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Propositional Logic

Delve into the intricacies of propositional logic via an engaging adventure game scenario in the Lion World. This exploration highlights key elements of logical reasoning, knowledge representation, and inference rules needed for intelligent action. From understanding the physical symbol system hypothesis to addressing limitations of conventional logic, this adventure exemplifies the need for a robust, expressive language. Engage with real-time decision-making as you navigate obstacles represented by percepts and actions, illustrating the functional application of logic in dynamic environments.

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Propositional Logic

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  1. Propositional Logic • An “adventure game” example • Thinking? LOGIC

  2. PSSS • The Physical Symbol System Hypothesis: A physical symbol system has the necessary and sufficient means for general intelligent action. Where a symbol is a designating pattern that can be combined with others to form another designating pattern. LOGIC

  3. Knowledge Representation • Key is problem formulation – • What happens when an n-dimensional array is insufficient? • Need a language that is • Expressive and concise • Unambiguous and independent of context • Has an inference procedure for new sentences LOGIC

  4. Inference Rules • And Elimination 1 2,  ...  n 1 • And Introduction 1, . . ., n 1 2,  ...  n LOGIC

  5. Inference Rules (cont’d) • Or Introduction i 1 2,  ...  i …  n • Double Negation Elimination   LOGIC

  6. Inference Rules (cont’d) • Modus Ponens (Implication Elimination) ,  (Chaining) ,   LOGIC

  7. Inference Rules (cont’d) • Unit Resolution:,   (cf.Modus Ponens) • Resolution:,    is true or false. If  is true,  is true. If  is false,  is true. LOGIC

  8. The Lion World Percepts: [Stench, Breeze, Glitter, Bump, Scream] Operators: [Right 90, Left 90, Forward, Grab, Shoot,Climb] LOGIC

  9. The Lion World (1,1) [none,none,none,none,none] ok A ok ok LOGIC

  10. The Lion World (1,1) [none,none,none,none,none] (2,1) [none,breeze,none,none,none] ok P? A ok B P? ok LOGIC

  11. The Lion World (1,1) [none,none,none,none,none] (2,1) [none,breeze,none,none,none] L? A (1,2) [stench,none,none,none,none] ok ok P? ok ok B P? LOGIC

  12. The Lion World (1,1) [none,none,none,none,none] (2,1) [none,breeze,none,none,none] (1,2) [stench,none,none,none,none] A L? ok ok (2,2) [none,none,none,none,none] (2,3)[Stench,none,Glitter,none,none] ok ok B P? LOGIC

  13. The Lion World • The Knowledge Base ¬ S1,1 , ¬ B1,1 P3,1 , B4,1 ¬ S2,1 , B2,1 ¬ S3,2 , B3,2 S1,2 , ¬ B1,2 ¬ S2,2 , ¬ B2,2 ¬ S3,3 , ¬ B3,3 Gl 1,4 ,¬ S1,4 , ¬ B1,4 ¬ S2,4 , ¬ B2,4 ,G 2.4 B3,4 , Gl 3,4 ¬ S1,3 , ¬ B1,3 , L 1,3 B4,3 S 2.3 , ¬ B 2.3 , Gl 2.3 P4,4 LOGIC

  14. Lion World Implications R1 : ¬ S1,1 → ¬ L1,2 /\ ¬ L2,1 R2 : ¬ S2,1 → ¬ L1,1 /\¬ L2,2 /\ ¬ L3,1 R3 : ¬ S1,2 → ¬ L1,1 /\ ¬ L2,2 /\¬ L1,3 R4 : S1,2 → L1,1 \/ L2,2 \/ L1,3 LOGIC

  15. Lion World Implications transformed into Conjunctive Normal Form (R1-R3) R1 : ¬ S1,1 → ¬ L1,2 /\ ¬ L2,1 R1 : ¬ ¬ S1,1 \/ (¬ L1,2 /\ ¬ L2,1) R1 : S1,1 \/ (¬ L1,2 /\ ¬ L2,1) R1: (S1,1 \/ ¬ L1,2 )/\ (S1,1 \/ ¬ L2,1 ) R1: (S1,1 \/ ¬ L1,2 ), (S1,1 \/ ¬ L2,1 ) LOGIC

  16. Lion World Implications transformed into Conjunctive Normal Form – R4 R4 : S1,2 → L1,1 \/ L2,2 \/ L1,3 R4 : ¬ S1,2 \/ (L1,1 \/ L2,2 \/ L1,3) R4: ¬ S1,2 \/ L1,1 \/ L2,2 \/ L1,3 LOGIC

  17. The Lion World (1,1) [none,none,none,none,none] ok A ok ok LOGIC

  18. Finding the Lion ¬ S1,1 , S1,1 \/ ¬L1,2 Unit Resolution ¬ L1,2 ¬ S1,1 , S1,1 \/ ¬L2,1 Unit Resolution ¬ L2,1 LOGIC

  19. The Lion World (1,2) [stench,none,none,none,none] L? A ok P? ok ok B P? LOGIC

  20. Finding the Lion S1,2 , ¬ S1,2 \/ L1,1 \/ L2,2 \/ L1,3 L1,1 \/ L2,2 \/ L1,3Unit Resolution L1,1 \/ L2,2 \/ L1,3 ,¬ L1,1 L2,2 \/ L1,3 Unit Resolution LOGIC

  21. Finding the Lion • How do we know ¬ L2,2 ? • L2,2 \/ L1,3 ,¬ L2,2 • L1,3 Unit Resolution LOGIC

  22. Avoiding the Lion • Don’t go forward if the lion is in front – A1,2 /\ NorthA /\ L1,3  ¬Forward • 64 rules (16 squares x 4 orientations) LOGIC

  23. Avoiding the Lion in the next move After the Agent moves, A1,2 is no longer true, now A2,3 is true. A2,3 /\ WestA /\ L1,3  ¬Forward LOGIC

  24. Limitations of Propositional Logic • Can’t express generalities • Need new propositions for each time stamp LOGIC

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