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Propositional Logic

Propositional Logic. An “adventure game” example Thinking?. PSSS. The Physical Symbol System Hypothesis: A physical symbol system has the necessary and sufficient means for general intelligent action.

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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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