / ABox Set of terminological axioms / TBox
Ontology
C 1 ⊓ … ⊓ C n ⊑ D ↦ C 1 ( x ) ∧ … ∧ C n ( 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 ⊑ ≤ 1 R . D ↦ C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z R 1 ∘ … ∘ R n ⊑ S ↦ R 1 ( x 0 , x 1 ) ∧ … ∧ R n ( x n − 1 , x n ) → R ( x 0 , x n )
R − ⊑ S ↦ R ( x , y ) → S ( y , x ) A ( a ) ↦ A ( a )
R ( a , b ) ↦ R ( a , b )
Datalog
Features ( 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
Datalog
Features ( 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 )
Datalog
Features ( 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
Datalog
Features ( 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
Datalog
Features ( 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
𝒫 = ⟨ℛ, ℱ⟩
Set of rules
Program
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /155
Datalog
Features ( 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
𝒫 = ⟨ℛ, ℱ⟩
Set of facts Set of rules
Program
Datalog
Features ( 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
𝒫 = ⟨ℛ, ℱ⟩
Set of rules
Program
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /156
TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
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)
Datalog Rule Set
Data-Independent Translation
DL Reasoner TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
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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)
Rule Engine
Datalog Rule Set
Data-Independent Translation
DL Reasoner TBox
Solving Assertion Retrieval with Datalog Rewritings
ABox
Entailed ABox
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
Rule Engine
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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
Rule Engine
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
Motivation
* Research expressivity
* Performance improvements
Evaluation
Reasoning with Rewritings
Konclude
RDFox
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
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
R 1 ∘ … ∘ R n ⊑ S → R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1511
Forest Model Property
C ⊑ ∃ R . D
C 1 ⊓ … ⊓ C n ⊑ D
∃ R . C ⊑ D C ⊑ ≤ 1 R . D
R ⊑ S
“Unnamed-to-Named” Consequences
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
Successor-to-predecessor
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1512
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G
a : C R
n : D
Successor-to-predecessor
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G
a : C R
n : D n ′ : E S
Successor-to-predecessor
, F
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G
a : C R
n : D n ′ : E
, G S
Successor-to-predecessor
C ( x ) → G ( x )
, F
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor Folding
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ 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 . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R
C ( x ) → G ( x )
, F
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor Folding
a : C b : D C ⊑ ∃ S . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R S
n : E
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ 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 . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R S , R
n : E
C ( x ) → G ( x )
, F
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor Folding
a : C b : D C ⊑ ∃ S . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R S , R
, D
n : E
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ 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 . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R S , R
, D n : E
C ( x ) → G ( x )
, F
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ S . E
∃ S . E ⊑ F
∃ R . F ⊑ G R
n : D n ′ : E S
Successor-to-predecessor Folding
a : C b : D C ⊑ ∃ S . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . D R , S
, E
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann
, F
/15 12
“Unnamed-to-Named” Consequences
C ⊑ ∃ R . D D ⊑ ∃ 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 . E
S ⊑ R E ⊑ D
C ⊑ ≤ 1 R . 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- ALCHIQ
Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯
Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯 For all R ⊑ S ∈ 𝒯,
R ( x , y ) → S ( x , y ) ∈ ℛ 𝒯
Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯 For all R ⊑ S ∈ 𝒯,
R ( x , y ) → S ( x , y ) ∈ ℛ 𝒯
For all C 1 ⊓ … ⊓ C n ⊑ D ∈ Ω(𝒯) C 1 ( x ) ∧ … ∧ C n ( x ) → D ( x ) ∈ ℛ 𝒯 Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯 For all R ⊑ S ∈ 𝒯,
R ( x , y ) → S ( x , y ) ∈ ℛ 𝒯
For all C 1 ⊓ … ⊓ C n ⊑ D ∈ Ω(𝒯) C 1 ( x ) ∧ … ∧ C n ( x ) → D ( x ) ∈ ℛ 𝒯 Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
Ω(𝒯) is the set of all axioms of one of the following forms entailed by 𝒯 .
C 1 ⊓ … ⊓ C n ⊑ D
C 1 ⊓ … ⊓ C n ⊑ ∃( R 1 ⊓ … ⊓ R m ) . ( D 1 ⊓ … ⊓ D k )
From Horn-SRIQ to Datalog David Carral, Larry González, and Patrick Koopmann /1513
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯 For all R ⊑ S ∈ 𝒯,
R ( x , y ) → S ( x , y ) ∈ ℛ 𝒯
For all C 1 ⊓ … ⊓ C n ⊑ D ∈ Ω(𝒯) C 1 ( x ) ∧ … ∧ C n ( x ) → D ( x ) ∈ ℛ 𝒯 Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
Ω(𝒯) is the set of all axioms of one of the following forms entailed by 𝒯 .
C 1 ⊓ … ⊓ C n ⊑ D
C 1 ⊓ … ⊓ C n ⊑ ∃( R 1 ⊓ … ⊓ R m ) . ( D 1 ⊓ … ⊓ D k )
Consequence-based Reasoning Calculi [IJCAI 2017] Yevgeny
[AAAI 2012] Eiter et al.
Computing IQ Rewritings for Horn- ALCHIQ
For all C ⊑ ≤ 1 R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z ∈ ℛ , For all C ⊑ ∀ R . D ∈ 𝒯,
C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒯 For all R ⊑ S ∈ 𝒯,
R ( x , y ) → S ( x , y ) ∈ ℛ 𝒯
For all C 1 ⊓ … ⊓ C n ⊑ D ∈ Ω(𝒯) C 1 ( x ) ∧ … ∧ C n ( x ) → D ( x ) ∈ ℛ 𝒯 Consider some Horn-ALCHIQ TBox 𝒯 .
Then, the rule set ℛ 𝒯 defined as follows is an IQ-preserving rewriting for 𝒯 .
Ω(𝒯) is the set of all axioms of one of the following forms entailed by 𝒯 .
C 1 ⊓ … ⊓ C n ⊑ D
C 1 ⊓ … ⊓ C n ⊑ ∃( R 1 ⊓ … ⊓ R m ) . ( D 1 ⊓ … ⊓ D k )
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
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