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Propositional Logic

Propositional Logic. Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4. sensors. environment. ?. agent. actuators. Knowledge base. Knowledge-Based Agent. Types of Knowledge. Procedural , e.g.: functions Such knowledge can only be used in one way -- by executing it

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Propositional Logic

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  1. Propositional Logic Russell and Norvig: Chapter 6Chapter 7, Sections 7.1—7.4

  2. sensors environment ? agent actuators Knowledge base Knowledge-Based Agent Propositional Logic

  3. Types of Knowledge • Procedural, e.g.: functionsSuch knowledge can only be used in one way -- by executing it • Declarative, e.g.: constraintsIt can be used to perform many different sorts of inferences Propositional Logic

  4. Logic Logic is a declarative language to: • Assert sentences representing facts that hold in a world W (these sentences are given the value true) • Deduce the true/false values to sentences representing other aspects of W Propositional Logic

  5. entail Sentences Sentences represent represent Conceptualization World W Facts about W Facts about W hold hold Connection World-Representation Propositional Logic

  6. Examples of Logics • Propositional calculusA  B  C • First-order predicate calculus( x)( y) Mother(y,x) • Logic of BeliefB(John,Father(Zeus,Cronus)) Propositional Logic

  7. Symbols of PL • Connectives: , , ,  • Propositional symbols, e.g., P, Q, R, … • True, False Propositional Logic

  8. Syntax of PL • sentence  atomic sentence | complex sentence • atomic sentence  Propositional symbol, True, False • Complex sentence  sentence | (sentence  sentence) | (sentence  sentence) | (sentence  sentence) Propositional Logic

  9. Syntax of PL • sentence  atomic sentence | complex sentence • atomic sentence  Propositional symbol, True, False • Complex sentence  sentence | (sentence  sentence) | (sentence  sentence) | (sentence  sentence) • Examples: • ((P  Q)  R) • (A  B)  (C) Propositional Logic

  10. Order of Precedence •  • Examples: •  A  B  C is equivalent to ((A)B)C • A B C is incorrect Propositional Logic

  11. Model • Assignment of a truth value – true or false – to every atomic sentence • Examples: • Let A, B, C, and D be the propositional symbols • m = {A=true, B=false, C=false, D=true} is a model • m’ = {A=true, B=false, C=false} is not a model • With n propositional symbols, one can define 2n models Propositional Logic

  12. What Worlds Does a Model Represent? A model represents any world in which a fact represented by a proposition A having the value True holds and a fact represented by a proposition B having the value False does not hold A model represents infinitely many worlds Propositional Logic

  13. BLOCK(A), BLOCK(B), BLOCK(C) ON(A,B), ON(B,C), ONTABLE(C) ON(A,B)  ON(B,C)  ABOVE(A,C) HUMAN(A), HUMAN(B), HUMAN(C) CHILD(A,B), CHILD(B,C), BLOND(C) CHILD(A,B)  CHILD(B,C)  GRAND-CHILD(A,C)  GRAND-CHILD(A,C) Compare!  ABOVE(A,C) Propositional Logic

  14. Semantics of PL • It specifies how to determine the truth value of any sentence in a model m • The truth value of True is True • The truth value of False is False • The truth value of each atomic sentence is given by m • The truth value of every other sentence is obtained recursively by using truth tables Propositional Logic

  15. Truth Tables Propositional Logic

  16. Truth Tables Propositional Logic

  17. Truth Tables Propositional Logic

  18. About  • ODD(5)  CAPITAL(Japan,Tokyo) • EVEN(5)  SMART(Sam) • Read A  B as:“If A IS True, then I claim that B is True, otherwise I make no claim.” Propositional Logic

  19. Example Model: A=True, B=False, C=False, D=True (A B  C)  D  A F F T T T Definition: If a sentence s is true in a model m, then m is said to be a model of s Propositional Logic

  20. A Small Knowledge Base • Battery-OK  Bulbs-OK  Headlights-Work • Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts • Engine-Starts Flat-Tire  Car-OK • Headlights-Work • Car-OK Sentences 1, 2, and 3  Background knowledge Sentences 4 and 5  Observed knowledge Propositional Logic

  21. Model of a KB • Let KB be a set of sentences • A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m Propositional Logic

  22. valid sentenceor tautology Satisfiability of a KB A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable KB1 = {P, QR} is satisfiableKB2 = {PP} is satisfiable KB3 = {P, P} is unsatisfiable Propositional Logic

  23. 3-SAT Problem • n propositional symbols P1,…,Pn • KB consists of p sentences of the form Qi  Qj  Qkwhere: • i  j  k are indices in {1,…,n} • Qi = Pi or Pi • 3-SAT: Is KB satisfiable? • 3-SAT is NP-complete Propositional Logic

  24. Logical Entailment • KB : set of sentences •  : arbitrary sentence • KB entails – written KB  – iff every model of KB is also a model of  Propositional Logic

  25. Logical Entailment • KB : set of sentences •  : arbitrary sentence • KB entails – written KB  – iff every model of KB is also a model of  • Alternatively, KB  iff • {KB,} is unsatisfiable • KB   is valid Propositional Logic

  26. Logical Equivalence • Two sentences  and  are logically equivalent – written    -- iff they have the same models, i.e.:   iff   and   Propositional Logic

  27. Logical Equivalence • Two sentences  and  are logically equivalent – written    -- iff they have the same models, i.e.:   iff   and   • Examples: • (  ) (  ) •    • ()    • ()   Propositional Logic

  28. Logical Equivalence • Two sentences  and  are logically equivalent – written    -- iff they have the same models, i.e.:   iff   and   • Examples: • (  ) (  ) •    • ()    • ()   • One can always replace a sentence by an equivalent one in a KB Propositional Logic

  29. Inference Rule • An inference rule {, } consists of 2 sentence patterns  and  called the conditions and one sentence pattern  called the conclusion  Propositional Logic

  30. Inference Rule • An inference rule {, } consists of 2 sentence patterns  and  called the conditions and one sentence pattern  called the conclusion • If  and  match two sentences of KB then the corresponding  can be inferred according to the rule  Propositional Logic

  31. {  , }   {, }  Example: Modus Ponens Propositional Logic

  32. {  , }   {, }  Example: Modus Ponens Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Battery-OK  Bulbs-OK Propositional Logic

  33. {  , }   {, }  Example: Modus Ponens Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Battery-OK  Bulbs-OK Propositional Logic

  34. {  , }   {, }  Example: Modus Ponens Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Battery-OK  Bulbs-OK Propositional Logic

  35. {  , }   {, }  Example: Modus Ponens Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Battery-OK  Bulbs-OK Headlights-Work Propositional Logic

  36. {  ,  }   Example: Modus Tolens Engine-Starts Flat-Tire  Car-OK Car-OK Propositional Logic

  37. {  ,  }   Example: Modus Tolens Engine-Starts Flat-Tire  Car-OKCar-OK (Engine-Starts Flat-Tire) Propositional Logic

  38. {  ,  }   Example: Modus Tolens Engine-Starts Flat-Tire  Car-OKCar-OK (Engine-Starts Flat-Tire)  Engine-Starts  Flat-Tire Propositional Logic

  39. Other Examples • {,}    • {,.}  • {,.}  • Etc …    Propositional Logic

  40. Inference • I: Set of inference rules • KB: Set of sentences • Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB Propositional Logic

  41. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Propositional Logic

  42. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Propositional Logic

  43. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Propositional Logic

  44. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Propositional Logic

  45. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Engine-Starts  Flat-Tire  (3+8) Propositional Logic

  46. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Engine-Starts  Flat-Tire  (3+8)  Engine-Starts  Flat-Tire Propositional Logic

  47. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Engine-Starts  Flat-Tire  (3+8) Propositional Logic

  48. Example Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Engine-Starts  Flat-Tire  (3+8) Flat-Tire  (11+12) Propositional Logic

  49. Soundness • An inference rule is sound if it generates only entailed sentences Propositional Logic

  50. Soundness • An inference rule is sound if it generates only entailed sentences • All inference rules previously given are sound, e.g.:modus ponens: {   , }   Propositional Logic

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