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Logical Agents

Logical Agents. ECE457 Applied Artificial Intelligence Spring 2008 Lecture #6. Outline. Logical reasoning Propositional Logic Wumpus World Inference Russell & Norvig, chapter 7. Logical Reasoning. Recall: Game-playing with imperfect information Partially-observable environment

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Logical Agents

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  1. Logical Agents ECE457 Applied Artificial Intelligence Spring 2008 Lecture #6

  2. Outline • Logical reasoning • Propositional Logic • Wumpus World • Inference • Russell & Norvig, chapter 7 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 2

  3. Logical Reasoning • Recall: Game-playing with imperfect information • Partially-observable environment • Need to infer about hidden information • Two new challenges • How to represent the information we have (knowledge representation) • How to use the information we have to infer new information and make decisions (knowledge reasoning) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 3

  4. Knowledge Representation • Represent facts about the environment • Many ways: ontologies, mathematical functions, … • Statements that are either true or false • Language • To write the statements • Syntax: symbols (words) and rules to combine them (grammar) • Semantics: meaning of the statements • Expressiveness vs. efficiency • Knowledge base (KB) • Contains all the statements • Agent can TELL it new statements (update) • Agent can ASK it for information (query) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 4

  5. Knowledge Representation • Example: Language of arithmetic • Syntax describes well-formed formulas (WFF) • X + Y > 7 (WFF) • X 7 @ Y + (not a WFF) • Semantics describes meanings of formulas • “X + Y > 7” is true if and only if the value of X and the value of Y summed together is greater than 7 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 5

  6. Knowledge Reasoning • Inference • Discovering new facts and drawing conclusions based on existing information • During ASK or TELL • “All humans are mortal” “Socrates is human” • Entailment • A sentence  is inferred from sentences  •  is true given that the  are true •  entails  •    ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 6

  7. Propositional Logic • Sometimes called “Boolean Logic” • Sentences are true (T) or false (F) • Words of the syntax include propositional symbols… • P, Q, R, … • P = “I’m hungry”, Q = “I have money”, R = “I’m going to a restaurant” • … and logical connectives • ¬ negation NOT •  conjunction AND •  disjunction OR •  implication IF-THEN •  biconditional IF AND ONLY IF ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 7

  8. Propositional Logic • Atomic sentences • Propositional symbols • True or false • Complex sentences • Groups of propositional symbols joined with connectives, and parenthesis if needed • (P  Q)  R • Well-formed formulas following grammar rules of the syntax ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 8

  9. Propositional Logic • Complex sentences evaluate to true or false • Using truth tables • Semantics ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 9

  10. Propositional Logic Semantics • Truth tables for all connectives • Given each possible truth value of each propositional symbol, we can get the possible truth values of the expression ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 10

  11. Propositional Logic Example • Propositional symbols: • A = “The car has gas” • B = “I can go to the store” • C = “I have money” • D = “I can buy food” • E = “The sun is shining” • F = “I have an umbrella” • G = “I can go on a picnic” • If the car has gas, then I can go to the store • A  B • I can buy food if I can go to the store and I have money • (B  C)  D • If I can buy food and either the sun is not shining or I have an umbrella, I can go on a picnic • (D  (¬E  F))  G ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 11

  12. ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 12

  13. 4 3 2 1 1 2 3 4 Wumpus World • 2D cave dividedin rooms • Gold • Glitters • Agent has to pick it up • Pits • Agent falls in and dies • Agent feels breeze near pit • Wumpus • Agent gets eaten and dies if Wumpus alive • Agent can kill Wumpus with arrow (will hear scream) • Agent smells stench near Wumpus (alive or dead) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 13

  14. 4 3 2 1 1 2 3 4 Wumpus World • Initial state: • (1,1) • Goal: • Get the gold and get back to (1,1) • Actions: • Turn 90°, move forward, shoot arrow, pick up gold • Cost: • +1000 for getting gold, -1000 for dying, -1 per action, -10 for shooting the arrow ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 14

  15. 4 3 2 1 1 2 3 4 Exploring the Wumpus World Pit? Pit? Wumpus? Pit? OK Pit? OK OK Pit? Wumpus? OK OK Pit? ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 15

  16. Wumpus World Logic • Propositional symbols • Pi,j = “there is a pit at (i,j)” • Bi,j = “there is a breeze at (i,j)” • Si,j = “there is a stench at (i,j)” • Wi,j = “there is a Wumpus at (i,j)” • Ki,j = “(i,j) is ok” • Rules • Bi,j  (Pi+1,j  Pi-1,j  Pi,j+1  Pi,j-1) • Si,j  (Wi+1,j  Wi-1,j  Wi,j+1  Wi,j-1) • Ki,j  (¬Wi,j  ¬Pi,j) • Have to be written out for every (i,j) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 16

  17. 4 3 2 1 1 2 3 4 Wumpus World KB • K1,1 • ¬B1,1 • ¬S1,1 • B1,1  (P2,1  P1,2) • S1,1  (W2,1  W1,2) • K2,1(¬W2,1¬P2,1) • K1,2(¬W1,2¬P1,2) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 17

  18. Wumpus World Inference 1. K1,1 3. ¬S1,1 2. ¬B1,1 1. K1,1 3. ¬S1,15. ¬P2,1 2. ¬B1,1 4. ¬P1,2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 18

  19. Wumpus World Inference 1. K1,1 3. ¬S1,15. ¬P2,1 7. ¬W2,1 2. ¬B1,1 4. ¬P1,2 6. ¬W1,2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 19

  20. Wumpus World Inference 1. K1,1 3. ¬S1,15. ¬P2,1 7. ¬W2,1 2. ¬B1,1 4. ¬P1,2 6. ¬W1,2 1. K1,1 3. ¬S1,15. ¬P2,1 7. ¬W2,1 9. K2,1 2. ¬B1,1 4. ¬P1,2 6. ¬W1,2 8. K1,2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 20

  21. 4 3 2 1 1 2 3 4 Wumpus World KB • K1,1 • ¬B1,1 • ¬S1,1 • ¬P1,2 • ¬P2,1 • ¬W1,2 • ¬W2,1 • K1,2 • K2,1 • B2,1 • P3,1 • ¬S2,1 • ¬W2,2 • ¬W3,1 • ¬B1,2 • ¬P1,3 • ¬P2,2 • S1,2 • W1,3 • K2,2 • B2,1 • P2,2  P3,1 • ¬S2,1 • ¬W2,2 • ¬W3,1 Wumpus? Pit? OK OK Wumpus? • ¬B1,2 • ¬P1,3 • ¬P2,2 • S1,2 • W1,3  W2,2 OK Pit? ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 21

  22. Inference with Truth Tables • Sound • Only infers true conclusions from true premises • Complete • Finds all facts entailed by KB • Time complexity = O(2n) • Checks all truth values of all symbols • Space complexity = O(n) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 22

  23. Inference with Rules • Speed up inference by using inference rules • Use along with logical equivalences • No need to enumerate and evaluate every truth value ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 23

  24. Rules and Equivalences • Inference rules • (α  β), αβ • (α  β) α • α, β(αβ) • (α  β), ¬βα • (αβ), (¬βγ)(α  γ) • Logical equivalences • (α  β)  (β α) • (α  β)  (β α) • ((α  β) γ) (α (βγ)) • ((α  β) γ) (α (βγ)) • ¬(¬α)  α • (α  β) (¬β ¬α) • (α  β) (¬α β) • (α  β) ((α  β) (β α)) • ¬(α  β) (¬α ¬β) • ¬(α  β) (¬α ¬β) • (α (βγ)) ((α  β)  (α  γ)) • (α (βγ)) ((α  β)  (α  γ)) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 24

  25. Wumpus World & Inference Rules KB: ¬B1,1 • B1,1  (P2,1  P1,2) • Biconditional elimination • (B1,1  (P2,1  P1,2))  ((P2,1  P1,2)  B1,1) • And elimination • (P2,1  P1,2)  B1,1 • Contraposition • ¬B1,1  ¬(P2,1  P1,2) • Modus Ponens • ¬(P2,1  P1,2) • De Morgan’s Rule • ¬P2,1¬P1,2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 25

  26. Resolution • Inference with rules is sound, but only complete if we have all the rules • Resolution rule is both sound and complete • (αβ), (¬βγ)(α  γ) • But it only works on disjunctions! • Conjunctive normal form (CNF) • Eliminate biconditionals: (αβ) ((αβ)(βα)) • Eliminate implications: (α  β) (¬α β) • Move/Eliminate negations: ¬(¬α)  α, ¬(α  β) (¬α ¬β), ¬(α  β) (¬α ¬β) • Distribute  over : (α (βγ)) ((αβ)  (αγ)) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 26

  27. CNF Example • B1,1  (P2,1  P1,2) • Eliminate biconditionals • (B1,1  (P2,1  P1,2))  ((P2,1  P1,2)  B1,1) • Eliminate implications • (¬B1,1P2,1  P1,2)  (¬(P2,1  P1,2)B1,1) • Move/Eliminate negations • (¬B1,1P2,1  P1,2)  ((¬P2,1¬P1,2)B1,1) • Distribute  over  • (¬B1,1P2,1  P1,2)  (¬P2,1 B1,1)(¬P1,2B1,1) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 27

  28. Resolution Algorithm • Given a KB • Need to answer a query α • KB  α ? • Proof by contradiction • Show that (KB  ¬α) is unsatisfiable • i.e. leads to a contradiction • If (KB  ¬α) is false, then (KB  α) must be true ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 28

  29. Resolution Algorithm • Convert (KB  ¬α) into CNF • For every pair of clauses that contain complementary symbols • Apply resolution to generate a new clause • Add new clause to KB • End when • Resolution gives the empty clause (KB  α) • No new clauses can be added (fail) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 29

  30. Wumpus World & Resolution • (¬B1,1P1,2  P2,1)  (¬P1,2  B1,1)  (¬P2,1  B1,1) • CNF form of B1,1  (P2,1  P1,2) • ¬B1,1 • Query: ¬P1,2 (¬B1,1 P1,2  P2,1)  (¬P2,1  B1,1)  (¬P1,2  B1,1)  ¬B1,1  P1,2 (¬B1,1 P1,2  P2,1)  (¬P2,1  B1,1)  B1,1  ¬B1,1  P1,2 (¬B1,1 P1,2  P2,1)  (¬P2,1  B1,1)  Empty clause!  P1,2 KB  ¬P1,2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 30

  31. Resolution Algorithm • Sound • Complete • Not efficient ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 31

  32. Horn Clauses • Resolution algorithm can be further improved by using Horn clauses • Disjunction clause with at most one positive symbol • ¬α ¬βγ • Can be rewritten as implication • (α  β)γ • Inference in linear time! • Using Modus Ponens • Forward or backward chaining ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 32

  33. Forward Chaining • Data-driven reasoning • Start with known symbols • Infer new symbols and add to KB • Use new symbols to infer more new symbols • Repeat until query proven or no new symbols can be inferred • Work forward from known data, towards proving goal • KB: α, β, δ, ε • (α  β)γ • (δε)λ • (λ γ) q ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 33

  34. Backward Chaining • Goal-driven reasoning • Start with query, try to infer it • If there are unknown symbols in the premise of the query, infer them first • If there are unknown symbols in the premise of these symbols, infer those first • Repeat until query proven or its premise cannot be inferred • Work backwards from goal, to prove needed information • KB: α, β, δ, ε • (λ γ) q • (δε)λ • (α  β)γ ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 34

  35. Forward vs. Backward • Forward chaining • Proves everything • Goes to work as soon as new information is available • Expands the KB a lot • Improves understanding of the world • Typically used for proving a world model • Backward chaining • Proves only what is needed for the goal • Does nothing until a query is asked • Expands the KB as little as needed • More efficient • Typically used for proofs by contradiction ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 35

  36. Assumptions • Utility-based agent • Environment • Fully observable / Partially observable (approximation) • Deterministic / Strategic / Stochastic • Sequential • Static / Semi-dynamic • Discrete / Continuous • Single agent / Multi-agent ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 36

  37. Assumptions Updated • Learning agent • Environment • Fully observable / Partially observable • Deterministic / Strategic / Stochastic • Sequential • Static / Semi-dynamic • Discrete / Continuous • Single agent / Multi-agent ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 37

  38. Exercise • If the unicorn is mythical, then it is immortal, but if it is not mythical then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned. • Is the unicorn • Magical? • Horned? • Mythical? ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 38

  39. Exercise: CNF • Propositional symbols • Mythical = “The unicorn is mythical” • Immortal = “The unicorn is immortal” • Mammal = “The unicorn is a mammal” • Horned = “The unicorn is horned” • Magical = “The unicorn is magical” • If the unicorn is mythical, then it is immortal • Mythical  Immortal • ¬Mythical  Immortal ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 39

  40. Exercise: CNF • Propositional symbols • Mythical = “The unicorn is mythical” • Immortal = “The unicorn is immortal” • Mammal = “The unicorn is a mammal” • Horned = “The unicorn is horned” • Magical = “The unicorn is magical” • If it is not mythical then it is a mortal mammal • ¬Mythical  (¬Immortal  Mammal) • Mythical  (¬Immortal  Mammal) • (Mythical  ¬Immortal)  (Mythical  Mammal) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 40

  41. Exercise: CNF • Propositional symbols • Mythical = “The unicorn is mythical” • Immortal = “The unicorn is immortal” • Mammal = “The unicorn is a mammal” • Horned = “The unicorn is horned” • Magical = “The unicorn is magical” • If the unicorn is either immortal or a mammal, then it is horned • (Immortal  Mammal)  Horned • ¬(Immortal  Mammal)  Horned • (¬Immortal  ¬Mammal)  Horned • (¬Immortal  Horned)  (¬Mammal  Horned) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 41

  42. Exercise: CNF • Propositional symbols • Mythical = “The unicorn is mythical” • Immortal = “The unicorn is immortal” • Mammal = “The unicorn is a mammal” • Horned = “The unicorn is horned” • Magical = “The unicorn is magical” • The unicorn is magical if it is horned • Horned  Magical • ¬Horned  Magical ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 42

  43. Exercise: KB, Queries • KB • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) • Negation of queries • ¬Magical • ¬Horned • ¬Mythical ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 43

  44. Exercise: Resolution, ¬Magical • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) ¬Magical • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) ¬Magical • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned) (¬Mammal  Horned)  ¬Horned  ¬Magical • (¬Mythical  Immortal) (Mythical  ¬Immortal)  (Mythical  Mammal) ¬Immortal  ¬Mammal ¬Horned ¬Magical • ¬Mythical  (Mythical  ¬Immortal)  Mythical  ¬Immortal  ¬Mammal  ¬Horned  ¬Magical ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 44

  45. Exercise: Resolution, ¬Horned • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) ¬Horned • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned) (¬Mammal  Horned)  (¬Horned  Magical) ¬Horned • (¬Mythical  Immortal) (Mythical  ¬Immortal)  (Mythical  Mammal) ¬Immortal  ¬Mammal (¬Horned  Magical)  ¬Horned • ¬Mythical  (Mythical  ¬Immortal)  Mythical  ¬Immortal  ¬Mammal  (¬Horned  Magical)  ¬Horned ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 45

  46. Exercise: Resolution, ¬Mythical • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) ¬Mythical • (¬Mythical  Immortal)  (Mythical  ¬Immortal) (Mythical  Mammal) (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) ¬Mythical • (¬Mythical  Immortal) ¬Immortal Mammal (¬Immortal  Horned)  (¬Mammal  Horned) (¬Horned  Magical) ¬Mythical • ¬Mythical  ¬Immortal  Mammal  (¬Immortal  Horned)  Horned (¬Horned  Magical)¬Mythical • ¬Mythical  ¬Immortal  Mammal  (¬Immortal  Horned)  Horned  Magical ¬Mythical ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 46

  47. Exercise: Resolution, Mythical • (¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  Mythical • (¬Mythical  Immortal) (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical) Mythical • Immortal (Mythical  ¬Immortal) (Mythical  Mammal)  (¬Immortal  Horned) (¬Mammal  Horned)  (¬Horned  Magical)  Mythical • Immortal  Mythical  (Mythical  Mammal)  Horned (¬Mammal  Horned)  (¬Horned  Magical) Mythical • Immortal  Mythical  (Mythical  Mammal)  Horned  (¬Mammal  Horned)  Magical  Mythical ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 47

  48. Exercise: Note • Previous two examples • (KB  ¬Mythical)  (Horned  Magical) • (KB  Mythical)  (Horned  Magical) • Therefore • KB  (Horned  Magical) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 48

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