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Realizing Default Logic over Description Logic Knowledge Bases Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner KBS Group, Institute of Information Systems, Vienna University of Technology ECSQARU 2009 July 2, 2009 KBS Knowledge-Based

  1. Realizing Default Logic over Description Logic Knowledge Bases Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner KBS Group, Institute of Information Systems, Vienna University of Technology ECSQARU 2009 — July 2, 2009 KBS Knowledge-Based Systems Group

  2. The need of common-sense reasoning on top of ontologies Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

  3. The need of common-sense reasoning on top of ontologies Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

  4. The need of common-sense reasoning on top of ontologies Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

  5. The need of common-sense reasoning on top of ontologies default reasoning on top of ontologies? Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

  6. The need of common-sense reasoning on top of ontologies default reasoning on top of ontologies? integrations of rules and ontologies : cq-programs Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

  7. Description Logic Knowledge Bases (DL-KBs) Syntax and Semantics Name Syntax Semantics ∆ I / ∅ ⊤ / ⊥ Top/Bottom C I ∩ D I C ⊓ D Intersection C I ∪ D I C ⊔ D Union ∆ I \ C I ¬ C Negation { a ∈ ∆ I | ∀ b . ( a , b ) ∈ R I → b ∈ C I } ∀ R . C Value restriction { a ∈ ∆ I | ∃ b . ( a , b ) ∈ R I ∧ b ∈ C I } ∃ R . C Existential quant. Modeling : TBox & ABox Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 3/17

  8. Description Logic Knowledge Bases (DL-KBs) Syntax and Semantics Name Syntax Semantics ∆ I / ∅ ⊤ / ⊥ Top/Bottom C I ∩ D I C ⊓ D Intersection C I ∪ D I C ⊔ D Union ∆ I \ C I ¬ C Negation { a ∈ ∆ I | ∀ b . ( a , b ) ∈ R I → b ∈ C I } ∀ R . C Value restriction { a ∈ ∆ I | ∃ b . ( a , b ) ∈ R I ∧ b ∈ C I } ∃ R . C Existential quant. Modeling : TBox & ABox Translation to first-order logic π x ( A ) = A ( x ) π x ( C ⊓ D ) = π x ( C ) ∧ π x ( D ) π x ( ∀ R . C ) = ∀ y . R ( x , y ) ⊃ π y ( C ) π x ( C ⊔ D ) = π x ( C ) ∨ π x ( D ) π x ( ∃ R . C ) = ∀ y . R ( x , y ) ∧ π y ( C ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 3/17

  9. Default Theories over DL-KBs ∆ = � L , D � similar to [Baader and Hollunder, 1995] Default rule α ( � β i ( � X ) Y i ) � �� � � �� � α 1 ( � X 1 ) ∧ · · · ∧ α k ( � X k ) : β 1 ( � β i , 1 ( � Y i , 1 ) ∧ · · · ∧ β i ,ℓ i ( � Y i ,ℓ i ) , . . . , β m ( � Y 1 ) , . . . , Y m ) γ 1 ( � Z 1 ) ∧ · · · ∧ γ n ( � Z n ) � �� � γ ( � Z ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

  10. Default Theories over DL-KBs ∆ = � L , D � similar to [Baader and Hollunder, 1995] Default rule α ( � β i ( � X ) Y i ) � �� � � �� � α 1 ( � X 1 ) ∧ · · · ∧ α k ( � X k ) : β 1 ( � β i , 1 ( � Y i , 1 ) ∧ · · · ∧ β i ,ℓ i ( � Y i ,ℓ i ) , . . . , β m ( � Y 1 ) , . . . , Y m ) γ 1 ( � Z 1 ) ∧ · · · ∧ γ n ( � Z n ) � �� � γ ( � Z ) Semantics: based on the Γ ∆ operator ◮ Let S be a set of assertions, then Γ ∆ ( S ) is the smallest set that ◮ contains Cn ( L ) ◮ is deductively closed ◮ if α ( � X ) ∈ Γ ∆ ( S ) and ¬ β i ( � ∈ S , then γ ( � Y i ) / Z ) ∈ Γ ∆ ( S ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

  11. Default Theories over DL-KBs ∆ = � L , D � similar to [Baader and Hollunder, 1995] Default rule α ( � β i ( � X ) Y i ) � �� � � �� � α 1 ( � X 1 ) ∧ · · · ∧ α k ( � X k ) : β 1 ( � β i , 1 ( � Y i , 1 ) ∧ · · · ∧ β i ,ℓ i ( � Y i ,ℓ i ) , . . . , β m ( � Y 1 ) , . . . , Y m ) γ 1 ( � Z 1 ) ∧ · · · ∧ γ n ( � Z n ) � �� � γ ( � Z ) Semantics: based on the Γ ∆ operator ◮ Let S be a set of assertions, then Γ ∆ ( S ) is the smallest set that ◮ contains Cn ( L ) ◮ is deductively closed ◮ if α ( � X ) ∈ Γ ∆ ( S ) and ¬ β i ( � ∈ S , then γ ( � Y i ) / Z ) ∈ Γ ∆ ( S ) ◮ E is an extension of ∆ iff Γ ∆ ( E ) = E Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

  12. Example ∆ = � L , D � � Flier ⊑ ¬ NonFlier , Penguin ⊑ Bird , � L = Penguin ⊑ NonFlier , Bird ( tweety ) � Bird ( X ) : Flier ( X ) � D = Flier ( X ) E = Cn ( L ∪ { Flier ( tweety ) } ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 5/17

  13. Example ∆ ′ = � L ′ , D � � Flier ⊑ ¬ NonFlier , Penguin ⊑ Bird , � L ′ = Penguin ⊑ NonFlier , Penguin ( tweety ) � Bird ( X ) : Flier ( X ) � D = Flier ( X ) E ′ = Cn ( L ′ ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 6/17

  14. Conjunctive Query Programs [Eiter et al. , 2008a] ◮ (union of) conjunctive queries: q ( X ) = { X | AirCraft ( X ) ∨ ( UFO ( X ) ∧ ¬ Hoax ( X )) } Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

  15. Conjunctive Query Programs [Eiter et al. , 2008a] ◮ (union of) conjunctive queries: q ( X ) = { X | AirCraft ( X ) ∨ ( UFO ( X ) ∧ ¬ Hoax ( X )) } ◮ cq-atom: DL [ AirCraft ⊎ isBoeing ; AirCraft ( X ) ∨ ( UFO ( X ) , ¬ Hoax ( X ))]( X ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

  16. Conjunctive Query Programs [Eiter et al. , 2008a] ◮ (union of) conjunctive queries: q ( X ) = { X | AirCraft ( X ) ∨ ( UFO ( X ) ∧ ¬ Hoax ( X )) } ◮ cq-atom: DL [ AirCraft ⊎ isBoeing ; AirCraft ( X ) ∨ ( UFO ( X ) , ¬ Hoax ( X ))]( X ) ◮ cq-rule: flying thing ( X ) ← thing ( X ) , DL [ AirCraft ⊎ isBoeing ; AirCraft ( X ) ∨ ( UFO ( X ) , ¬ Hoax ( X ))]( X ) . Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

  17. Conjunctive Query Programs [Eiter et al. , 2008a] ◮ (union of) conjunctive queries: q ( X ) = { X | AirCraft ( X ) ∨ ( UFO ( X ) ∧ ¬ Hoax ( X )) } ◮ cq-atom: DL [ AirCraft ⊎ isBoeing ; AirCraft ( X ) ∨ ( UFO ( X ) , ¬ Hoax ( X ))]( X ) ◮ cq-rule: flying thing ( X ) ← thing ( X ) , DL [ AirCraft ⊎ isBoeing ; AirCraft ( X ) ∨ ( UFO ( X ) , ¬ Hoax ( X ))]( X ) . ◮ cq-program: KB = ( L , P ) — based on answer set semantics Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

  18. Answer Set Semantics of cq-Programs generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b 1 , . . . , b m , not c 1 , . . . , not c n . Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

  19. Answer Set Semantics of cq-Programs generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b 1 , . . . , b m , not c 1 , . . . , not c n . ◮ Interpretation I ⊆ HB P Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

  20. Answer Set Semantics of cq-Programs generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b 1 , . . . , b m , not c 1 , . . . , not c n . ◮ Interpretation I ⊆ HB P ◮ I is an answer set of P if I is the least model of the reduct P I Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

  21. Answer Set Semantics of cq-Programs generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b 1 , . . . , b m , not c 1 , . . . , not c n . ◮ Interpretation I ⊆ HB P ◮ I is an answer set of P if I is the least model of the reduct P I P I is constructed by ◮ removing rules r ∈ P such that c i ∈ I ◮ removing all c i and nonmonotonic cq-atoms from remaining rules Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

  22. Transformation Ω (inspired by [Eiter et al. , 2008b]) D is a set of defaults of form β i ( � Y i ) � �� � α ( � X ) : β 1 ( � β i , 1 ( � Y i , 1 ) ∧ · · · ∧ β i ,ℓ i ( � Y i ,ℓ i ) , . . . , β m ( � Y 1 ) , . . . , Y m ) γ 1 ( � Z 1 ) ∧ · · · ∧ γ n ( � Z n ) � �� � γ ( � Z ) Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

  23. Transformation Ω (inspired by [Eiter et al. , 2008b]) D is a set of defaults of form β i ( � Y i ) � �� � α ( � X ) : β 1 ( � β i , 1 ( � Y i , 1 ) ∧ · · · ∧ β i ,ℓ i ( � Y i ,ℓ i ) , . . . , β m ( � Y 1 ) , . . . , Y m ) γ 1 ( � Z 1 ) ∧ · · · ∧ γ n ( � Z n ) � �� � γ ( � Z ) R = { in γ i ( � Z i ) ← in γ ( � ◮ Concluding rules Z ) | 1 ≤ i ≤ n } Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

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