Query Inseparability for Description Logic Knowledge Bases Elena Botoeva 1 Roman Kontchakov 2 Vladislav Ryzhikov 1 Frank Wolter 3 Michael Zakharyaschev 2 1 Faculty of Computer Science, Free University of Bozen-Bolzano, Italy 2 Department of Computer Science, Birkbeck, University of London, UK 3 Department of Computer Science, University of Liverpool, UK July 21 KR 2014 Vienna Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 1/16
Query Answering Over Knowledge Bases Size Color Brand Category Choose size ▽ Choose color ▽ Choose brand ▽ ⊲ Product Sandals: (3 products found) ⊲ Shoes ⋆ Sandals ⋆ Heeled ⋆ Platform ⋆ Classic . . . heel sand wedge sand brown sand Heeled Platform Classic ⊲ Clothing e 79 e 69 e 50 . . . Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 2/16
Query Answering Over Knowledge Bases Size Color Brand Category Choose size ▽ Choose color ▽ Choose brand ▽ ⊲ Product Sandals: (3 products found) ⊲ Shoes ⋆ Sandals ⋆ Heeled ⋆ Platform ⋆ Classic . . . heel sand wedge sand brown sand Heeled Platform Classic ⊲ Clothing e 79 e 69 e 50 . . . Viewed as a knowledge base ( T , A ) and a query q : T Product A Shoes Sandals Heeled Platform Classic Heeled Platform heel sand wedge sand brown sand Classic . . . . . . Clothing q ( x ) ← Sandals ( x ) . . . Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 2/16
Query Answering Over Knowledge Bases Size Reset Color Brand Reset Category 37-38 ▽ Choose color ▽ Geox ▽ ⊲ Product Sandals: (1 product found) ⊲ Shoes ⋆ Sandals ⋆ Heeled ⋆ Platform ⋆ Classic . . . heel sand Heeled ⊲ Clothing e 79 . . . Viewed as a knowledge base ( T , A ) and a query q : T Product A Shoes Sandals Heeled Platform Classic Heeled Platform heel sand wedge sand brown sand Classic . . . . . . Sandals ( x ) , hasSize ( x , 37) , hasBrand ( x , geox) Clothing ∨ q ( x ) ← . . . Sandals ( x ) , hasSize ( x , 38) , hasBrand ( x , geox) Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 2/16
Motivation: Module Extraction . . . . . . . . . B A C . . . E D . . . Give me all B and D such that . . . knowledge base K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 3/16
Motivation: Module Extraction . . . . . . . . . B A C . . . E D . . . Give me all B and D such that . . . knowledge base K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 3/16
Motivation: Module Extraction . . . Module Extraction . . . . . . B module K ′ A C . . . E D . . . Give me all B and D such that . . . knowledge base K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 3/16
Motivation: Module Extraction . . . Module Extraction . . . . . . B module K ′ A C . . . E D . . . Give me all B and D such that . . . knowledge base K b 1 d 1 b 2 d 2 . . . Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 3/16
Motivation: Knowledge Exchange mapping M Γ G Λ L Π P target schema Σ 2 source schema Σ 1 I want a translation of K 1 source KB K 1 in Σ 2 to ask queries Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 4/16
Motivation: Knowledge Exchange mapping M Γ G Λ L Universal UCQ-solution Π P target schema Σ 2 source schema Σ 1 I want a translation of K 1 source KB K 1 target KB K 2 in Σ 2 to ask queries K 2 Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 4/16
Σ-Query Entailment and Inseparability for KBs • K 1 Σ -query entails K 2 if K 2 | = q ( � a ) implies K 1 | = q ( � a ), for each CQ q ( � x ) over Σ and each tuple � a ⊆ ind( K 2 ). Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 5/16
Σ-Query Entailment and Inseparability for KBs • K 1 Σ -query entails K 2 if K 2 | = q ( � a ) implies K 1 | = q ( � a ), for each CQ q ( � x ) over Σ and each tuple � a ⊆ ind( K 2 ). • K 1 and K 2 are Σ -query inseparable , K 1 ≡ Σ K 2 , if K 1 Σ-query entails K 2 and K 2 Σ-query entails K 1 Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 5/16
Σ-Query Entailment and Inseparability for KBs • K 1 Σ -query entails K 2 if K 2 | = q ( � a ) implies K 1 | = q ( � a ), for each CQ q ( � x ) over Σ and each tuple � a ⊆ ind( K 2 ). • K 1 and K 2 are Σ -query inseparable , K 1 ≡ Σ K 2 , if K 1 Σ-query entails K 2 and K 2 Σ-query entails K 1 Then, • K ′ ⊆ K is a Σ -module of K if K ′ ≡ Σ K . Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 5/16
Σ-Query Entailment and Inseparability for KBs • K 1 Σ -query entails K 2 if K 2 | = q ( � a ) implies K 1 | = q ( � a ), for each CQ q ( � x ) over Σ and each tuple � a ⊆ ind( K 2 ). • K 1 and K 2 are Σ -query inseparable , K 1 ≡ Σ K 2 , if K 1 Σ-query entails K 2 and K 2 Σ-query entails K 1 Then, • K ′ ⊆ K is a Σ -module of K if K ′ ≡ Σ K . • K 2 is a universal UCQ-solution for K 1 under M if K 2 ≡ Σ 2 K 1 ∪ M . Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 5/16
Horn Description Logics Description Logics (DLs) represent knowledge in terms of concepts (unary predicates) and roles (binary predicates). Horn- ALCHI DL-Lite H Horn- ALCH Horn- ALCI horn OWL 2 QL OWL 2 EL DL-Lite H ELH Horn- ALC DL-Lite horn core P 1 ⊑ P 2 A 1 ⊑ ∀ P . A 2 , A ⊑ ⊥ ⊓ R 1 ⊑ R 2 EL DL-Lite core A , ∃ P . A , ⊓ A , ∃ R , R := P | P − Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 6/16
Horn Description Logics Description Logics (DLs) represent knowledge in terms of concepts (unary predicates) and roles (binary predicates). Horn- ALCHI AC 0 PTime DL-Lite H Horn- ALCH Horn- ALCI horn OWL 2 QL OWL 2 EL DL-Lite H ELH Horn- ALC DL-Lite horn core P 1 ⊑ P 2 A 1 ⊑ ∀ P . A 2 , A ⊑ ⊥ ⊓ R 1 ⊑ R 2 EL DL-Lite core A , ∃ P . A , ⊓ A , ∃ R , R := P | P − Data complexity of CQ-answering Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 6/16
How We Tackle Σ-Query Entailment We rely on two fundamental instruments: 1 Materialisation , as an abstract way to characterize all answers to CQs over a KB. A materialisation of a KB K is an interpretation M such that K | = q ( � a ) iff M | = q ( � a ) , for each CQ q ( � x ) and each tuple � a ⊆ ind( K ). 2 Reachability Games , as a technique for obtaining upper-bounds. Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 7/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Let K = �T , A� a B A = { B ( a ) } Materialisation M of K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Let K = �T , A� a B P A = { B ( a ) } T = { B ⊑ ∃ P . ∃ R . ( ∃ S ⊓ ∃ Q ) R Q S � ∀ x . B ( x ) → ∃ y , z , u , v . P ( x , y ) , R ( y , z ) , S ( z , u ) , Q ( z , v ) � Materialisation M of K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Let K = �T , A� a B P A = { B ( a ) } T = { B ⊑ ∃ P . ∃ R . ( ∃ S ⊓ ∃ Q ) R ∃ S − ⊑ ∃ T . ∃ S Q S T S � ∀ x . B ( x ) → ∃ y , z , u , v . P ( x , y ) , R ( y , z ) , S ( z , u ) , Q ( z , v ) � � � ∀ x . ∃ u . S ( u , x ) → ∃ y , z . T ( x , y ) , S ( y , z ) Materialisation M of K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Let K = �T , A� a B P A = { B ( a ) } T = { B ⊑ ∃ P . ∃ R . ( ∃ S ⊓ ∃ Q ) R ∃ S − ⊑ ∃ T . ∃ S Q S T S � ∀ x . B ( x ) → ∃ y , z , u , v . P ( x , y ) , R ( y , z ) , S ( z , u ) , Q ( z , v ) � � � ∀ x . ∃ u . S ( u , x ) → ∃ y , z . T ( x , y ) , S ( y , z ) T · · · Materialisation M of K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
Materialisations Horn DLs enjoy materialisations (chase, canonical models). Let K = �T , A� a B P A = { B ( a ) } T = { B ⊑ ∃ P . ∃ R . ( ∃ S ⊓ ∃ Q ) R ∃ S − ⊑ ∃ T . ∃ S Q ∃ Q − ⊑ ∃ Q } S Q T · · · S � ∀ x . B ( x ) → ∃ y , z , u , v . P ( x , y ) , R ( y , z ) , S ( z , u ) , Q ( z , v ) � � � ∀ x . ∃ u . S ( u , x ) → ∃ y , z . T ( x , y ) , S ( y , z ) T � � ∀ x . ∃ y . Q ( y , x ) → ∃ z . Q ( x , z ) · · · Materialisation M of K Elena Botoeva(FUB) Query Inseparability for Description Logic Knowledge Bases 8/16
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