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OpenHPI 4.9 - Tableaux Algorithm

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OpenHPI 4.9 - Tableaux Algorithm

  1. 1. Semantic Web TechnologiesLecture 4: Knowledge Representations I 09: Tableaux Algorithm 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 4: Knowledge Representations I Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  3. 3. 3 09 Tableaux AlgorithmOpen HPI - Course: Semantic Web Technologies - Lecture Potsdam Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität 4: Knowledge Representations I
  4. 4. Tableaux Algorithm • For automated reasoning, we need syntactic algorithms to4 check the consistency of logical assertions • To apply resolution, formulas have to be in clausal form • The method of analytical tableaux is based on disjunctive normal form • invented by Dutch logician Evert Willem Beth in 1955 and Evert Willem Beth simplified by Raymond Smullyan. (1908-1964) Raymond Merrill Smullyan Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  5. 5. Tableaux Algorithm • For automated reasoning, we need syntactic algorithms to4 check the consistency of logical assertions • To apply resolution, formulas have to be in clausal form • The method of analytical tableaux is based on disjunctive normal form • invented by Dutch logician Evert Willem Beth in 1955 and Evert Willem Beth simplified by Raymond Smullyan. (1908-1964) • Basic Idea of Tableaux Algorithm (similar to Resolution): Raymond Merrill Smullyan Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  6. 6. Tableaux Algorithm • For automated reasoning, we need syntactic algorithms to4 check the consistency of logical assertions • To apply resolution, formulas have to be in clausal form • The method of analytical tableaux is based on disjunctive normal form • invented by Dutch logician Evert Willem Beth in 1955 and Evert Willem Beth simplified by Raymond Smullyan. (1908-1964) • Basic Idea of Tableaux Algorithm (similar to Resolution): • Proof algorithm (decision procedure) to check the consistency of a logical formula by inferring that its negation is a contradiction (proof by refutation). Raymond Merrill Smullyan Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  7. 7. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with a logical formula.5 • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  8. 8. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with a logical formula.5 • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; • a branch of the path represents a disjunction. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  9. 9. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with a logical formula.5 • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; • a branch of the path represents a disjunction. r (q ∧ r) ∨ (p ∧ ¬ r) ∨ (q ∧ r) (p ∧ ¬ r) ∨ r (p ∧ ¬ r) r q r p ¬ r Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  10. 10. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with6 a logical formula. • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; • a branch of the path represents a disjunction. (2) The tree is created by successive application of the Tableaux Extension Rules. (3) A path in the Tableaux is closed, Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  11. 11. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with6 a logical formula. • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; • a branch of the path represents a disjunction. (2) The tree is created by successive application of the Tableaux Extension Rules. (3) A path in the Tableaux is closed, • if along the path as well X as ¬X for a formula X occurs (X doesn‘t have to be atomic) or Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  12. 12. Tableaux Algorithm for Propositional Logic (1) Construct Decision Tree, where each node is marked with6 a logical formula. • A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; • a branch of the path represents a disjunction. (2) The tree is created by successive application of the Tableaux Extension Rules. (3) A path in the Tableaux is closed, • if along the path as well X as ¬X for a formula X occurs (X doesn‘t have to be atomic) or • if false occurs . Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  13. 13. Tableaux Algorithm for Propositional Logic (4) A tableaux is fully expanded, if no more extension rules7 can be applied. (5) A tableaux is called closed, if all its paths are closed. (6) A Tableaux Proof for a formula X is a closed tableaux for ¬X. • The selection of the tableaux extension rules to be applied in the tableaux is not deterministic. • There are heuristics for the propositional logic tableaux to select which extension rules to be applied best Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  14. 14. Tableaux Extension Rules - PL8 • for PL: ¬¬X ¬T ¬F X F T • for conjunctive Formula (α-Rules): α X∧Y ¬(X∨Y) ¬(X Y) α1 X ¬X X α2 Y ¬Y ¬Y • for disjunctive formula (β-Rules): β X∨Y ¬(X∧Y) (X Y) β1 | β2 X | Y ¬X | ¬Y ¬X | Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  15. 15. Tableaux Algorithm (PL) - Example (1):9 α-Rule ¬(X Y) X ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  16. 16. Tableaux Algorithm (PL) - Example (1):9 proof: ((q ∧ r) (¬q ∨ r)) α-Rule ¬(X Y) X ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  17. 17. Tableaux Algorithm (PL) - Example (1):9 proof: ((q ∧ r) (¬q ∨ r)) α-Rule (1) ¬((q ∧ r) (¬q ∨ r)) ¬(X Y) X ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  18. 18. Tableaux Algorithm (PL) - Example (1):9 proof: ((q ∧ r) (¬q ∨ r)) α-Rule (1) ¬((q ∧ r) (¬q ∨ r)) ¬(X Y) X (2) α,1: (q ∧ r) ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  19. 19. Tableaux Algorithm (PL) - Example (1):9 proof: ((q ∧ r) (¬q ∨ r)) α-Rule (1) ¬((q ∧ r) (¬q ∨ r)) ¬(X Y) X (2) α,1: (q ∧ r) ¬Y (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  20. 20. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule X ∧ Y X Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  21. 21. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule (4) α,2: q X ∧ Y X Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  22. 22. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule (4) α,2: q X ∧ Y (5) α,2: r X Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  23. 23. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule (4) α,2: q X ∧ Y (5) α,2: r X Y (6) α,3: q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  24. 24. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule (4) α,2: q X ∧ Y (5) α,2: r X Y (6) α,3: q (7) α,3: ¬r Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  25. 25. Tableaux Algorithm (PL) - Example (1):10 proof: ((q ∧ r) (¬q ∨ r)) (1) ¬((q ∧ r) (¬q ∨ r)) (2) α,1: (q ∧ r) (3) α,1: ¬(¬q ∨ r) = q ∧ ¬r α-Rule (4) α,2: q X ∧ Y (5) α,2: r X Y (6) α,3: q (7) α,3: ¬r • path is closed • tableaux is closed Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  26. 26. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) X ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  27. 27. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  28. 28. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  29. 29. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y (3|α from 1) ¬((p q) (p r)) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  30. 30. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  31. 31. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  32. 32. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  33. 33. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))11 α-Rule ¬(X Y) (1) ¬((p (q r)) ((p q) (p r))) X (2|α from 1) (p (q r)) ¬Y (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  34. 34. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (X Y) ¬X | Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  35. 35. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) ¬X | Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  36. 36. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) (10|β from 9) ¬q | (11|β from 9) r ¬X | Y Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  37. 37. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) (10|β from 9) ¬q | (11|β from 9) r ¬X | Y (12|β from 4) ¬p | (13|β from 4) q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  38. 38. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) (10|β from 9) ¬q | (11|β from 9) r ¬X | Y (12|β from 4) ¬p | (13|β from 4) q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  39. 39. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) (10|β from 9) ¬q | (11|β from 9) r ¬X | Y (12|β from 4) ¬p | (13|β from 4) q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  40. 40. Tableaux Algorithm (PL) - Example (2): proof: (p (q r)) ((p q) (p r))12 (1) ¬((p (q r)) ((p q) (p r))) (2|α from 1) (p (q r)) (3|α from 1) ¬((p q) (p r)) (4|α from 3) (p q) (5|α from 3) ¬(p r) (6|α from 5) p (7|α from 5) ¬r β-Rule (8|β from 2) ¬p | (9|β from 2) (q r) (X Y) (10|β from 9) ¬q | (11|β from 9) r ¬X | Y (12|β from 4) ¬p | (13|β from 4) q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  41. 41. Tableaux Algorithm Extensions for FOL • as for propositional logic - X and Y stand for arbitrary13 (FOL) formulas • Additional Rules for quantified formulas : γ δ γ[t] δ[c] • γ for universally quantified formulas, δ existentially quantified formulas, with: γ γ[t] δ δ[c] ∀x.Φ Φ[x←t] ∃x.Φ Φ[x←c] ¬∃x.Φ ¬Φ[x←t] ¬∀x.Φ ¬Φ[x←c] • t is an arbitrary ground term (i.e. doesn‘t contain variables that are bound in Φ), • c is a „new“ constant Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  42. 42. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] ¬∃x.Φ ¬Φ[x←t] δ δ[c] ∃x.Φ Φ[x←c] ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  43. 43. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] δ δ[c] ∃x.Φ Φ[x←c] ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  44. 44. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) δ δ[c] ∃x.Φ Φ[x←c] ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  45. 45. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] ∃x.Φ Φ[x←c] ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  46. 46. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  47. 47. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  48. 48. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  49. 49. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) (7|γ from 4) ¬P(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  50. 50. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) (7|γ from 4) ¬P(c) (8|γ from 2) P(c)∨Q(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  51. 51. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) (7|γ from 4) ¬P(c) (8|γ from 2) P(c)∨Q(c) (9|β from 8) P(c) | (10|β from 8) Q(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  52. 52. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) (7|γ from 4) ¬P(c) (8|γ from 2) P(c)∨Q(c) (9|β from 8) P(c) | (10|β from 8) Q(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  53. 53. Tableaux Algorithm (FOL) - Example(3):14 γ γ[t] Proof: (∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x)) ∀x.Φ Φ[x←t] (1) ¬((∀x.P(x)∨Q(x)) (∃x.P(x))∨(∀x.Q(x))) ¬∃x.Φ ¬Φ[x←t] (2|α from 1) (∀x.P(x)∨Q(x)) (3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x))) δ δ[c] (4|α from 3) ¬(∃x.P(x)) ∃x.Φ Φ[x←c] (5|α from 3) ¬(∀x.Q(x)) ¬∀x.Φ ¬Φ[x←c] (6|δ from 5) ¬Q(c) (7|γ from 4) ¬P(c) (8|γ from 2) P(c)∨Q(c) (9|β from 8) P(c) | (10|β from 8) Q(c) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  54. 54. 15 Lecture 5: Knowledge Representations II Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam

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