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OpenHPI 5.2 - DL Inference and Reasoning

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OpenHPI 5.2 - DL Inference and Reasoning

  1. 1. Semantic Web TechnologiesLecture 5: Knowledge Representations II 02: DL Inference and Reasoning Dr. Harald Sack Hasso Plattner Institute for IT Systems Engineering University of Potsdam Spring 2013 This file is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0)
  2. 2. 2 Lecture 5: Knowledge Representations II Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  3. 3. 3 02 DL Inference and ReasoningOpen HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II
  4. 4. Open World Assumption - OWA4 • When we have an empty DL ontology, everything is possible • We then constrain an ontology iteratively, making it more restrictive as we go • We state what is not possible, what is forbidden or excluded Sheep ⊑ Animal ⊓ ∀hasLimbs.Leg Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  5. 5. Open World Assumption - OWA5 • Can Sheep fly? Sheep ⊑ Animal ⊓ ∀hasLimbs.Leg • No idea, but probably yes (according to our knowledge base) • In the OWA, unless we have a statement (or we can infer) “sheep can/cannot fly” we return “don’t know” • In the real world, we are used to deal with incomplete information Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  6. 6. Open World Assumption - OWA6 • In the Semantic Web we expect people to extend our own models (but we don‘t worry in advance how) • The OWA assumes incomplete information by default • Therefore, we can intentionally underspecify our models and allow others to reuse and extend Sheep ⊑ Animal ⊓ ∀hasLimbs.Leg ⊓ canFly b le on possi extensi further Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  7. 7. Closed World Assumption - CWA7 • Closed World Systems require a place to put everything • You can’t say anything until there’s somewhere to say it, as e.g. a slot on a frame, field on an OO class, column in a DB • In Close World Systems, we state what is possible and have to specify all knowledge • the CWA holds that anything that cannot be shown to be true is false; no explicit declaration of falsehood is needed. ! can‘ t fly Sheep Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  8. 8. Open World vs. Closed World Assumption8 • OWA: Open World Assumption The existence of further individuals is possible, if they are not explicitly excluded. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  9. 9. Open World vs. Closed World Assumption8 • OWA: Open World Assumption The existence of further individuals is possible, if they are not explicitly excluded. • CWA: Closed World Assumption It is assumed that the knowledge base contains all individuals. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  10. 10. Open World vs. Closed World Assumption8 • OWA: Open World Assumption The existence of further individuals is possible, if they are not explicitly excluded. • CWA: Closed World Assumption It is assumed that the knowledge base contains all individuals. if we assume that we know no idea since everything about are all children we do not know Bill then all of his of Bill male? all children of Bill children are male child(Bill,Bob) ? ⊨ ∀child.Man(Bill) DL answers PROLOG answers Man(Bob) don‘t know yes Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  11. 11. Open World vs. Closed World Assumption8 • OWA: Open World Assumption The existence of further individuals is possible, if they are not explicitly excluded. • CWA: Closed World Assumption It is assumed that the knowledge base contains all individuals. if we assume that we know no idea since everything about are all children we do not know Bill then all of his of Bill male? all children of Bill children are male child(Bill,Bob) ? ⊨ ∀child.Man(Bill) DL answers PROLOG answers Man(Bob) don‘t know yes now we know ≤1 child.⊤(Bill) ? ⊨ ∀child.Man(Bill) yes everything about Bill‘s children Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  12. 12. Problems of Inference (1)9 • Global (In)Consistency of the knowledge base • Does the knowledge base make sense? KB ⊨ !? • Class(in)consistency C ≡ !? • Must class C be empty? • Class inclusion (Subsumption) C ⊑ D? • Structuring the knowledge base • Class equivalency C ≡ D? • Are two classes the same? Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  13. 13. Problems of Inference (2)10 • Class disjointness C ⊓ D = !? • Are two classes disjunctive? • Class membership C(a)? • Is individual a contained in class C? • Instance generation (Retrieval) „find all x with C(x)“ • Find all (known!) Individuals of class C. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  14. 14. Description Logics and Inference11 • Decidability: for each inference problem there always exists an algorithm that terminates in finite time • DLs are fragments of FOL, therefore (in principle) FOL inference algorithms (Resolution, Tableaux) can be applied. • But FOL algorithms do not always terminate! • Problem: Find algorithms that always terminate! • There might be no „naive“ solutions! Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  15. 15. Decidability and DL12 • FOL inference algorithms (Tableaux algorithm and Resolution) must be adapted for DLs • (for the lecture we will restrict to ALC tableaux algorithm) • Tableaux algorithm and resolution show unsatisfiability of a theory (knowledge base) • Adaption of entailment problems to the detection of contradictions in the knowledge base, i.e. proof of the unsatisfiability of the knowledge base! Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  16. 16. Reduction to Unsatisfiability (1)13 • Class (in)consistency C ≡ ! • iff KB⋃{C(a)} unsatisfiable (a new) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  17. 17. Reduction to Unsatisfiability (1)13 • Class (in)consistency C ≡ ! • iff KB⋃{C(a)} unsatisfiable (a new) • Class inclusion (Subsumption) C ⊑ D Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  18. 18. Reduction to Unsatisfiability (1)13 • Class (in)consistency C ≡ ! • iff KB⋃{C(a)} unsatisfiable (a new) • Class inclusion (Subsumption) C ⊑ D • iff KB⋃{(C⊓¬D)(a)} unsatisfiable (a new) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  19. 19. Reduction to Unsatisfiability (1)13 • Class (in)consistency C ≡ ! • iff KB⋃{C(a)} unsatisfiable (a new) • Class inclusion (Subsumption) C ⊑ D • iff KB⋃{(C⊓¬D)(a)} unsatisfiable (a new) • Class equivalency C ≡ D Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  20. 20. Reduction to Unsatisfiability (1)13 • Class (in)consistency C ≡ ! • iff KB⋃{C(a)} unsatisfiable (a new) • Class inclusion (Subsumption) C ⊑ D • iff KB⋃{(C⊓¬D)(a)} unsatisfiable (a new) • Class equivalency C ≡ D • iff C ⊑ D and D ⊑ C Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  21. 21. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  22. 22. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) • Class membership C(a) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  23. 23. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) • Class membership C(a) • iff KB⋃{¬C(a)} unsatisfiable Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  24. 24. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) • Class membership C(a) • iff KB⋃{¬C(a)} unsatisfiable • Instance generation (Retrieval) „find all x with C(x)“ Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  25. 25. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) • Class membership C(a) • iff KB⋃{¬C(a)} unsatisfiable • Instance generation (Retrieval) „find all x with C(x)“ • Check class membership for all individuals Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  26. 26. Reduction to Unsatisfiability (2)14 • Class disjointness C ⊓ D = ! • iff KB⋃{(C⊓D)(a)} unsatisfiable (a new) • Class membership C(a) • iff KB⋃{¬C(a)} unsatisfiable • Instance generation (Retrieval) „find all x with C(x)“ • Check class membership for all individuals • but: efficiency...? Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  27. 27. 15 03 Tableaux Algorithm for ALCOpen HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II

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