Local Closed World Semantics:Grounded Circumscription for OWL   Kunal Sengupta Adila Krisnadhi Pascal Hitzler    Kno.e.sis...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contribution•   Decidability•   Algorith...
OWA and CWA• Open World Assumption (OWA)     – If a statement is not known to be true, it is not       assumed to be false...
OWL Example                      Paper(paper1)• KB =                Paper(paper2)                hasAuthor(paper1, author1...
OWL Example                      Paper(paper1)• KB =                Paper(paper2)                                         ...
Local Closed World                                                                               Closed                   ...
Solution?• Local closed world Assumption     – Combination of OWA and CWA.     – Allow ontology engineers to close parts o...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contribution•   Decidability•   Algorith...
Circumscription• Circumscription for FOL [McCarthy 80]• Minimisation: Extension of minimized predicates as  small as possi...
Circumscription• Preference relation <CP on Interpretations I = (I, I)• Choose the preferred model. i.e minimal. compari...
Circumscription • Minimizing the extensions of closed predicates   (classes and properties).                       Problem...
Grounded Circumscription• Allow only named individuals in the  extensions of minimized predicates.• We say the pair (K,M) ...
Grounded Circumscription• Allow only named• A GC-model of (K,M): individuals in the   • Is a classical minimized  extensio...
GC- Example• I and J two models of KB (Assuming UNA)• hasAuthorI = { (paper1I, author1I),                 (paper1I, author...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contribution•   Decidability•   Algorith...
Contribution• Grounded circumscription semantics – An  intuitive approach to Local Closed World  Assumption.• Decidable ev...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contributions•   Decidability•   Algorit...
Decidability (Sketch)• Underlying DL is decidable• Finite number of named individuals• A GC-model can be constructed by   ...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contributions•   Decidability•   Algorit...
Algorithm (GC-satisfiability)• GC-satisfiability : A tableau procedure for  testing GC-KB (K,M) satisfiability• Task – To ...
Key• It suffices to show that there is a grounded model to  check GC-satisfiability.• Grounded Model: A model of GC-KB (K,...
New Expansion Rules• Grounding closed predicates.• Rule for C 2 M: If a variable node x, with C 2 L(x) then  choose a nomi...
New Tableau RulesISWC2011         K. Sengupta, A.Krisnadhi, and P.Hitzler   kunal@knoesis.org
GC-Satisfiability• Start with initial graph (Abox).• Apply expansion rules exhaustively.• If there is a inconsistency free...
Beyond Satisfiability• Instance checking, concept satisfiability, and  concept subsumption.• Reducing other inference prob...
Tableau2• Initialization: Abox and Nodes from a  consistent completion graph from GC-sat  checker.• Expansion rules same a...
Finding GC-model              Start           GC-Sat Tableau              No           Grounded            Model          ...
Finding GC-model              Start                                  Found                                 Grounded       ...
Finding GC-model              Start                                  Found                                 Grounded       ...
Finding GC-model              Start                                  Found                                 Grounded       ...
Inference Problems• Instance Checking C(a): Invoke the GC-Model  Finder algorithm and verify if C 2 L(a) for all  GC-Model...
Outline•   Local Closed World Assumption•   Grounded Circumscription Semantics•   Contributions•   Decidability•   Algorit...
Conclusion and Outlook• Conclusion     –     A new approach to LCWA, Grounded circumscription.     –     Decidable     –  ...
Thanks!ISWC2011   K. Sengupta, A.Krisnadhi, and P.Hitzler   kunal@knoesis.org
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Gc semantics- iswc2011

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Gc semantics- iswc2011

  1. 1. Local Closed World Semantics:Grounded Circumscription for OWL Kunal Sengupta Adila Krisnadhi Pascal Hitzler Kno.e.sis Center, Wright State University, Dayton, OH. {kunal, adila, pascal}@knoesis.org
  2. 2. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contribution• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  3. 3. OWA and CWA• Open World Assumption (OWA) – If a statement is not known to be true, it is not assumed to be false. – Knowledge is considered incomplete. – OWL• Closed world assumption (CWA) – If there is no proof for a statement to be true, it is false. – Knowledge is assumed to be complete. – Logic programming, databases etc.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  4. 4. OWL Example Paper(paper1)• KB = Paper(paper2) hasAuthor(paper1, author1) hasAuthor(paper1, author2) hasAuthor(paper2, author3) > v 8hasAuthor.Author• :hasAuthor(paper1, author3) is not a consequence.• Because of OWA, can’t rule out hasAuthor(paper1, auther3)• (·2 hasAuthor.Author)(paper1) is not a consequence.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  5. 5. OWL Example Paper(paper1)• KB = Paper(paper2) There is a Model in hasAuthor(paper1, author1) which author3 is an hasAuthor(paper1, author2) author of paper1. hasAuthor(paper2, author3) > v 8hasAuthor.Author• :hasAuthor(paper1, author3) is not a consequence.• Because of OWA, can’t rule out hasAuthor(paper1, auther3)• (·2 hasAuthor.Author)(paper1) is not a consequence.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  6. 6. Local Closed World Closed Predicates Paper hasAuthor Author Reviewer Conference Journal Issue publishedInISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  7. 7. Solution?• Local closed world Assumption – Combination of OWA and CWA. – Allow ontology engineers to close parts of the KB. – E.g. We can mark the class Author and the property hasAuthor as closed in the last example. – :hasAuthor(paper1, author3) – (·2 hasAuthor.Author)(paper1)ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  8. 8. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contribution• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  9. 9. Circumscription• Circumscription for FOL [McCarthy 80]• Minimisation: Extension of minimized predicates as small as possible.• CircCP(KB), Circumscription Pattern (M,V,F)• Circumpscription in DLs [Bonatti, Lutz, Wolter: JAIR 2009]ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  10. 10. Circumscription• Preference relation <CP on Interpretations I = (I, I)• Choose the preferred model. i.e minimal. comparing interpretations by their extensions for minimized predicates• A circumscriptive model of a KB is a model of KB which is minimal w.r.t <CP relationISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  11. 11. Circumscription • Minimizing the extensions of closed predicates (classes and properties). Problems • Preference relation <CP on Interpretations I = (I, I) comparing interpretations by their extensions for minimized predicates• Extensions of minimized predicates may contain unknown individuals.• Undecidable in the presence of non-empty Tbox and minimized properties [Bonatti, Lutz, Wolter: JAIR 2009].• High Complexity for expressive DLs. • A circumscriptive model of a KB is a model of KB which is minimal w.r.t <CP relation ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  12. 12. Grounded Circumscription• Allow only named individuals in the extensions of minimized predicates.• We say the pair (K,M) is a GC-KB K w.r.t the set of minimized predicates M in K.• Preference relation for comparing two modelsISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  13. 13. Grounded Circumscription• Allow only named• A GC-model of (K,M): individuals in the • Is a classical minimized extensions ofmodel of K, predicates. • Extensionspair (K,M) is predicatesK w.r.t theonly• We say the of minimized a GC-KB consist of set named individuals (and pairs), and of Is a minimal model with respect to the preference • minimized predicates M in K.• Preference relation for comparing two models relationISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  14. 14. GC- Example• I and J two models of KB (Assuming UNA)• hasAuthorI = { (paper1I, author1I), (paper1I, author2I), (paper1I, author3I), (paper2I, author3I)}• hasAuthorJ = { (paper1J, author1J), (paper1J, author2J), (paper2J, author3J)}• hasAuthorJ ½ hasAuthorI• J ÁM I, I is not a GC-Model of (K,M)ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  15. 15. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contribution• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  16. 16. Contribution• Grounded circumscription semantics – An intuitive approach to Local Closed World Assumption.• Decidable even with minimized/closed roles.• A Tableau procedure to reason with GC knowledge bases.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  17. 17. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contributions• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  18. 18. Decidability (Sketch)• Underlying DL is decidable• Finite number of named individuals• A GC-model can be constructed by – Assigning a minimal set of named individuals to each minimized classes. – A minimal set of pairs of named individuals to minimized Roles .• Since we have a finite set to choose from the problem of finding a GC-model is decidable.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  19. 19. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contributions• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  20. 20. Algorithm (GC-satisfiability)• GC-satisfiability : A tableau procedure for testing GC-KB (K,M) satisfiability• Task – To check if GC-KB (K,M) has a GC- model.• Reduced to checking for grounded model (not necessarily minimal).• Modify exiting Tableau and add expansion rules to ground minimized predicates.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  21. 21. Key• It suffices to show that there is a grounded model to check GC-satisfiability.• Grounded Model: A model of GC-KB (K,M) such that, the extensions of the minimized predicates contain only named individuals.• GC-model: A grounded model which is also a minimal model of the GC-KB (K,M)ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  22. 22. New Expansion Rules• Grounding closed predicates.• Rule for C 2 M: If a variable node x, with C 2 L(x) then choose a nominal node and merge the labels (grounding), disregard node x.• Rule for R 2 M: If R 2 L(x,y) and at least one of x, y is a variable, then ground the variable nodes by choosing a nominal node.• NOTE: These rules are not applied to blocked nodes in the graph.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  23. 23. New Tableau RulesISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  24. 24. GC-Satisfiability• Start with initial graph (Abox).• Apply expansion rules exhaustively.• If there is a inconsistency free completion graph, then GC-KB is GC-satisfiable.• Blocking• Termination.• Sound and complete.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  25. 25. Beyond Satisfiability• Instance checking, concept satisfiability, and concept subsumption.• Reducing other inference problems to GC- satisfiability is not straight forward.• GC-satisfiability just looks for grounded models.• Tableau2: Try to find a smaller model.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  26. 26. Tableau2• Initialization: Abox and Nodes from a consistent completion graph from GC-sat checker.• Expansion rules same as GC-sat but 9R.C rule does not add new nodes.• Preference clash - if a completion graph represents a bigger model than initial model .ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  27. 27. Finding GC-model Start GC-Sat Tableau No Grounded Model No GC-Model EndISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  28. 28. Finding GC-model Start Found Grounded Model I GC-Sat Tableau Tableau2 No Grounded Model No GC-Model EndISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  29. 29. Finding GC-model Start Found Grounded Model I GC-Sat Tableau Tableau2 No Grounded No Smaller Model Model Found No GC-Model End I is a GC-ModelISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  30. 30. Finding GC-model Start Found Grounded Model I GC-Sat Tableau Tableau2 No Smaller Grounded Model No Smaller Model Found Model Found No GC-Model End I is a GC-ModelISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  31. 31. Inference Problems• Instance Checking C(a): Invoke the GC-Model Finder algorithm and verify if C 2 L(a) for all GC-Models.• Concept satisfiability: Invoke the GC-Model Finder algorithm and verify if C 2 L(a) for at least one named individual in all GC-Models.• Subsumption: Reducible to Concept satisfiability.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  32. 32. Outline• Local Closed World Assumption• Grounded Circumscription Semantics• Contributions• Decidability• Algorithms• ConclusionISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  33. 33. Conclusion and Outlook• Conclusion – A new approach to LCWA, Grounded circumscription. – Decidable – Reducing one reasoning task to other is not trivial. – Algorithm for reasoning with GC.• Future work: – Find smarter reasoning algorithms. – Complexity analysis for all OWL fragments. – Implementation for use in real world.ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org
  34. 34. Thanks!ISWC2011 K. Sengupta, A.Krisnadhi, and P.Hitzler kunal@knoesis.org

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