From Horn- SRIQ to Datalog: A Data-Independent Transformation that - PowerPoint PPT Presentation

From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment David Carral , Larry González, and Patrick Koopmann 1 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Introduction 2 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

⃗ ⃗ ⃗ ⃗ Syntax Datalog Horn- SRIQ Rules TBox Axioms C 1 ⊓ … ⊓ C n ⊑ D C ⊑ ∃ R . D P 1 ( x 1 ) ∧ … ∧ P n ( x n ) → Q ( y ) ∃ R . C ⊑ D C ⊑ ≤ 1 R . D R − ⊑ S Formulas R 1 ∘ … ∘ R n ⊑ S Facts ABox Axioms (or Facts) P ( c ) C ( a ) R ( a , b ) Programs Ontologies Theories O = ( T , F ) P = ( R , F ) 3 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

From Horn- SRIQ to Datalog Definition. A rule set R is an AR-rewriting for a TBox T iff, for all fact sets F , * the ontology ( T , F ) and the program ( R , F ) are equi-satisfiable and, * for all facts 𝞫 over the signature of T , ( T , F ) entails 𝞫 iff ( R , F ) entails 𝞫 . Can we compute AR-rewritings ? * Reasoning in Description Logics by a Reduction to Disjunctive Datalog. Hustadt, Motik, and Sattler. In Journal of Autom. Reasoning 2007. ALCHOIQ * The Combined Approach to Query Answering in Horn- . Carral, Dragoste, and Krötzsch. In KR 2018. What about Horn- SRIQ ? Yes! Wait… but why is this interesting? 4 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Evaluation 5 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Reasoning with Rewritings RDFox Konclude TBox size: 485 TBox size: 304 Rewriting size: 549 Rewriting size: 367 Time: 221s Time: 182s 6 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Size of Rewritings - MOWLCorpus: TBoxes with less 1000 axioms and containing role chain axioms - 187 TBoxes: 121 computed rewritings w/o OOM errors 7 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

From Horn- ALCHIQ to Datalog R 1 ∘ … ∘ R n ⊑ S → R ⊑ S 8 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Forest Model Property C 1 ⊓ … ⊓ C n ⊑ D ∃ R . C ⊑ D C ⊑ ∃ R . D C ⊑ ≤ 1 R . D R ⊑ S 9 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

“Unnamed-to-Named” Consequences Successor-to-predecessor Folding n : E , D C ⊑ ∃ S . E n ′ � : E C ⊑ ∃ R . D S ⊑ R D ⊑ ∃ S . E E ⊑ D S S , R ∃ S . E ⊑ F F ⊑ ≤ 1 R . D ∃ R . F ⊑ G n : D , F R , S b : D , E a : C , F R C ( x ) → G ( x ) C ( x ) ∧ F ( x ) ∧ R ( x , y ) ∧ D ( y ) → S ( x , y ) a : C , G C ( x ) ∧ F ( x ) ∧ R ( x , y ) ∧ D ( y ) → E ( y ) 10 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Computing AR-Rewritings for Horn- ALCHIQ Definition. Consider some Horn- TBox 𝒰 . ALCHIQ The rule set ℛ 𝒰 , which is an AR-preserving rewriting for 𝒰 , is defined as follows : 3. For all R − ⊑ S ∈ 𝒰 , 1. For all C ⊑ ∀ R . D ∈ 𝒰 , 2. For all R ⊑ S ∈ 𝒰 , C ( x ) ∧ R ( x , y ) → D ( y ) ∈ ℛ 𝒰 R ( x , y ) → S ( x , y ) ∈ ℛ 𝒰 R ( y , x ) → S ( x , y ) ∈ ℛ 𝒰 4. For all C 1 ⊓ … ⊓ C n ⊑ D ∈ Ω ( 𝒰 ), C 1 ( x ) ∧ … ∧ C n ( x ) → D ( x ) ∈ ℛ 𝒰 Successor-to-predecessor 5. For all C ⊑ ≤ 1 R . D ∈ 𝒰 , Folding C ( x ) ∧ R ( x , y ) ∧ D ( y ) ∧ R ( x , z ) ∧ D ( z ) → y ≈ z ∈ ℛ 𝒰 , C ( x ) ∧ C 1 ( x ) ∧ … ∧ C n ( x ) ∧ R ( x , y ) ∧ D ( y ) → E ( y ) ∈ ℛ 𝒰 if C 1 ⊓ … ⊓ C n ⊑ ∃ R . ( D ⊓ E ) ∈ Ω ( 𝒰 ), and C ( x ) ∧ C 1 ( x ) ∧ … ∧ C n ( x ) ∧ R ( x , y ) ∧ D ( y ) → S ( x , y ) ∈ ℛ 𝒰 if C 1 ⊓ … ⊓ C n ⊑ ∃ ( R ⊓ S ) . D ∈ Ω ( 𝒰 ) Remarks Definition. Ω ( 𝒰 ) is the set of all axioms of 𝒰 * is exponential in ℛ 𝒰 either of the following forms entailed by 𝒰 . Ω ( 𝒰 ) * Compute using consequence-based C 1 ⊓ … ⊓ C n ⊑ D Query Rewriting for Horn- plus Rules. SHIQ C 1 ⊓ … ⊓ C n ⊑ ∃ ( R 1 ⊓ … ⊓ R m ) . ( D 1 ⊓ … ⊓ D k ) Eiter, Ortiz, Simkus, Tran, and Xiao. In AAAI 2012. 11 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

From Horn- SRIQ to Datalog 12 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Complex Roles and NFA } 𝒰 = { V ∘ X ∘ Y ⊑ R , R ∘ S ∘ T ⊑ R , W ⊑ X , R ∘ R ⊑ R q 3 𝒪 𝒰 ( R ) : S T ϵ R i R f R q 1 q 2 V X Y W 13 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Box Pushing q 3 S T A ⊑ ∀ R . B ∈ 𝒰 ϵ BP ( 𝒰 ) ⊇ 𝒰 ∪ { R i R f R A ⊑ B i R , B f R ⊑ B , q 1 q 2 V X Y B i R ⊑ ∀ R . B f R , W B f R ∀ S . B q 3 , B q 3 ⊑ ∀ T . B f R , B i R ⊑ ∀ V . B q 1 , B q 1 ⊑ ∀ X . B q 2 , B q 2 ⊑ ∀ Y . B f R , } B q 1 ⊑ ∀ W . B q 2 , B f R ⊑ B i R 14 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Computing “AR-Rewritings” for Horn- SRIQ Definition. Consider some Horn- TBox 𝒰 . SRIQ 1. For all roles R in 𝒰 , compute the NFA 𝒪 𝒰 ( R ) . 2. Compute the TBox 𝒰′ � which results from adding all the axioms obtained via "box pushing", and then removing all axioms with role chains . 3. Compute the AR-rewriting ℛ 𝒰′ � for the TBox 𝒰′ � (as defined in previous slides). 4. The rule set ℛ 𝒰′ � can be used to solve class retrieval "in place" of 𝒰 . Remarks 𝒰′ � 𝒰 * is kind of an “AR-rewriting” for , but only for class assertions! 𝒰′ � ALCHIQ * is a Horn- TBox 15 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

“Unnamed-to-Named” Role Consequences Unnamed Paths V i V f V R − C ⊑ ∃ R . ⊤ Y W R S R − ⊑ S q 1 q 2 q 3 Y ( x , y ) → V q 1 ( x , y ) Y ∘ R ∘ S ∘ W ⊑ V V q 2 ( x , y ) ∧ S ( y , z ) → V q 3 ( x , z ) V q 1 ( x , y ) ∧ R ( y , z ) → V q 2 ( x , z ) S V q 3 ( x , y ) ∧ W ( y , z ) → V f V ( x , z ) R C ( x ) → V q 1 , q 3 ( x ) Y W V q 1 ( x , y ) ∧ V q 1 , q 3 ( y ) → V q 3 ( x , z ) a c b : C V f V ( x , y ) → V ( x , y ) V 16 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

“Unnamed-to-Named” Role Consequences Unnamed Paths V i X f X R − C ⊑ ∃ R . ⊤ Y W R S R − ⊑ S q 1 q 2 q 3 Y ( x , y ) → V q 1 ( x , y ) Y ∘ R ∘ S ∘ W ⊑ V V q 2 ( x , y ) ∧ S ( y , z ) → V q 3 ( x , z ) V q 1 ( x , y ) ∧ R ( y , z ) → V q 2 ( x , z ) V f V , V V q 3 ( x , y ) ∧ W ( y , z ) → V f V ( x , z ) V q 1 C ( x ) → V q 1 , q 3 ( x ) Y W V q 1 ( x , y ) ∧ V q 1 , q 3 ( y ) → V q 3 ( x , z ) a c b : C , V q 1, q 3 V q 3 V f V ( x , y ) → V ( x , y ) 17 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Computing AR-Rewritings for Horn- SRIQ Definition. Consider some Horn- TBox 𝒰 . SRIQ The rule set ℛ 𝒰 , which is an AR-preserving rewriting for 𝒰 , is defined as follows : A. Let 𝒰 + = 𝒰 ∪ { X ⊑ ∀ R . Y ∣ R a role in 𝒰 } where X and Y are fresh class names. B. Let 𝒰 × be the TBox that results from extending 𝒰 + with all axioms obtained via "box pushing" and then removing every axiom with role chains . C. Add all of the rules in the AR-rewriting of 𝒰 × to ℛ 𝒰 (computed as shown in previous slides). D. For all roles R in 𝒰 , all states q and q ′ � in 𝒪 𝒰 ( R ), and all sets of concepts C 1 , …, C n , if C 1 ⊓ … ⊓ C n ⊓ Y q ⊑ Y q ′ � ∈ Ω ( 𝒰 × ), then add C 1 ( x ) ∧ … ∧ C n ( x ) → R q , q ′ � ( x ) ∈ ℛ 𝒰 . Unnamed paths E. For all roles R occurring in 𝒰 , for all transitions i R → * S q ∈ 𝒪 𝒰 ( R ) with i R the initial state, add S ( x , y ) → R q ( x , y ) ∈ ℛ 𝒰 , for all states q in 𝒪 𝒰 ( R ), add R i R , q ( x ) → R q ( x , x ) ∈ ℛ 𝒰 , for all transitions q → * S q ′ � ∈ 𝒪 𝒰 ( R ), add R q ( x , y ) ∧ S ( y , z ) → R q ′ � ( x , z ) ∈ ℛ 𝒰 , for all states q and q ′ � in 𝒪 𝒰 ( R ), add R q ( x , y ) ∧ R q , q ′ � ( y ) → R q ′ � ( x , y ) ∈ ℛ 𝒰 , and add R f R ( x , y ) → R ( x , y ) with f R the final state. 18 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment

Conclusion 19 Carral , González, and Koopmann From Horn- SRIQ to Datalog: A Data-Independent Transformation that Preserves Assertion Entailment