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OpenHPI 4.7 - Resolution

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OpenHPI 4.7 - Resolution

  1. 1. Semantic Web TechnologiesLecture 4: Knowledge Representations I 07: Resolution 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 07 Resolution (PL)Open HPI - Course: Semantic Web Technologies - Lecture Potsdam Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität 4: Knowledge Representations I
  4. 4. A Calculator Machine for Logic304 ■ Recall: ■ A formula F is a logical consequence of a theory (knowledge base) T, i.e., T ⊨ F, iff all models of T are also models of F ■ Problem: ■ We have to consider all possible interpretations. ■ How do we do this in practice? Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  5. 5. A Calculator Machine for Logic305 ■ Problem: ■ We have to consider all possible interpretations. ■ How do we do this in practice? ■ Therefore, logical consequence is implemented via syntactical methods (= Calculus). Gottfried Wilhelm Leibniz (1646-1716) ■ For the logical calculus, there must be proven: ■ Correctness: every syntactic entailment is also a semantic entailment, if T ⊢ F then T ⊨ F ■ Completeness: all semantic entailments are also syntactic entailments, if T ⊨ F then T ⊢ F Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  6. 6. Resolution □ Resolution is based on simple deduction rules and is a special form of enumeration306 □ Instead of proofing that a formula is a tautology, it deducts a logical contradiction from its negation Theory equivalent assertions {F1,…,Fn} with F0 as logical Consequence {F1,…,Fn} ⊨ F0 F1 ∧… ∧ Fn → F0 is a tautology John Alan Robinson ¬(F1 ∧… ∧ Fn → F0) is a contradiction G1 ∧ …∧ Gk is a contradiction □ The resolution procedure allows the entailment of a contradiction from G1 ∧ …∧ Gk. John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam Communications of the ACM, 5:23–41, 1965.
  7. 7. Resolution (Propositional Logic) ■ If (p1∨…∨pk∨p∨¬q1∨…∨¬ql)∧(r1∨…∨rm∨¬p∨¬s1∨…∨¬sn)307 is true, then: ■ One of both p, ¬p has to be wrong. ■ Therefore: One of the other Literals must be true, i.e. p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn must be true. ■ Therefore: If p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn is a contradiction, then (p1∨…∨pk∨p∨¬q1∨…∨¬ql)∧(r1∨…∨rm∨¬p∨¬s1∨…∨¬sn) is also a contradiction. Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  8. 8. Resolution (Propositional Logic)308 (p1∨…∨pk∨p∨¬q1∨…∨¬ql) ∧ (r1∨…∨rm∨¬p∨¬s1∨…∨¬sn) K1 K2 Resolution step {K1,K 2} ⊨ K3 p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn K3 ■ two clauses K1 and K2 are transformed into a new one K3 ■ End of the resolution procedure: ■ If clauses are resolved that consist only of an atom and the negated atom, then a new „empty clause“ ⊥ can be resolved. Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  9. 9. Resolution (Propositional Logic)309 • How to deduce a contradiction from a set M of clauses: 1. Select two clauses from M and create a new clause K via a resolution step. 2. If K =⊥ , then a contradiction has been found. 3. If K ≠⊥ , K is added to the set M, continue with step 1. • The Resolution Calculus (for propositional logic) is correct and complete Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  10. 10. Resolution (Propositional Logic)3010 • How to proof that a formula ψ is a logical consequence of a Knowledge Base (Theory) Φ of clauses, i.e. Φ ⊨ ψ ? 1. Compute the negation of Φ → ψ, which is a contradiction: ¬(Φ → ψ) ≣ Φ ∧ ¬ ψ 2. Determine the clausal form of Φ ∧ ¬ ψ 3. Select two clauses from Φ ∧ ¬ ψ and create a new clause K via a resolution step. 4. If K =⊥ , then a contradiction has been found. It holds that Φ ⊨ ψ. q.e.d. 5. If K ≠⊥ , K is added to the set Φ ∧¬ ψ, continue with step 3. Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  11. 11. Example30 • If it rains, Jane brings her umbrella11 knowledge r ! u base • If Jane has an umbrella, she doesnt get wet Φ u ! ¬w • If it doesnt rain, Jane doesnt get wet ¬r ! ¬w • Now, prove that Jane doesnt get wet formula ¬w ψ Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  12. 12. Example 1. Transform knowledge base into clausal form30 • r ! u = ¬r ∨ u = {¬r,u}12 knowledge • u ! ¬w = ¬u ∨ ¬w = {¬u,¬w} base • ¬r ! ¬w = r ∨ ¬w = {r,¬w} Φ 2. Add negated formula to knowledge base • ¬¬w = w formula ψ 1.{¬r,u} new 2.{¬u,¬w} knowledge 3.{r,¬w} base Φ∧¬ψ 4. {w} Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  13. 13. Example 3. Start Resolution3013 1.{¬r,u} new 2.{¬u,¬w} knowledge 3.{r,¬w} base Φ∧¬ψ 4. {w} (3,4) {r,¬w} {w} 5. {r} Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  14. 14. Example 3. Continue Resolution3014 1.{¬r,u} new 2.{¬u,¬w} knowledge 3.{r,¬w} base Φ∧¬ψ 4. {w} 5. {r} (2,4) {¬u,¬w} {w} 6. {¬u} Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  15. 15. Example 3. Continue Resolution3015 1.{¬r,u} new 2.{¬u,¬w} knowledge 3.{r,¬w} base Φ∧¬ψ 4. {w} 5. {r} 6. {¬u} (1,5) {¬r,u} {r} 7. {u} Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  16. 16. Example 3. Continue Resolution3016 1.{¬r,u} new 2.{¬u,¬w} knowledge 3.{r,¬w} base Φ∧¬ψ 4. {w} 5. {r} 6. {¬u} 7. {u} (6,7) {¬u} {u} {⊥} 4. We have found a contradiction in Φ∧¬ψ, therefore it holds that Φ ⊨ ψ , q.e.d. Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  17. 17. 17 08 Resolution (FOL)Open HPI - Course: Semantic Web Technologies - Lecture Potsdam Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität 4: Knowledge Representations I

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