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Propositional Logic

Propositional Logic. Reading: C. 7.4-7.8, C. 8. Announcements. Read discussion board frequently Questions answered New posts of client-server Today: version posted with improved IO on display and timing Mid-term evaluation on courseworks Complete by next Tuesday (1 week)

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Propositional Logic

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  1. Propositional Logic Reading: C. 7.4-7.8, C. 8

  2. Announcements • Read discussion board frequently • Questions answered • New posts of client-server • Today: version posted with improved IO on display and timing • Mid-term evaluation on courseworks • Complete by next Tuesday (1 week) • Written homework • Do not do “predicate logic” on problem 10.5 (will be part of next assignment) • Should read “but use semantic networks and KL-one type ” .. Do not extend the representation itself. • Note that section 10.6 covers semantic networks and description logics (another name for KL-one type)

  3. Logic: Outline • Propositional Logic • Inference in Propositional Logic • First-order logic • Inference in FOL

  4. Agents that reason logically • A logic is a: • Formal language in which knowledge can be expressed • A means of carrying out reasoning in the language • A Knowledge base agent • Tell: add facts to the KB • Ask: query the KB

  5. Towards General-Purpose AI • Problem-specific AI (e.g., Roomba) • Specific data structure • Need special implementation • Can be fast • General –purpose AI (e.g., logic-based) • Flexible and expressive • Generic implementation possible • Can be slow

  6. Language Examples • Programming languages • Formal, not ambiguous • Lacks expressivity (e.g., partial information) • Natural Language • Very expressive, but ambiguous: • Flying planes can be dangerous. • The teacher gave the boys an apple. • Inference possible, but hard to automate • Good representation language • Both formal and can express partial information • Can accommodate inference

  7. Components of a Formal Logic • Syntax: symbols and rules for combining themWhat you can say • Semantics: Specification of the way symbols (and sentences) relate to the worldWhat it means • Inference Procedures: Rules for deriving new sentences (and therefore, new semantics) from existing sentencesReasoning

  8. Semantics • A possible world (also called a model) is an assignment of truth values to each propositional symbol • The semantics of a logic defines the truth of each sentence with respect to each possible world • A model of a sentence is an interpretation in which the sentence evaluates to True • E.g., TodayIsTuesday -> ClassAI is true in model {TodayIsTuesday=True, ClassAI=True} • We say {TodayIsTuesday=True, ClassAI=True} is a model of the sentence

  9. Exercise: Semantics What is the meaning of these two sentences? • If Shakespeare ate Crunchy-Wunchies for breakfast, then Sally will go to Harvard • If Shakespeare ate Cocoa-Puffs for breakfast, then Sally will go to Columbia

  10. Examples • What are the models of the following sentences? • KB1: TodayIsTuesday -> ClassAI • KB2: TodayIsTuesday -> ClassAI, TodayIsTuesday

  11. Proof by refutation • A complete inference procedure • A single inference rule, resolution • A conjunctive normal form for the logic

  12. Example: Wumpus World • Agent in [1,1] has no breeze • KB = R2Λ R4 = (B1,1<->(P1,2) V P2,1)) Λ⌐B1,1 • Goal: show ⌐P1,2

  13. Conversion Example

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