Acyclicity Conditions and their Application to Query Answering in Description Logics

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Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules, so the chase is not guaranteed to terminate. Several acyclicity conditions provide sufficient conditions for chase termination. In this paper, we present two novel such conditions—modelfaithful acyclicity (MFA) and model-summarising acyclicity (MSA)—that generalise many of the acyclicity conditions known so far in the literature. Materialisation provides the basis for several widely-used OWL 2 DL reasoners. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2 DL; furthermore, some systems go beyond OWL 2 RL, but they provide no termination guarantees. In this paper we investigate whether various acyclicity conditions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 DL ontologies are MSA, and that the facts obtained via materialisation are not too large. Thus, our results suggest that principled extensions to materialisationbased OWL 2 DL reasoners may be practically feasible.

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Acyclicity Conditions and their Application to Query Answering in Description Logics

  1. 1. ACYCLICITY C ONDITIONS AND THEIR A PPLICATION TO Q UERY A NSWERING IN D ESCRIPTION L OGICS Bernardo Cuenca Grau, Ian Horrocks, Markus Krötzsch,Clemens Kupke, Despoina Magka, Boris Motik, Zhe Wang Department of Computer Science, University of Oxford June 14, 2012
  2. 2. O UTLINE 1 M OTIVATION 2 MFA AND MSA 3 Q UERYING ACYCLIC DL O NTOLOGIES 4 E XPERIMENTAL R ESULTS1
  3. 3. O NTOLOGICAL Q UERY A NSWERING Key reasoning task for DL and rule-based applications ABox  A Query Q    T ∪A ⊨  Q ? + TBox  T2
  4. 4. O NTOLOGICAL Q UERY A NSWERING Key reasoning task for DL and rule-based applications Answering CQs over DLs high computational complexity ABox  A Query Q    T ∪A ⊨  Q ? + TBox  T2
  5. 5. O NTOLOGICAL Q UERY A NSWERING Key reasoning task for DL and rule-based applications Answering CQs over DLs high computational complexity Materialisation-based paradigm: chase ABox using TBox and evaluate Q in the computed ABox ABox  A Query Q  Materia­ lised Chase  ChaseT (A)⊨Q ? ABox   TBox  T2
  6. 6. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head3
  7. 7. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) DL-equivalent A ∃R.B3
  8. 8. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) DL-equivalent A ∃R.B Existential rules fundamental for several KR formalisms:3
  9. 9. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) DL-equivalent A ∃R.B Existential rules fundamental for several KR formalisms: 1 Schema constraints in databases 2 Data transformation rules in data exchange 3 Foundation for Datalog± family of KR languages 4 Ubiquitous in Description Logics3
  10. 10. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) DL-equivalent A ∃R.B Existential rules fundamental for several KR formalisms: 1 Schema constraints in databases 2 Data transformation rules in data exchange 3 Foundation for Datalog± family of KR languages 4 Ubiquitous in Description Logics Chase termination is undecidable for existential rules3
  11. 11. E XISTENTIAL RULES Positive, function-free, FOL implications with existentially quantified variables in the head E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) DL-equivalent A ∃R.B Existential rules fundamental for several KR formalisms: 1 Schema constraints in databases 2 Data transformation rules in data exchange 3 Foundation for Datalog± family of KR languages 4 Ubiquitous in Description Logics Chase termination is undecidable for existential rules CQ answering is undecidable for existential rules3
  12. 12. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable4
  13. 13. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets4
  14. 14. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .4
  15. 15. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules4
  16. 16. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination4
  17. 17. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination Yields finite materialisation4
  18. 18. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination Yields finite materialisation Plus I No restriction on the shape of rules (unlike guarded rules)4
  19. 19. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination Yields finite materialisation Plus I No restriction on the shape of rules (unlike guarded rules) II Materialised ABoxes can be stored as databases4
  20. 20. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination Yields finite materialisation Plus Minus I No restriction on the shape of I Only sets of rules with models rules (unlike guarded rules) of bounded size II Materialised ABoxes can be stored as databases4
  21. 21. TACKLING U NDECIDABILITY 1 Identify groups of rules for which query answering is decidable Guarded rules, sticky rules, bounded treewidth sets 2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT, 2002], super-weak acyclicity [Marnette, PODS, 2009], joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . . Acyclic set of rules Guarantees chase termination Yields finite materialisation Plus Minus I No restriction on the shape of I Only sets of rules with models rules (unlike guarded rules) of bounded size II Materialised ABoxes can be II Acyclicity conditions might be stored as databases too restrictive4
  22. 22. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011]5
  23. 23. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . .5
  24. 24. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination5
  25. 25. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination Approaches taken: 1 Saturate only non-existential rules (OWL 2 RL)5
  26. 26. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination Approaches taken: 1 Saturate only non-existential rules (OWL 2 RL): can miss answers 5
  27. 27. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination Approaches taken: 1 Saturate only non-existential rules (OWL 2 RL): can miss answers 2 Apply existential rules in a restricted way5
  28. 28. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination Approaches taken: 1 Saturate only non-existential rules (OWL 2 RL): can miss answers 2 Apply existential rules in a restricted way: can still miss answers and/or not terminate 5
  29. 29. M ATERIALISATION - BASED R EASONING Answering CQs over expressive DLs is expensive, e.g. EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph and Simkus, 2011] For Horn ontologies, consequences can be precomputed, stored and used for query evaluation, e.g. by the RDF repositories Sesame, Jena, OWLIM, DLE-Jena,. . . Risk of non-termination Approaches taken: 1 Saturate only non-existential rules (OWL 2 RL): can miss answers 2 Apply existential rules in a restricted way: can still miss answers and/or not terminate Suggestion: materialise ABoxes only over acyclic TBoxes Always complete Provably terminating 5
  30. 30. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA6
  31. 31. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA6
  32. 32. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete6
  33. 33. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions6
  34. 34. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard6
  35. 35. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete6
  36. 36. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies6
  37. 37. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA)6
  38. 38. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large6
  39. 39. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large × 5 bigger on average for ontologies with depth 5 (= most ontologies)6
  40. 40. R ESULTS OVERVIEW 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large Materialisation-based reasoning beyond OWL 2 RL might be practically feasible6
  41. 41. O UTLINE 1 M OTIVATION 2 MFA AND MSA 3 Q UERYING ACYCLIC DL O NTOLOGIES 4 E XPERIMENTAL R ESULTS7
  42. 42. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → ∃y1 .R(u, y1 ) ∧ B(y1 ) r2 : B(v) → ∃y2 .R(v, y2 ) ∧ C(y2 ) r3 : R(w, z) ∧ B(z) → A(w)8
  43. 43. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → ∃y1 .R(u, y1 ) ∧ B(y1 ) r2 : B(v) → ∃y2 .R(v, y2 ) ∧ C(y2 ) A ≡ ∃R.B r3 : R(w, z) ∧ B(z) → A(w) B ∃R.C8
  44. 44. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → ∃y1 .R(u, y1 ) ∧ B(y1 ) r2 : B(v) → ∃y2 .R(v, y2 ) ∧ C(y2 ) r3 : R(w, z) ∧ B(z) → A(w)8
  45. 45. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → ∃y2 .R(v, y2 ) ∧ C(y2 ) r3 : R(w, z) ∧ B(z) → A(w)8
  46. 46. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → ∃y2 .R(v, y2 ) ∧ C(y2 ) r3 : R(w, z) ∧ B(z) → A(w)8
  47. 47. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w)8
  48. 48. E XISTING ACYCLICITY C ONDITIONS E XAMPLE R r1 : A(u) → R(u, f (u)) ∧ B(f (u)) A, B, C ∗ r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w)8
  49. 49. E XISTING ACYCLICITY C ONDITIONS E XAMPLE R r1 : A(u) → R(u, f (u)) ∧ B(f (u)) A, B, C ∗ r2 : B(v) → R(v, g(v)) ∧ C(g(v)) R R r3 : R(w, z) ∧ B(z) → A(w) B f (∗) C g(∗)8
  50. 50. E XISTING ACYCLICITY C ONDITIONS E XAMPLE R r1 : A(u) → R(u, f (u)) ∧ B(f (u)) A, B, C ∗ r2 : B(v) → R(v, g(v)) ∧ C(g(v)) R R r3 : R(w, z) ∧ B(z) → A(w) B f (∗) C g(∗) R C g(f (∗))8
  51. 51. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w) Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  52. 52. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w) Move(f (u)) = {R|2 , B|1 Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  53. 53. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w) Move(f (u)) = {R|2 , B|1 , R|1 Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  54. 54. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w) Move(f (u)) = {R|2 , B|1 , R|1 , A|1 } Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  55. 55. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) r3 : R(w, z) ∧ B(z) → A(w) PosB (u) = {A|1 } Move(f (u)) = {R|2 , B|1 , R|1 , A|1 } Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  56. 56. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) f (u) r3 : R(w, z) ∧ B(z) → A(w) PosB (u) = {A|1 } ⊆ Move(f (u)) = {R|2 , B|1 , R|1 , A|1 } Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  57. 57. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) f (u) r3 : R(w, z) ∧ B(z) → A(w) not in JA PosB (u) = {A|1 } ⊆ Move(f (u)) = {R|2 , B|1 , R|1 , A|1 } Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check8
  58. 58. E XISTING ACYCLICITY C ONDITIONS E XAMPLE r1 : A(u) → R(u, f (u)) ∧ B(f (u)) r2 : B(v) → R(v, g(v)) ∧ C(g(v)) f (u) r3 : R(w, z) ∧ B(z) → A(w) not in JA PosB (u) = {A|1 } ⊆ Move(f (u)) = {R|2 , B|1 , R|1 , A|1 } Joint acyclicity 1 Tracks value generation and propagation to detect cyclic creation of terms 2 Polynomial time to check May overestimate rule applicability8
  59. 59. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’9
  60. 60. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) R(w, z) ∧ B(z) → A(w)9
  61. 61. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R(w, z) ∧ B(z) → A(w)9
  62. 62. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z)9
  63. 63. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle9
  64. 64. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle9
  65. 65. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) R A, B, C ∗ B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle9
  66. 66. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) R A, B, C ∗ B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R, S, D R(w, z) ∧ B(z) → A(w) R, S, D S(x, y) → D(x, y) f (∗) g(∗) D(x, y) ∧ S(y, z) → D(x, z) B, Ff C, Fg Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle9
  67. 67. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) R A, B, C ∗ B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R, S, D R(w, z) ∧ B(z) → A(w) R, S, D S(x, y) → D(x, y) f (∗) g(∗) D(x, y) ∧ S(y, z) → D(x, z) B, Ff C, Fg Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle R, S, D Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle g(f (∗)) C, Fg ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle9
  68. 68. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) R A, B, C ∗ B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R, S, D R(w, z) ∧ B(z) → A(w) R, S, D S(x, y) → D(x, y) D f (∗) g(∗) D(x, y) ∧ S(y, z) → D(x, z) B, Ff C, Fg Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle R, S, D Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle g(f (∗)) C, Fg ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle9
  69. 69. M ODEL - FAITHFUL ACYCLICITY Track rule applications more ‘faithfully’ E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u)) R A, B, C ∗ B(v) → R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg (g(v)) R, S, D R(w, z) ∧ B(z) → A(w) R, S, D S(x, y) → D(x, y) D f (∗) g(∗) D(x, y) ∧ S(y, z) → D(x, z) B, Ff C, Fg Ff (x) ∧ D(x, y) ∧ Ff (y) → Cycle R, S, D Fg (x) ∧ D(x, y) ∧ Fg (y) → Cycle g(f (∗)) C, Fg ∗ For Σ a set of rules, Σ is MFA if IΣ ∪ MFA(Σ) |= Cycle Set of rules that correspond to DL subsumptions {A ≡ ∃R.B, B ∃R.C} is MFA9
  70. 70. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ10
  71. 71. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols )10
  72. 72. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols ) 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity 2EXPTIME-complete10
  73. 73. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols ) 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity 2EXPTIME-complete 3 Rules from Horn-SRI EXPTIME-hard10
  74. 74. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols ) 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity 2EXPTIME-complete 3 Rules from Horn-SRI EXPTIME-hard 4 Rules from Horn-SHIQ PSPACE-complete10
  75. 75. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols ) 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity 2EXPTIME-complete 3 Rules from Horn-SRI EXPTIME-hard 4 Rules from Horn-SHIQ PSPACE-complete Existing acyclicity conditions can be checked in PTIME10
  76. 76. C OST OF C HECKING MFA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) 2EXPTIME-complete (tree with branching factor |x| and height the total number of function symbols ) 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity 2EXPTIME-complete 3 Rules from Horn-SRI EXPTIME-hard 4 Rules from Horn-SHIQ PSPACE-complete Existing acyclicity conditions can be checked in PTIME Isn’t computational complexity too high?10
  77. 77. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough11
  78. 78. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) R(w, z) ∧ B(z) → A(w)11
  79. 79. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, f (u)) ∧ B(f (u)) B(v) → R(v, g(v)) ∧ C(g(v)) R(w, z) ∧ B(z) → A(w)11
  80. 80. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) R(w, z) ∧ B(z) → A(w)11
  81. 81. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w)11
  82. 82. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle11
  83. 83. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle ∗ For Σ a set of rules, Σ is MSA if IΣ ∪ MSA(Σ) |= Cycle11
  84. 84. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) R A, B, C ∗ S(x, y) → D(x, y) D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle ∗ For Σ a set of rules, Σ is MSA if IΣ ∪ MSA(Σ) |= Cycle11
  85. 85. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) R A, B, C ∗ S(x, y) → D(x, y) R, S, D R, S, D D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle c1 c2 Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle B, Fc1 C, Fc2 ∗ For Σ a set of rules, Σ is MSA if IΣ ∪ MSA(Σ) |= Cycle11
  86. 86. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) R A, B, C ∗ S(x, y) → D(x, y) R, S, D R, S, D D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle c1 c2 R, S, D Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle B, Fc1 C, Fc2 ∗ For Σ a set of rules, Σ is MSA if IΣ ∪ MSA(Σ) |= Cycle11
  87. 87. M ODEL - SUMMARISING ACYCLICITY Track rule applications just ‘faithfully’ enough E XAMPLE A(u) → R(u, c1 ) ∧ B(c1 ) ∧ S(u, c1 ) ∧ Fc1 (c1 ) B(v) → R(v, c2 ) ∧ C(c2 ) ∧ S(v, c2 ) ∧ Fc2 (c2 ) R(w, z) ∧ B(z) → A(w) R A, B, C ∗ S(x, y) → D(x, y) R, S, D R, S, D D(x, y) ∧ S(y, z) → D(x, z) Fc1 (x) ∧ D(x, y) ∧ Fc1 (y) → Cycle c1 c2 R, S, D Fc2 (x) ∧ D(x, y) ∧ Fc2 (y) → Cycle B, Fc1 C, Fc2 ∗ For Σ a set of rules, Σ is MSA if IΣ ∪ MSA(Σ) |= Cycle Set of rules that correspond to DL subsumptions {A ≡ ∃R.B, B ∃R.C} is still MSA11
  88. 88. C OST OF C HECKING MSA Testing model-faithful acyclicity for a set of rules Σ12
  89. 89. C OST OF C HECKING MSA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) EXPTIME-complete12
  90. 90. C OST OF C HECKING MSA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) EXPTIME-complete 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity coNP-complete12
  91. 91. C OST OF C HECKING MSA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) EXPTIME-complete 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity coNP-complete 3 Rules from Horn-SHIQ PTIME-complete12
  92. 92. C OST OF C HECKING MSA Testing model-faithful acyclicity for a set of rules Σ 1 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) (no restriction) EXPTIME-complete 2 Rules of the form ϕ(x, z) → ∃y.ψ(x, y) with predicates of bounded arity coNP-complete 3 Rules from Horn-SHIQ PTIME-complete Horn-SHIQ TBoxes can be checked in PTIME for MSA before potential materialisation-based query answering12
  93. 93. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY JA SWA MSA MFA13
  94. 94. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY Our contributions: JA SWA MSA MFA13
  95. 95. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY Our contributions: 1 MSA strictly subsumes SWA JA SWA MSA MFA13
  96. 96. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY Our contributions: 1 MSA strictly subsumes SWA 2 MFA strictly subsumes MSA JA SWA MSA MFA E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) B(x) → ∃y.S(x, y) ∧ T(y, x) A(z) ∧ S(z, x) → C(x) C(z) ∧ T(z, x) → A(x) MFA but not MSA13
  97. 97. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY Our contributions: 1 MSA strictly subsumes SWA 2 MFA strictly subsumes MSA JA SWA MSA MFA E XAMPLE A(x) → ∃y.R(x, y) ∧ B(y) B(x) → ∃y.S(x, y) ∧ T(y, x) A(z) ∧ S(z, x) → C(x) C(z) ∧ T(z, x) → A(x) MFA but not MSA MSA and MFA coincide in experimental evaluation of DL ontologies13
  98. 98. O UTLINE 1 M OTIVATION 2 MFA AND MSA 3 Q UERYING ACYCLIC DL O NTOLOGIES 4 E XPERIMENTAL R ESULTS14
  99. 99. T RANSLATING DL S INTO RULES Axioms of normalised Horn-SRIQ ontologies can be converted to (existential) rules A ∃R.B A(x) → ∃y.R(x, y) ∧ B(y) A ≤ 1 R.B A(z) ∧ R(z, x1 ) ∧ B(x1 ) ∧ R(z, x2 ) ∧ B(x2 ) → x1 ≈x2 A B C A(x) ∧ B(x) → C(x) A ∀R.B A(z) ∧ R(z, x) → B(x) R S R(x1 , x2 ) → S(x1 , x2 ) R ◦ S T R(x1 , z) ∧ S(z, x2 ) → T(x1 , x2 )15
  100. 100. T RANSLATING DL S INTO RULES Axioms of normalised Horn-SRIQ ontologies can be converted to (existential) rules A ∃R.B A(x) → ∃y.R(x, y) ∧ B(y) A ≤ 1 R.B A(z) ∧ R(z, x1 ) ∧ B(x1 ) ∧ R(z, x2 ) ∧ B(x2 ) → x1 ≈x2 A B C A(x) ∧ B(x) → C(x) A ∀R.B A(z) ∧ R(z, x) → B(x) R S R(x1 , x2 ) → S(x1 , x2 ) R ◦ S T R(x1 , z) ∧ S(z, x2 ) → T(x1 , x2 ) Equality is handled with a modification of the singularisation [Marnette, PODS, 2009] technique15
  101. 101. DL Q UERY A NSWERING UNDER ACYCLICITY Answering conjunctive queries for the DL Horn-SHIQ is EXPTIME-complete [Eiter et al., 2008]16
  102. 102. DL Q UERY A NSWERING UNDER ACYCLICITY Answering conjunctive queries for the DL Horn-SHIQ is EXPTIME-complete [Eiter et al., 2008] Does acyclicity affect complexity for DL Query Answering?16
  103. 103. DL Q UERY A NSWERING UNDER ACYCLICITY Answering conjunctive queries for the DL Horn-SHIQ is EXPTIME-complete [Eiter et al., 2008] Does acyclicity affect complexity for DL Query Answering? 1 Horn-SHIQ TBox T and ABox A T is MFA Q Boolean conjunctive query Deciding T ∪ A |= Q is PSPACE-complete16
  104. 104. DL Q UERY A NSWERING UNDER ACYCLICITY Answering conjunctive queries for the DL Horn-SHIQ is EXPTIME-complete [Eiter et al., 2008] Does acyclicity affect complexity for DL Query Answering? 1 Horn-SHIQ TBox T and ABox A T is MFA Q Boolean conjunctive query Deciding T ∪ A |= Q is PSPACE-complete 2 Horn-SRI TBox T and ABox A T is weakly acyclic F set of facts Deciding T ∪ A |= F is EXPTIME-hard16
  105. 105. O UTLINE 1 M OTIVATION 2 MFA AND MSA 3 Q UERYING ACYCLIC DL O NTOLOGIES 4 E XPERIMENTAL R ESULTS17
  106. 106. ACYCLICITY T ESTS Checked 149 DL ontologies for WA, JA, MSA, MFA18
  107. 107. ACYCLICITY T ESTS Checked 149 DL ontologies for WA, JA, MSA, MFA Existential rules Total MSA JA WA 100 70 64 64 64 100–1K 33 30 30 23 1K–5K 20 14 14 12 5K–12K 14 11 6 6 12K–160K 12 5 3 3 All sizes 149 124 117 10818
  108. 108. ACYCLICITY T ESTS Checked 149 DL ontologies for WA, JA, MSA, MFA Existential rules Total MSA JA WA 100 70 64 64 64 100–1K 33 30 30 23 1K–5K 20 14 14 12 5K–12K 14 11 6 6 12K–160K 12 5 3 3 All sizes 149 124 117 108 MSA and MFA coincide w.r.t. the test ontologies18
  109. 109. ACYCLICITY T ESTS Checked 149 DL ontologies for WA, JA, MSA, MFA Existential rules Total MSA JA WA 100 70 64 64 64 100–1K 33 30 30 23 1K–5K 20 14 14 12 5K–12K 14 11 6 6 12K–160K 12 5 3 3 All sizes 149 124 117 108 MSA and MFA coincide w.r.t. the test ontologies 83% were found MSA18
  110. 110. ACYCLICITY T ESTS Checked 149 DL ontologies for WA, JA, MSA, MFA Existential rules Total MSA JA WA 100 70 64 64 64 100–1K 33 30 30 23 1K–5K 20 14 14 12 5K–12K 14 11 6 6 12K–160K 12 5 3 3 All sizes 149 124 117 108 MSA and MFA coincide w.r.t. the test ontologies 83% were found MSA 7 large and expressive OBO ontologies MSA but not JA (only two of them were ELHr and DL-Lite)18
  111. 111. M ATERIALISATION T ESTS Computed materialisation of acyclic TBoxes19
  112. 112. M ATERIALISATION T ESTS Computed materialisation of acyclic TBoxes Depth # generated size materialisation size max avg max avg 5 82 27 2 35 5 5–9 13 37 11 41 13 10–80 14 281 51 283 53 Depth = length of function symbol nesting # facts generated by existential rules generated size = # facts in initial ABox # facts in materialisation materialisation size = # facts in initial ABox19
  113. 113. M ATERIALISATION T ESTS Computed materialisation of acyclic TBoxes Depth # generated size materialisation size max avg max avg 5 82 27 2 35 5 5–9 13 37 11 41 13 10–80 14 281 51 283 53 Depth = length of function symbol nesting # facts generated by existential rules generated size = # facts in initial ABox # facts in materialisation materialisation size = # facts in initial ABox For ontologies with small depths materialisation seems practically feasible19
  114. 114. M ATERIALISATION T ESTS Computed materialisation of acyclic TBoxes Depth # generated size materialisation size max avg max avg 5 82 27 2 35 5 5–9 13 37 11 41 13 10–80 14 281 51 283 53 Depth = length of function symbol nesting # facts generated by existential rules generated size = # facts in initial ABox # facts in materialisation materialisation size = # facts in initial ABox For ontologies with small depths materialisation seems practically feasible19
  115. 115. S UMMARY OF THE RESULTS 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large × 5 bigger on average for ontologies with depth 5 (= most ontologies)20
  116. 116. S UMMARY OF THE RESULTS 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large Materialisation-based reasoning beyond OWL 2 RL might be practically feasible20
  117. 117. S UMMARY OF THE RESULTS 1 More general acyclicity conditions: MSA and MFA 2 Complexity analysis for checking MSA and MFA Horn-SHIQ bounded arity no restriction MSA PTime-complete coNP-complete ExpTime-complete MFA PSpace-complete 2ExpTime-complete 2ExpTime-complete 3 DL query answering under acyclicity conditions Horn-SRI T in WA: T ∪ A |= F is ExpTime-hard Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete 4 Experimental evaluation on DL ontologies 83% ontologies found acyclic (78% JA) materialised ABoxes not too large Materialisation-based reasoning beyond OWL 2 RL might be practically feasible Thank you! Questions?!?20

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