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Updating TBoxes in DL-Lite

Updating TBoxes in DL-Lite. D. Zheleznyakov , D. Calvanese, E. Kharlamov, W. Nutt Free University of Bozen-Bolzano DL 2010, May 6th. 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

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Updating TBoxes in DL-Lite

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  1. Updating TBoxes in DL-Lite D. Zheleznyakov, D. Calvanese,E. Kharlamov, W. Nutt Free University of Bozen-Bolzano DL 2010, May 6th

  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. Inference in DLs • Traditional inference tasks for static DL KBs:– checking concept satisfiability– computing concept and role hierarchies– query answering • Ontologies may evolve in time: interest in ontology evolution has appeared 3/31

  4. DLs for Web Services • DLs provide foundation for OWL • Services: software systems supportingmachine-to-machine interoperation • Services access data through ontologies • Services can be specified using ontologies • To reflect changes, we need: ∙ABox evolution ∙TBox evolution 4/31

  5. DL-Lite Ontologies • DL-Lite is a tractable fragment of OWL 2 • TBox reasoning in DL-Lite is polynomial • We consider DL-LiteFR:∙ Concept and role inclusions∙ Concept disjointness∙ Domain and range of roles∙ Mandatory participation to roles∙ Role functionality∙ Interaction between role functionality and role inclusion is restricted to ensure polynomiality 5/31

  6. Example of DL-Lite TBox Concepts: PermStaff PermStaff Manager AreaManager TopManager TBox: Manager ⊑ PermStaff AreaManager ⊑ Manager Manager AreaManager TopManager 6/31

  7. Updating DL-Lite Ontologies • We study updates for DL-Lite KBs • ABox updates:– Initially studied in [De Giacomo&al:2006, 2009]– Revised and extended in [Calvanese&al:2010] • TBox updates:– TBox evolution studied in [Qi,Du:2009]– Topic of this talk is TBox updates We do not consider ABoxes 7/31

  8. PermStaff Employee Manager Manager Project AreaManager TopManager AreaManager TopManager Updating DL TBoxes Old TBox: New TBox: • New TBox should be consistent • New TBox should entail U • New TBox should minimally differ from Old TBox U: 8/31

  9. 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

  10. Desiderata for Updates We want an update operator such that: • Closure under updates:Updated TBox should be expressible in DL-Lite • Efficiency:Update result should be computable in PTIME 10/31

  11. Principles of TBox Updates (1) U: AreaManager ⊑ ¬ PermStaff PermStaff ⊨ AreaManager ⊑ PermStaff TBox IF new TBox has both AreaManager ⊑ PermStaff and AreaManager ⊑ ¬ PermStaff Manager THEN AreaManagerM=∅for every modelM of the new TBox Satisfiability Preservation: Updates should preserve satisfiability of every atomic concept and role in the TBox (i.e., updates preserve full satisfiability) AreaManager TopManager 11/31

  12. Principles of TBox Updates (2) U: AreaManager ⊑ ¬ PermStaff PermStaff ⊨ AreaManager ⊑ PermStaff TBox We may want to forbidto changesome parts of the TBox. We declare a fragment Tpr⊆ TBoxas protectedE.g., Tpr = {Manager ⊑ PermStaff}. Manager Protection: Update should not affect the protected part of the TBox AreaManager TopManager 12/31

  13. Principles of TBox Updates • Preservation of full satisfiability • Protection  We accept update iff the resulting TBox is fully satisfiable and contains both the protected part and the update • Minimal change principleWe discuss it later 13/31

  14. 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

  15. PermStaff Manager AreaManager TopManager Manager Manager Project Project TopManager TopManager Model-BasedSemantics (MBS) TBox: Mod(TBox): U: ✓ ✓ ✓ ✓ 15/31

  16. PermStaff Manager AreaManager TopManager Manager Project TopManager Model-BasedSemantics (MBS) TBox: Mod(TBox): Minimaldistance U: Mod(U): ✓ ✓ ✗ ✓ 15/31

  17. PermStaff Manager Employee AreaManager TopManager Manager Project AreaManager TopManager Model-BasedSemantics (MBS) TBox: Mod(TBox): T’: ? ✓ ✓ ✗ ✓ Mod(T’): 15/31

  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 • Distance under Winslett’s Semantics:based on symmetric difference and set inclusion 16/31

  19. Winslett's Semantics When distance(I, J) < distance(I, K) ? I: A(John), A(Frank),B(Mary) distance(I, J) distance(I, K) A(John)B(Mary) K: A(John)B(Jane) J: 17/31

  20. Winslett's Semantics When distance(I, J) < distance(I, K) ? I: A(John),A(Frank),B(Mary) diff(I, J) = { A(Frank) } distance(I, J) distance(I, K) A(John)B(Mary) K: A(John)B(Jane) J: 17/31

  21. Winslett's Semantics When distance(I, J) < distance(I, K) ? I: A(John),A(Frank),B(Mary) diff(I, J) = { A(Frank) } diff(I, K) = { A(Frank), B(Mary), B(Jane) } 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) A(John)B(Mary) K: A(John)B(Jane) J: 17/31

  22. Winslett's Semantics. Example PermStaff TopManager ⊑ Manager U: What should the updated TBox be? The expectation: like in the picture Is it so under Winslett’s semantics? Manager AreaManager TopManager 18/31

  23. Winslett's Semantics. Example FrankMary PermStaff TopManager ⊑ Manager U: What should the updated TBox be? ✓ ⊨ TopManager ⊑ Manager new TBox: ✗ ⊨ Manager ⊑ PermStaff ? FrankMary` Manager ✓ ⊨ AreaManager ⊑ Manager ? ⊨ AreaManager ⊑ PermStaff ? ✓ Anything else? John John Mary AreaManager TopManager 18/31

  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 them off Manager We cannot add any otherDL-Lite assertion into the new TBox,otherwise, we cut off some relevant models AreaManager TopManager 18/31

  25. Drawbacks of Winslett’s Sem. • Result of update under Winslett’s semantics is inexpressible in DL-Lite • We have to unexpectedly drop assertions(Manager ⊑ PermStaff) Every MBS may have similar problems  Consider Formula-Based semantics 19/31

  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) TBox: FBS: closeness is measuredbetween sets of formulas How? T1: Satisfiable ✓ • We take a satisfiable subsetTmax⊆ TBox, which is maximal wrt: ∙ cardinality, or ∙ set inclusion, or ∙ some preferences T2: Unsatisfiable ✗ • Result is: Tmax∪U U: T3: Satisfiable • In general, Tmaxis not unique! • There are: T1max, T2max, … ✓ • What to do with all of them?This depends on the approach we choose 21/31

  28. WIDTIO Approach. Example We take only those formulas that appear in every Tmax: The result is: U∪ ∩Tjmax PermStaff j U: AreaManager ⊑ ¬ PermStaff Manager AreaManager ⊑ PermStaff TBox: { Manager ⊑ PermStaff } = T1max { AreaManager ⊑ Manager } = T2max TopManager AreaManager 22/31

  29. Cross-Product Approach. Example The output is a disjunction of TBoxes,one for each Tmax: The result is: U∪ {∨Tjmax} PermStaff j U: AreaManager ⊑ ¬ PermStaff Manager AreaManager ⊑ PermStaff TBox: OR { Manager ⊑ PermStaff } = T1max { AreaManager ⊑ Manager } = T2max TopManager AreaManager 23/31

  30. PermStaff PermStaff Manager Manager TopManager TopManager AreaManager AreaManager Cross-Product Approach. Example The output is a disjunction of TBoxes,one for each Tmax: The result is: U∪ {∨Tjmax} j U: AreaManager ⊑ ¬ PermStaff AreaManager ⊑ PermStaff TBox: OR { Manager ⊑ PermStaff } = T1max { AreaManager ⊑ Manager } =T2max 23/31

  31. Formula-Based Semantics • WIDTIO approach: – Loses too much information – We have proved that computing the result is NP-complete • Cross-product approach: – “Keeps” too much information – Disjunction is inexpressible in DL-Lite 24/31

  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 Tmax to take? • Bold approach: – Takes on board only one Tmax • A maximal one by cardinality.NP-Hard • A maximal one by set inclusion.Polynomial • A maximal one by some preferences 26/31

  34. Bold Semantics. Algorithm • Start with empty TBox • Add assertions from U • Add assertions of the old TBoxone by one,if no unsatisfiability appears PermStaff U: AreaManager ⊑ ¬ PermStaff ✓ Manager TBox: AreaManager ⊑ Manager ? ✓ Manager ⊑ PermStaff ✗ ? AreaManager ⊑ PermStaff ✗ ? TopManager AreaManager 27/31

  35. Bold Semantics. Algorithm • Start with empty TBox • Add assertions from U • Add assertions of the old TBoxone by one,if no unsatisfiability appears PermStaff U: AreaManager ⊑ ¬ PermStaff ✓ Manager TBox: AreaManager ⊑ Manager ✓ Manager ⊑ PermStaff ✗ AreaManager ⊑ PermStaff ✗ The algorithm is non-deterministic U: AreaManager ⊑ ¬ PermStaff ✓ TopManager AreaManager TBox: AreaManager ⊑ Manager ? ✗ Manager ⊑ PermStaff ✓ ? AreaManager ⊑ PermStaff ✗ ? 27/31

  36. Polynomiality of the Algorithm • Building the closure of the TBox:polynomial • Going through all the assertions:polynomial • Checking full satisfiability:Theorem: Checking full satisfiability for DL-LiteFR is polynomial in the size of (TBox∪ U) 28/31

  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 discussed principles for TBox updates • Both MBS and FBS have drawbacks forDL-Lite TBox updates • We proposed a new semantics • We developed a polynomial time algorithmto compute update under the new semantics 30/31

  39. Future work • Combining ABox and TBox updates • Use of preferences to manage non-determinism • Implementing update algorithms • Extend it to more expressive DLs 31/31

  40. Thank you! ONTORULE ProjectONTOlogies Meets Business RULesFP 7 grant, ICT-231875http://ontorule-project.eu/ Webdam Project Foundations of Web Data Management ERC FP7 grant, agreement n. 226513http://webdam.inria.fr/

  41. References • [De Giacomo&al:2006] On the update of description logic ontologies at the instance level. In: Proc. of the 21st Nat. Conf. on Artificial Intelligence (AAAI 2006). 1271–1276 • [Calvanese&al:2010] Updating ABoxes in DL-Lite. In: Proc. of the 4th Alberto Mendelzon Workshop on Foundations of Data Management (AWS 2010) • [Qi,Du:2009] Model-based revision operators forterminologies in description logics.In:Proc. of the 21st Int. Joint Conf.on ArtificialIntelligence (IJCAI 2009).891–897

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