1. Updating TBoxes in DL-Lite
D. Zheleznyakov

2. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

3. Description Logics (DLs)
Formalism to represent structered knowledge
Traditinal inference tasks for static DL KBs: – concept satisfiability – concept, role hierarchies
More recently – query answering
Web services are getting more important

4. Web Services (?)
There are many things that might be called Web Services
We use the following meaning: software system designed to support interoperable machine-to-machine interaction over a network

5. DLs for Web Services Services access data through ontologies
Services can be specified using ontologies
There are needs:
to enable services do data modification
 ABox changes
to modify web services
 TBox changes

6. Ontology Changes
There are several types of ontology changes: – Revision – Update – Smth. – and such

7. Updating DL-Lite Ontologies
We study updates for DL-Lite KBs: it is the most tractable family of OWL 2
ABox updates: – Prelim./limited studied in [De Giacomo&al:2006] (?) – We revised and extended it [Calvanese&al:2010]
TBox updates: – Only TBox revision studied in [Qi,Du:2009] – Topic of this talk is TBox updates

8. Ontologies Concepts: PermStaff Manager AreaManager TopManager TBox:
Manager ⊑ PermStaff
Manager
AreaManager ⊑ Manager
ABox: ∅
We considered TBox updates only for KBs with empty ABoxes
AreaManager
TopManager

9. Updating Ontologies O: Mod(O): U: ✓ ✓ ✓ ✓ PermStaff Manager
AreaManager
TopManager U: ✓ ✓ ✓ ✓

10. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

11. Tractable Closure under Updates
We want an update operator such that:
Results are expressible in DL-Lite: we require updated KBs to be expressible in DL-Lite
Results computation is tractable: we require PTIME complexity

12. Principles of TBox Updates
⊨ AreaManager ⊑ PermStaff
PermStaff U:
AreaManager ⊑ ¬ PermStaff
IF new TBox ⊨
AreaManager ⊑ PermStaff ⊨
AreaManager ⊑ ¬ PermStaff
Manager
THEN AreaManagerM=∅ ∀M – model of the new TBox
Satisfiability Preservation: IF AM≠∅ before update, THEN AM≠∅ after update
(A is a atomic concept or role)
AreaManager
TopManager

13. Principles of TBox Updates
Manager ⊑ PermSatff AreaManager ⊑ Manager
PermStaff U:
AreaManager ⊑ ¬ PermStaff
Assume it is forbidden to change some parts of TBox.  There is a protected fragment Tpr ⊆ TBox E.g., Tpr = {Manager ⊑ PermSatff}.
Manager
Protection: We accept update iff Tpr and U together are fully satisfiable
AreaManager
TopManager

14. Principles of TBox Updates
Satisfiability Preservation: IF AM≠∅ before update, THEN AM≠∅ after update
Protection: We accept update iff protected part and U together are fully satisfiable
Moreover, we reject any update that enforces us to drop protected part

15. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

16. Model-Based Semantics (MBS)
Mod(O):
PermStaff
Manager
AreaManager
TopManager
Minimal distance U:
Mod(U): ✓ ✓ ✓ ✓

17. Model-Based Semantics (MBS)
Mod(O):
PermStaff
Manager
AreaManager
TopManager
Employee
Manager
AreaManager
TopManager
Project
O’: ? ✓ ✓ ✓ ✓
Mod(O’):

18. Winslett's Semantics
What does minimal distance mean? This depends on semantics.
Winslett’s semantics:
Well known
There are works on ABox update under Winslett’s semantics
Representative of MBS

19. Winslett's Semantics When distance(I, J) < distance(I, K) ? I:
AI={ John, Frank } BI={ Mary }
distance(I, J)
distance(I, K) J:
AJ={ John } BJ={ Mary } K:
AK={ John } BK=∅

20. Winslett's Semantics When distance(I, J) < distance(I, K) ? I:
AI={ John, Frank } BI={ Mary }
diff(I, J) = { {Frank}, ∅ }
distance(I, J)
distance(I, K) J:
AJ={ John } BJ={ Mary } K:
AK={ John } BK=∅

21. Winslett's Semantics When distance(I, J) < distance(I, K) ? I:
AI={ John, Frank } BI={ Mary }
diff(I, J) = { {Frank}, ∅ }
diff(I, K) = { {Frank}, {Mary} }
diff(I, J) ⊂ diff(I, K)
So, distance(I, J) < distance(I, K) iff diff(I, J) ⊂ diff(I, K)
distance(I, J)
distance(I, K) J:
AJ={ John } BJ={ Mary } K:
AK={ John } BK=∅

22. Winslett's Semantics. Example
PermStaff U:
TopManager ⊑ Manager
What should the updated result be?
The expectation: like in the picture
Is it so under Winslett’s semantics?
Manager
AreaManager
TopManager

23. Winslett's Semantics. Example
Frank
PermStaff U:
TopManager ⊑ Manager
Winslett’s semantics:
new TBox ⊨ U
Mimimal change in models
Frank
Manager
What is a new TBox here?
new TBox:
⊨ TopManager ⊑ Manager ✓
⊨ Manager ⊑ PermStuff ? ✗
⊨ AreaManager ⊑ Manager ? ✓
⊨ AreaManager ⊑ PermStaff ✓ ?
John
John
AreaManager
TopManager
Anything else?

24. Winslett's Semantics. Example
PermStaff
This TBox has irrelevant models that cannot be obtained from any model of the old TBox.
We should add something into the new TBox to cut off them
Manager
We cannot add any other DL-Lite assertion into the new TBox, otherwise, we cut off
some relevant models
AreaManager
TopManager

25. Winslett's Semantics
We have to drop important assertions (Manager ⊑ PermStuff)
Every MBS has such a problem
Result of update under Winslett’s semantics is inexpressible in DL-Lite.
 Consider Formula-Based semantics

26. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

27. Formula-Based Semantics (FBS)
FBS: closeness is measured btw set of formulas
How?
O1: O:
Satisfiable
PermStaff
Manager
AreaManager
TopManager
PermStaff
Manager ✓
Manager
AreaManager
TopManager
We take such a subset Omax ⊆ O, which is maximal by:
cardinality, or
set inclusion, or
some preferences
AreaManager
TopManager
O2:
Unsatisfiable ✗
The result is: Omax ∪ U
O3:
Satisfiable U:
Omax is not unique!
There are: O1max, O2max, … ✓
What to do with all of them? Depends on an approach

28. WIDTIO Approach. Example
We take only those formulas that appear in every Omax:
The result is: U ∪ ∩ Ojmax
PermStaff j U:
AreaManager ⊑ ¬ PermStaff
Manager
TBox:
AreaManager ⊑ PermStaff
Manager ⊑ PermStaff ⊈ O1max
AreaManager ⊑ Manager ⊈ O2max
AreaManager
TopManager

29. Cross-Product Approach. Example
The output is a disjunction of KBs, one KB for each Omax:
The result is: U ∪ {∨ Ojmax}
PermStaff j U:
AreaManager ⊑ ¬ PermStaff
Manager
TBox:
AreaManager ⊑ PermStaff OR
Manager ⊑ PermStaff ⊈ O1max
AreaManager ⊑ Manager ⊈ O2max
AreaManager
TopManager

30. Cross-Product Approach. Example
The output is a disjunction of KBs, one KB for each Omax:
The result is: U ∪ {∨ Ojmax}
PermStaff
Manager
AreaManager
TopManager
PermStaff
Manager
AreaManager
TopManager j U:
AreaManager ⊑ ¬ PermStaff
TBox:
AreaManager ⊑ PermStaff OR
Manager ⊑ PermStaff ⊈ O1max
AreaManager ⊑ Manager ⊈ O2max

31. Formula-Based Semantics
WIDTIO approach: – Loses too much information
Cross-product approach: – “Keeps” too much information – Inexpressible in DL-Lite

32. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

33. Bold Semantics Bold approach: – Takes on board only one Omax
Which Omax to take?
A maximal one by cardinality. NP-Hard
A maximal one by set inclusion. Polynomial
A maximal one by some preferences

34. Bold Semantics. Example
Start with empty TBox
Add assertions from U
Add assertions from TBox one by one, if no unsatisfiability appears
PermStaff U:
AreaManager ⊑ ¬ PermStaff ✓
Manager
TBox:
AreaManager ⊑ Manager ? ✓
Manager ⊑ PermStaff ? ✗
AreaManager ⊑ PermStaff ? ✗
The result is not unique
AreaManager
TopManager

35. Bold Semantics. Example
Start with empty TBox
Add assertions from U
Add assertions from TBox one by one, if no unsatisfiability appears
PermStaff U:
AreaManager ⊑ ¬ PermStaff ✓
Manager
TBox:
AreaManager ⊑ Manager ✓
Manager ⊑ PermStaff ✗
AreaManager ⊑ PermStaff ✗
The result is not unique U:
AreaManager ⊑ ¬ PermStaff ✓
AreaManager
TopManager
TBox:
AreaManager ⊑ Manager ? ✗
Manager ⊑ PermStaff ✓ ?
AreaManager ⊑ PermStaff ✗ ?

36. Checking Full Satisfiability

37. Outline I. Introduction
II. Requirements And Principles of TBox Updates
III. Review of Model-Based Semantics
IV. Review of Formula-Based Semantics
V. Bold semantics
VI. Conclusion

38. Conclusion We proposed two principles for DL KB updates
Model-based approaches: not good for TBox updates
Formula-based approaches: WIDTIO and CP are not applicable to DL-Lite KBs

39. Conclusion We proposed new semantics: Bold Semantics
We proposed polynomial time algorithm to compute update under Bold semantics

40. Thank you!

41. References
[De Giacomo&al:2006]
[Calvanese&al:2010]
[Qi,Du:2009]