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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krtzsch Knowledge-Based Systems Group at The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krtzsch

  1. The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch Knowledge-Based Systems Group at The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 1

  2. The Combined Approach Conjunctive query TBox ABox The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

  3. The Combined Approach 1. Materialisation Step Conjunctive query TBox ABox The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

  4. The Combined Approach 1. Materialisation Step Conjunctive query TBox Datalog Rule Set ABox The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

  5. The Combined Approach 1. Materialisation Step Conjunctive query TBox Datalog Rule Set Datalog ABox Rule Engine The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

  6. The Combined Approach 1. Materialisation Step Conjunctive query TBox Datalog Rule Set Datalog ABox Rule Engine The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

  7. The Combined Approach 1. Materialisation Step Conjunctive query TBox Datalog Rule Set Datalog ABox Rule * Complete for CQA * May not be a model Engine * Sound (and complete) for instance queries The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 2 /15

  8. The Combined Approach 2. Filtration Step 1. Materialisation Step Conjunctive query TBox Datalog Rule Set Datalog ABox Rule * Complete for CQA * May not be a model Engine * Sound (and complete) for instance queries The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 2 /15

  9. The Combined Approach 2. Filtration Step 1. Materialisation Step Conjunctive query Ans. 1 Ans. 2 TBox Datalog Rule Set Ans. 3 Ans. 4 Datalog ABox Rule * Complete for CQA * May not be a model Engine * Sound (and complete) for instance queries The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 2 /15

  10. The Combined Approach 2. Filtration Step 1. Materialisation Step Conjunctive query Ans. 1 Ans. 2 TBox Datalog Rule Set Ans. 3 Ans. 4 Datalog ABox Rule * Complete for CQA * May not be a model Engine * Sound (and complete) for instance queries The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 2 /15

  11. The Combined Approach for Horn-ALCHOIQ C 1 ⊓ … ⊓ C n ⊑ D R ⊑ S ∃ R . C ⊑ D R 1 ⊓ … ⊓ R n ⊑ ⊥ C ⊑ ∃ R . D R ( a , b ) C ⊑ ≤ 1 R . D A ( a ) C ⊑ { a } The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 3

  12. The Materialisation Step The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 4

  13. Materialisation Step: EL 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 ) → R ( x , t D ) ∧ D ( t D ) R ⊑ S ↦ R ( x , y ) → S ( x , y ) C ( a ) ↦ C ( a ) R ( a , b ) ↦ R ( a , b ) The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 5 /15

  14. Materialisation Step: EL 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 ) → R ( x , t D ) ∧ D ( t D ) R ⊑ S ↦ R ( x , y ) → S ( x , y ) C ( a ) ↦ C ( a ) R ( a , b ) ↦ R ( a , b ) The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 5 /15

  15. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E ∃ R . E ⊑ F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

  16. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E ∃ R . E ⊑ F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

  17. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E ∃ R . E ⊑ F d 1 : D R a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

  18. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D R a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

  19. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

  20. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) D ⊑ ∃ S . C ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C a : C, F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  21. Materialisation Step: EL C ⊑ ∃ R . D C ( a ) d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C a : C, F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  22. Materialisation Step: EL d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  23. Materialisation Step: EL d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  24. Materialisation Step: EL d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C d 2 : C, F S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C a : C, F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  25. Materialisation Step: EL d 5 : D R d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C, F d 2 : C S ∃ R . E ⊑ F d 1 : D d 1 : D, E R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  26. Materialisation Step: EL d 5 : D d 6 : C S R d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C, F d 2 : C S ∃ R . E ⊑ F d 1 : D d 1 : D, E R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  27. Materialisation Step: EL d 5 : D, E d 5 : D d 6 : C S R d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C, F d 2 : C S ∃ R . E ⊑ F d 1 : D d 1 : D, E R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  28. Materialisation Step: EL d 5 : D, E d 5 : D d 6 : C S R d 4 : C, F d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C d 2 : C, F S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C a : C, F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  29. Materialisation Step: EL d 5 : D, E d 5 : D d 6 : C d 7 : D S R R d 4 : C d 4 : C, F S C ⊑ ∃ R . D C ( a ) d 3 : D d 3 : D, E D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C d 2 : C, F S ∃ R . E ⊑ F d 1 : D, E d 1 : D R a : C a : C, F The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  30. Materialisation Step: EL d 5 : D, E d 5 : D d 6 : C d 7 : D d 8 : C S R S R d 4 : C, F d 4 : C S C ⊑ ∃ R . D C ( a ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C, F d 2 : C S ∃ R . E ⊑ F d 1 : D d 1 : D, E R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

  31. Materialisation Step: EL d 5 : D, E d 5 : D d 6 : C d 7 : D d 8 : C C ( x ) → R ( x , t D ) ∧ D ( t D ) S R S R D ( x ) → S ( x , t C ) ∧ C ( t C ) d 4 : C, F d 4 : C S ( x , y ) ∧ C ( y ) → E ( x ) S C ⊑ ∃ R . D C ( a ) R ( x , y ) ∧ E ( y ) → F ( x ) d 3 : D, E d 3 : D D ⊑ ∃ S . C R ∃ S . C ⊑ E d 2 : C d 2 : C, F S ∃ R . E ⊑ F d 1 : D d 1 : D, E R a : C, F a : C The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch 6 /15

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