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.

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. O UTLINE
1 M OTIVATION
2 MFA AND MSA
3 Q UERYING ACYCLIC DL O NTOLOGIES
4 E XPERIMENTAL R ESULTS
1

3. O NTOLOGICAL Q UERY A NSWERING
Key reasoning task for DL and rule-based applications
ABox
A
Query Q
T ∪A ⊨ Q ?
+
TBox
T
2

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
T
2

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
T
2

6. E XISTENTIAL RULES
Positive, function-free, FOL implications with existentially
quantified variables in the head
3

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.B
3

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. 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
3

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 rules
3

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 rules
3

12. TACKLING U NDECIDABILITY
1 Identify groups of rules for which query answering is
decidable
4

13. TACKLING U NDECIDABILITY
1 Identify groups of rules for which query answering is
decidable
Guarded rules, sticky rules, bounded treewidth sets
4

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. 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
4

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 termination
4

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 materialisation
4

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. 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 databases
4

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 databases
4

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 restrictive
4

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. 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. 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
5

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. 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. 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
5

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. 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. R ESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA
6

31. R ESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA
2 Complexity analysis for checking MSA and MFA
6

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-complete
6

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 conditions
6

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-hard
6

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-complete
6

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 ontologies
6

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. 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
6

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. 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 feasible
6

41. O UTLINE
1 M OTIVATION
2 MFA AND MSA
3 Q UERYING ACYCLIC DL O NTOLOGIES
4 E XPERIMENTAL R ESULTS
7

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. 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.C
8

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. 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. 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. 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. 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. 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. 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. 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 check
8

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 check
8

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 check
8

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 check
8

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 check
8

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 check
8

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 check
8

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 applicability
8

59. M ODEL - FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
9

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. 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. 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. 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
9

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) |= Cycle
9

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) |= Cycle
9

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) |= Cycle
9

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) |= Cycle
9

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) |= Cycle
9

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 MFA
9

70. C OST OF C HECKING MFA
Testing model-faithful acyclicity for a set of rules Σ
10

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. 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
10

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-hard
10

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-complete
10

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 PTIME
10

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. M ODEL - SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
11

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. 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. 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. 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. 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
11

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) |= Cycle
11

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) |= Cycle
11

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) |= Cycle
11

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) |= Cycle
11

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 MSA
11

88. C OST OF C HECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
12

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-complete
12

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-complete
12

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-complete
12

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 answering
12

93. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY
JA SWA MSA MFA
13

94. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY
Our contributions:
JA SWA MSA MFA
13

95. ACYCLICITY C ONDITIONS (PARTIAL ) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
JA SWA MSA MFA
13

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 MSA
13

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
ontologies
13

98. O UTLINE
1 M OTIVATION
2 MFA AND MSA
3 Q UERYING ACYCLIC DL O NTOLOGIES
4 E XPERIMENTAL R ESULTS
14

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. 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] technique
15

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. 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. 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
16

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-hard
16

105. O UTLINE
1 M OTIVATION
2 MFA AND MSA
3 Q UERYING ACYCLIC DL O NTOLOGIES
4 E XPERIMENTAL R ESULTS
17

106. ACYCLICITY T ESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
18

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 149 124 117 108
18

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 ontologies
18

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 MSA
18

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. M ATERIALISATION T ESTS
Computed materialisation of acyclic TBoxes
19

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 ABox
19

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 feasible
19

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 feasible
19

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. 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
20

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?!?
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