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Chapter 5

Chapter 5. Knowledge Representation & Reasoning (Part 2) Propositional Logic. Knowledge Representation & Reasoning . Propositional Logic: Inference rules An inference rule is sound if the conclusion is true in all cases where the premises are true.  Premise _____  Conclusion.

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Chapter 5

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  1. Chapter 5 Knowledge Representation & Reasoning (Part 2)Propositional Logic

  2. Knowledge Representation & Reasoning Propositional Logic: Inference rules An inference rule is sound if the conclusion is true in all cases where the premises are true.  Premise _____  Conclusion

  3. Knowledge Representation & Reasoning Propositional Logic: An inference rule: Modus Ponens • From an implication and the premise of the implication, you can infer the conclusion.     Premise ___________  Conclusion Example: “raining implies soggy courts”, “raining” Infer: “soggy courts”

  4. Knowledge Representation & Reasoning Propositional Logic: An inference rule:Modus Tollens • From an implication and the premise of the implication, you can infer the conclusion.    ¬  Premise ___________ ¬  Conclusion Example: “raining implies soggy courts”, “courts not soggy” Infer: “not raining”

  5. Knowledge Representation & Reasoning Propositional Logic: Equivalence rules • Two sentences are logically equivalent iff they are true in the same models: α ≡ ß iff α╞ β and β╞ α.

  6. Knowledge Representation & Reasoning Resolution Unit Resolution inference rule: l1 … lk, m l1 … li-1li+1 … lk where liand m are complementary literals.

  7. Knowledge Representation & Reasoning Resolution Full resolution inference rule: l1… lk, m1 … mn l1… li-1li+1…lkm1…mj-1mj+1...mn where liand m are complementary literals.

  8. Knowledge Representation & Reasoning Resolution For simplicity let’s consider clauses of length two: l1l2, ¬l2l3 l1l3 To derive the soundness of resolutionconsider the values l2cantake: • If l2isTrue, thensincewe know that ¬l2l3holds, it must be the case thatl3 isTrue. • If l2isFalse, thensincewe know thatl1l2holds, it must be the case thatl1isTrue.

  9. Knowledge Representation & Reasoning Resolution 1. Properties of the resolution rule: • Sound • Complete (yields to a complete inference algorithm). 2. The resolution rule forms the basis for a family of complete inference algorithms. 3. Resolution rule is used to either confirm or refute a sentence but it cannot be used to enumerate true sentences.

  10. Knowledge Representation & Reasoning Resolution 4. Resolution can be applied only to disjunctions of literals. How can it lead to a complete inference procedure for all propositional logic? 5. Turns out any knowledge base can be expressed as a conjunction of disjunctions (conjunctive normal form, CNF). E.g., (A  ¬B)  (B  ¬C  ¬D)

  11. Knowledge Representation & Reasoning Resolution: Inference procedure 6. Inference procedures based on resolution work by using the principle of proof by contradiction: To show that KB ╞ αwe show that (KB ¬α) isunsatisfiable The process: 1. convertKB ¬α to CNF 2. resolutionruleisapplied to the resulting clauses.

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