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Updating TBoxes in DL-Lite

Updating TBoxes in DL-Lite. D. Zheleznyakov. Outline. I. Introduction II. Requirements A nd P rinciples of TBox U pdates III. Review of Model-Based Semantics IV. Review of Formula-Based Semantics V. Bold semantics VI. Conclusion. Description Logics (DLs).

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Updating TBoxes in DL-Lite

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  1. Updating TBoxes in DL-Lite D. Zheleznyakov

  2. Outline I. Introduction II.Requirements And Principles ofTBox 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 PermStaff Manager AreaManager TopManager TBox: Manager ⊑ PermStaff AreaManager ⊑ Manager Manager ABox: ∅ We considered TBox updates only for KBs with empty ABoxes AreaManager TopManager

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

  10. Outline I. Introduction II.Requirements And Principles ofTBox 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 TBox: ⊨ AreaManager ⊑ PermStaff PermStaff AreaManager ⊑ ¬ PermStaff U: AreaManager ⊑ PermStaff IF new TBox ⊨ ⊨ 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 TBox: Manager ⊑ PermSatffAreaManager ⊑ Manager PermStaff AreaManager ⊑ ¬ PermStaff U: Assume it is forbidden to changesome parts of TBox. There is a protected fragmentTpr⊆ TBoxE.g., Tpr = {Manager ⊑ PermSatff}. Manager Protection: We accept update iffTpr 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 andU together are fully satisfiable Moreover, we reject any update that enforces us to drop protected part

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

  16. PermStaff Manager AreaManager TopManager Model-BasedSemantics (MBS) O: Mod(O): Minimaldistance U: Mod(U): ✓ ✓ ✓ ✓

  17. PermStaff Manager Employee AreaManager TopManager Manager Project AreaManager TopManager Model-BasedSemantics (MBS) O: Mod(O): 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) AJ={ John }BJ={ Mary } K: AK={ John }BK=∅ J:

  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) AJ={ John }BJ={ Mary } K: AK={ John }BK=∅ J:

  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) AJ={ John }BJ={ Mary } K: AK={ John }BK=∅ J:

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

  24. Winslett's Semantics. Example PermStaff This TBox has irrelevant modelsthat cannot be obtainedfrom any model of the old TBox. • We should add something into the new TBoxto cut off them Manager We cannot add any otherDL-Lite assertion into the new TBox,otherwise, we cut off some relevant models AreaManager TopManager

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

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

  27. PermStaff PermStaff Manager Manager AreaManager TopManager AreaManager TopManager Manager AreaManager TopManager Formula-Based Semantics (FBS) FBS: closeness is measuredbtw set of formulas How? O1: O: Satisfiable ✓ • We take such a subset Omax⊆O, which is maximal by: • cardinality, or • set inclusion, or • some preferences O2: Unsatisfiable ✗ The result is: Omax∪U O3: Satisfiable U: Omaxis 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 AreaManager ⊑ PermStaff TBox: Manager ⊑ PermStaff⊈ O1max AreaManager ⊑ Manager⊈ O2max TopManager AreaManager

  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 AreaManager ⊑ PermStaff TBox: OR Manager ⊑ PermStaff⊈ O1max AreaManager ⊑ Manager⊈ O2max TopManager AreaManager

  30. PermStaff PermStaff Manager Manager TopManager TopManager AreaManager AreaManager Cross-Product Approach. Example The output is a disjunction of KBs,one KB for each Omax: The result is: U∪ {∨Ojmax} j U: AreaManager ⊑ ¬ PermStaff AreaManager ⊑ PermStaff TBox: 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 ofTBox Updates III. Review of Model-Based Semantics IV. Review of Formula-Based Semantics V. Bold semantics VI. Conclusion

  33. Bold Semantics • Which Omax to take? • Bold approach: – Takes on board only one Omax • 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 TopManager AreaManager

  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 ✓ TopManager AreaManager TBox: AreaManager ⊑ Manager ? ✗ Manager ⊑ PermStaff ✓ ? AreaManager ⊑ PermStaff ✗ ?

  36. Checking Full Satisfiability

  37. Outline I. Introduction II.Requirements And Principles ofTBox 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 applicableto DL-Lite KBs

  39. Conclusion • We proposed new semantics:Bold Semantics • We proposed polynomial time algorithmto compute update under Bold semantics

  40. Thank you!

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

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