Presentation slides for my JIST 2014 presentation on Revisiting defaults in description logics - and their role in aligning ontologies. Awarded second best paper award

1. Introduction
Contribution
Free defaults
Conclusion
Revisiting default description logics { and their
role in aligning ontologies
Kunal Sengupta 1 Pascal Hitzler 1 Krzysztof Janowicz 2
1Wright State University, Dayton OH 45435, USA
2University of California, Santa Barbara, USA
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

2. Introduction
Contribution
Free defaults
Conclusion
1 Introduction
Motivation
Examples
Proposed Solution
2 Contribution
3 Free defaults
Semantics
Decidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

3. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
1 Introduction
Motivation
Examples
Proposed Solution
2 Contribution
3 Free defaults
Semantics
Decidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

4. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, World
wide web
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

5. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, World
wide web
How to have ontology mappings that respect heterogeneity?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

6. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, World
wide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

7. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, World
wide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Why, you ask?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

8. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
9a:hasSpouse:a:Female v a:Male
9a:hasSpouse:a:Male v a:Female
a:hasWife(a:john; a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

9. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
9a:hasSpouse:a:Female v a:Male
9a:hasSpouse:a:Male v a:Female
a:hasWife(a:john; a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ?
Ontology B
symmetric(b:hasSpouse)
b:hasSpouse(b:mike; b:david)
b:Male(b:david)
b:Male(b:mike)
b:Female(b:anna)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

10. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
9a:hasSpouse:a:Female v a:Male
9a:hasSpouse:a:Male v a:Female
a:hasWife(a:john; a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ?
Mappings
a:hasSpouse b:hasSpouse
a:Male b:Male
a:Female b:Female
Ontology B
symmetric(b:hasSpouse)
b:hasSpouse(b:mike; b:david)
b:Male(b:david)
b:Male(b:mike)
b:Female(b:anna)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

11. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
owl:sameAs example
Ontology A
a:Airport(a:kennedy)
a:Airport v a:Place
Ontology B
b:President(b:kennedy)
b:President v b:Person
b:Place u b:Person v ?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

12. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
owl:sameAs example
Ontology A
a:Airport(a:kennedy)
a:Airport v a:Place
Ontology B
b:President(b:kennedy)
b:President v b:Person
b:Place u b:Person v ?
Mappings
a:Place b:Place
owl:sameAs(a:kennedy,b:kennedy)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

13. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Using defaults for ontology mapping
Use statements like b:hasSpouse vd a:hasSpouse to denote
mappings
For each pair that satis

14. es b:hasSpouse assume it satis

15. es
a:hasSpouse unless it causes an inconsistency
Assuming a:hasSpouse(b:mike, b:david) causes an
inconsistency
The pair (b:mike, b:david) is treated as an exception to the
statement b:hasSpouse vd b:hasSpouse
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

16. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Solution
Use defaults to denote mappings, such that exceptions are
allowed.
b:hasSpouse : a:hasSpouse
a:hasSpouse
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

17. Introduction
Contribution
Free defaults
Conclusion
Motivation
Examples
Proposed Solution
Solution
Use defaults to denote mappings, such that exceptions are
allowed.
b:hasSpouse : a:hasSpouse
a:hasSpouse
But DLs + Defaults = Undecidable logics [Baader, Hollunder
95].
Workaround: Defaults apply only to named individuals
[Baader, Hollunder 95].
What about un-named individuals?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

18. Introduction
Contribution
Free defaults
Conclusion
1 Introduction
Motivation
Examples
Proposed Solution
2 Contribution
3 Free defaults
Semantics
Decidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

19. Introduction
Contribution
Free defaults
Conclusion
Contribution
Free defaults
Application of defaults not limited to named individuals
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

20. Introduction
Contribution
Free defaults
Conclusion
Contribution
Free defaults
Application of defaults not limited to named individuals
Defaults with role inclusions are also decidable.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

21. Introduction
Contribution
Free defaults
Conclusion
Contribution
Free defaults
Application of defaults not limited to named individuals
Defaults with role inclusions are also decidable.
A new, more powerful language to de

22. ne mapping between
ontologies.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

23. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
1 Introduction
Motivation
Examples
Proposed Solution
2 Contribution
3 Free defaults
Semantics
Decidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

24. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Syntax
A new operator vd that represents default inclusions
C vd D is a default class inclusion
A default-knowledge-base is denoted as (KB; ), where KB is
a DL knowledge base and is a set of default axioms
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

25. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Semantics (Intuitive)
C vd D
Every named individual in C is also in D unless it causes an
inconsistency
Every un-named individual in C is also in D (it behaves
exactly same as v)
Exceptions occur only in named individuals
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

26. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Default satisfying individuals
A named individual a is said to satisfy a default C vd D if:
1 a 2 CI;DI, or
2 a 2 (:C)I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

27. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Default satisfying individuals
A named individual a is said to satisfy a default C vd D if:
1 a 2 CI;DI, or
2 a 2 (:C)I
Key: We want to maximize the sets of individuals that satisfy each
default
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

28. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Interpretation
An interpretation I for a default-knowledge-base (KB; )
1 (I, :I) as usual
2 Additionally, I denotes the tuple (XI
1 ; : : : ;XI
n )
3 XI
i is the set of interpreted named individuals satisfying the
i th default Ci vd Di
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

29. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Preference relation KB;
I, J are two interpretations of default-knowledge-base (KB; ).
Then I KB; J if:
1 aI = aJ for all a 2 KB
i XJ
2 XI
i for all 1 i j j
i XJ
3 XI
i for some 1 i j j
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

30. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Example
Knowledge Base
Bird(a)
Bird(b)
Penguin(c)
Penguin v Bird
Penguin u Fly v ?
= fBird vd Flyg
Interpretation I
I = fa; b; cg
BirdI = fa; b; cg
PenguinI = fcg
Fly I = fag
I = (fag)
Interpretation J
J = fa; b; cg
BirdJ = fa; b; cg
PenguinJ = fcg
FlyJ = fa; bg
J = (fa; bg)
J KB; I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

31. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Example
Knowledge Base
Bird(a)
Bird(b)
Penguin(c)
Penguin v Bird
Penguin u Fly v ?
= fBird vd Flyg
Interpretation I
I = fa; b; cg
BirdI = fa; b; cg
PenguinI = fcg
Fly I = fag
I = (fag)
Interpretation J
J = fa; b; cg
BirdJ = fa; b; cg
PenguinJ = fcg
FlyJ = fa; bg
J = (fa; bg)
J KB; I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

32. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
d-model
An interpretation I of (KB; ) is called a default model or d-model
if
I satis

33. es all axioms of KB
CI
i DI
i , for all un-named individuals
I is maximal with respect to the preference relation KB;
A default-knowledge-base that has a d-model is called d-satis

34. able.
Note: Reiter's normal defaults are always satis

35. able.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

36. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
d-entailment
An axiom is d-entailed by a default-knowledge-base if it is
true in all the d-models.
We follow skeptical reasoning.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

37. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Decidability
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

38. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
IndKB = fa; bg
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

39. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
IndKB = fa; bg
P(IndKB) = ffag; fbg; fa; bgg
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

40. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB) : : :n-1 times P(IndKB), where n is
the cardinality of
IndKB = fa; bg
P(IndKB) = ffag; fbg; fa; bgg
Pn(IndKB) =
f(fag; fag); (fag; fbg); (fbg; fag); : : : ; (fa; bg; fa; bg)g, for
j j= 2
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

41. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB) : : :n-1 times P(IndKB), where n is
the cardinality of
IndKB = fa; bg
P(IndKB) = ffag; fbg; fa; bgg
Pn(IndKB) =
f(fag; fag); (fag; fbg); (fbg; fag); : : : ; (fa; bg; fa; bg)g, for
j j= 2
Note: Pn(IndKB) is a set of n-tuples that contains all the
possible combinations for I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

42. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Knowledge base re-writing
Consider some P = (X1; : : : ;Xn) 2 Pn(IndKB). Then KBP is
obtained by adding the following axioms to KB, for each Ci vd Di
Intuition
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

43. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Knowledge base re-writing
Consider some P = (X1; : : : ;Xn) 2 Pn(IndKB). Then KBP is
obtained by adding the following axioms to KB, for each Ci vd Di
1 Xi (Ci u Di u fa1; : : : ; akg) t (:Ci u fa1; : : : ; akg)
Intuition
Recall, an individual a satis

44. es a default C vd D if
a 2 CI;DI or a 2 (:C)I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

45. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Knowledge base re-writing
Consider some P = (X1; : : : ;Xn) 2 Pn(IndKB). Then KBP is
obtained by adding the following axioms to KB, for each Ci vd Di
1 Xi (Ci u Di u fa1; : : : ; akg) t (:Ci u fa1; : : : ; akg)
2 Ci u :fa1; : : : ; akg v Di
Intuition
Recall, an individual a satis

46. es a default C vd D if
a 2 CI;DI or a 2 (:C)I
Unknowns always satisfy the defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

47. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Decidability of d-satis

48. ability
Lemma
Let (KB; ) be a default-knowledge-base, if KBP is classically
satis

49. able for some P 2 Pn(IndKB), then (KB; ) is d-satis

50. able.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

51. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Decidability of d-satis

52. ability
Lemma
Let (KB; ) be a default-knowledge-base, if KBP is classically
satis

53. able for some P 2 Pn(IndKB), then (KB; ) is d-satis

54. able.
Theorem
The task of determining d-satis

55. ability of default-knowledge-bases
is decidable.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

56. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Decidability of d-entailment tasks
For non-montonic logics such as free-defaults entailment tasks
are not directly reducible to satis

57. ability check
We need to check all models for d-entailments
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

58. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Proof sketch
KBP is constructed using P 2 Pn(IndKB)
Pn(IndKB) is

59. nite
An order can be computed on Pn(IndKB) based on maximality
of its elements (

60. nite time)
Construct a set MaxP consisting of maximal classically
satis

61. able KBPs
MaxP is generated in

62. nite time due to the

63. niteness of
Pn(IndKB)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

64. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Proof sketch
KBP is constructed using P 2 Pn(IndKB)
Pn(IndKB) is

65. nite
An order can be computed on Pn(IndKB) based on maximality
of its elements (

66. nite time)
Construct a set MaxP consisting of maximal classically
satis

67. able KBPs
MaxP is generated in

68. nite time due to the

69. niteness of
Pn(IndKB)
Theorem
A DL axiom is entailed by a default-knowledge-base (KB; ) i it
is classically entailed by every KBP 2 MaxP
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

70. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Role defaults
Note: Similar construction can be used to show decidability of role
defaults under the semantics of free-defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

71. Introduction
Contribution
Free defaults
Conclusion
Semantics
Decidability
Back to the example
Now, the mapping can be denoted as
b:hasSpouse vd a:hasSpouse, a:hasSpouse vd b:hasSpouse
b:hasSpouseI = f(b:david; b:mike); (b:mike; b:david);
(a:john; a:marry); (a:marry; a:john)g
a:hasSpouseI = f((a:john; a:marry); (a:marry; a:john)g
Notice that heterogeneity is still respected and at the same
time relations can be carried over
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

72. Introduction
Contribution
Free defaults
Conclusion
1 Introduction
Motivation
Examples
Proposed Solution
2 Contribution
3 Free defaults
Semantics
Decidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

73. Introduction
Contribution
Free defaults
Conclusion
Conclusion
Drawback
The free-defaults don't work when un-named individuals are
implicit exceptions
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

74. Introduction
Contribution
Free defaults
Conclusion
Conclusion
Drawback
The free-defaults don't work when un-named individuals are
implicit exceptions
Future work
Investigation of decidability of more general defaults in DLs
Eecient algorithmization of d-entailment tasks
Implementation
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)

75. Introduction
Contribution
Free defaults
Conclusion
Questions?
DaSe Lab for Data Semantics at Wright state university
Twitter : @DaSeLab, @sengupta kunal
Website : http://wright.edu/~sengupta.4
Topics : Semantic Web, Knowledge representation and
reasoning, Description Logics, Rules, Ontology alignment,
Applications ...
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)