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Propositional Logic

Propositional Logic. Reading: C. 7.4-7.8, C. 8. Logic: Outline. Propositional Logic Inference in Propositional Logic First-order logic. 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

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Propositional Logic

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

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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

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

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

  11. 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

  12. Conversion Example

  13. Resolution of Example

  14. Inference Properties • Inference method A is sound (or truth-preserving) if it only derives entailed sentences • Inference method A is complete if it can derive any sentence that is entailed • A proof is a record of the progress of a sound inference algorithm.

  15. Other Types of Inference • Model Checking • Forward chaining with modus ponens • Backward chaining with modus ponens

  16. Model Checking • Enumerate all possible worlds • Restrict to possible worlds in which the KB is true • Check whether the goal is true in those worlds or not

  17. Wumpus Reasoning • Percepts: {nothing in 1,1; breeze in 2,1} • Assume agent has moved to [2,1] • Goal: where are the pits? • Construct the models of KB based on rules of world • Use entailment to determine knowledge about pits

  18. Constructing the KB

  19. Properties of Model Checking • Sound because it directly implements entailment • Complete because it works for any KB and sentence to prove αand always terminates • Problem: there can be way too many worlds to check • O(2n) when KB and α have n variables in total

  20. Inference as Search • State: current set of sentences • Operator: sound inference rules to derive new entailed sentences from a set of sentences • Can be goal directed if there is a particular goal sentence we have in mind • Can also try to enumerateevery entailed sentence

  21. Example

  22. Complexity • N propositions; M rules • Every possible fact can be establisehd with at most N linear passes over the database • Complexity O(NM) • Forward chaining with Modus Ponens is complete for Horn logic

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