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Logical Entailment and Knowledge Representation in AI

Explore logical entailment, well-formed propositions, semantic properties, and model-finding refutation in knowledge representation. Learn how to apply these concepts in AI. Understand the domain of "How Dorothy Ended Up in Oz" with logical reasoning. Dive into the intriguing world of knowledge representation and logical thinking.

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Logical Entailment and Knowledge Representation in AI

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  1. CSE4/563: Knowledge RepresentationRecitation September 20, 2010 “In activities other than purely logical thought, our minds function much faster than any computer yet devised.” --Daniel Crevier

  2. HW1 Questions/Concerns? http://www.cse.buffalo.edu/~shapiro/Courses/CSE563/2010/Homeworks/hw1.pdf

  3. Logical Entailment • A1,…, An ╞L B

  4. Logical Entailment • A1,…, An ╞L B • Example: • A1 = Tom drives  Betty is the passenger • A2= Tom drives • A1, A2 ╞ Betty is the passenger

  5. Logical Entailment • A1,…, An ╞L B • Example: • A1 = Tom drives  Betty is the passenger • A2= Tom drives • A1, A2 ╞ Betty is the passenger • ╞ B

  6. Logical Entailment • A1,…, An ╞L B • Example: • A1 = Tom drives  Betty is the passenger • A2= Tom drives • A1, A2 ╞ Betty is the passenger • ╞ B • Example: ╞ (P or ~P)

  7. Well Formed Propositions • What is a WFP?

  8. Well Formed Propositions • Every atomic proposition is a wfp. • If P is a wfp, then so is (~P). • If P and Q are wfps, then so are • (P ^ Q) • (P v Q) • (P => Q) • (P <=> Q) • 4. Nothing else is a wfp.

  9. Semantic Properties of WFPs • Satisfiable? • Contingent? • Valid? • Unsatisfiable? • Contradictory?

  10. Model-Finding Refutation • TDB=>TD, TD=>BP, ~(TP^BP) ╞ ~(TDB ^ TP)

  11. ‘How Dorothy Ended Up In Oz’ Domain If there is a cyclone Dorothy should hide in the cellar. If Toto jumps from Dorothy’s arms and hides, she should find him. If Dorothy takes time to find Toto, or if she is too slow, she won’t make it to the cellar before the cyclone hits. Dorothy ends up in Oz if she doesn’t make it to the cellar in time. Assume: There is a cyclone and Toto jumps from Dorothy’s arms and hides. Prove: Dorothy ends up in Oz.

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