CS 4700: Artificial Intelligence

Transcription

1 CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 8

2 Today Informed Search (R&N Ch 3,4) Adversarial search (R&N Ch 5) Adversarial Search (R&N Ch 5) Homework due Tuesday, February 28

3 Office Hours This week: Friday (tomorrow) 1:30-2:30pm Starting next week: Tuesdays 1:30-2:30pm

4 Homework 1 Clarifications 1: Brief definition of intelligence, like a dictionary 2b, 3b: Draw the search space Means all states and operators Does not depend on the search method used on it. 3: You can empty the jugs. 5d: Was h 1 s = a g(s) + b h(s) Should have been f(s) = a g(s) + b h(s)

5 Iterative Deepening Search A Initial State B C D Goal

6 Iterative Deepening Search Depth Bound = 1 A DFS with depth bound = 1 B C D

7 Iterative Deepening Search Depth Bound = 1 A 1 DFS with depth bound = 1 B C D

8 Iterative Deepening Search Depth Bound = 2 A 1 B C DFS with depth bound = 2 D

9 Iterative Deepening Search Depth Bound = 2 A 1,2 B C DFS with depth bound = 2 D

10 Iterative Deepening Search Depth Bound = 2 A 1,2 3 B C DFS with depth bound = 2 D

11 Iterative Deepening Search Depth Bound = 2 A 1,2 3 B C 4 DFS with depth bound = 2 D

12 Iterative Deepening Search Depth Bound = 3 A 1,2 3 B C 4 D DFS with depth bound = 3

13 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3 B C 4 D DFS with depth bound = 3

14 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3,6 B C 4 D DFS with depth bound = 3

15 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3,6 B C 4,7 D DFS with depth bound = 3

16 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3,6 B C 4,7 C gets placed on closed list! D DFS with depth bound = 3

17 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3,6 B C 4,7 D DFS with depth bound = 3

18 Iterative Deepening Search Depth Bound = 3 A 1,2,5 3,6 B C 4,7 C already on closed list, so D is not reached! D DFS with depth bound = 3

19 Iterative Deepening Search Depth Bound = 4 A 1,2,5 3,6 B C 4,7 D 8 DFS with depth bound = 4

20 Iterative Deepening Search Depth Bound = 4 A 1,2,5,8 3,6 B C 4,7 D 8 DFS with depth bound = 4

21 Iterative Deepening Search Depth Bound = 4 A 1,2,5,8 3,6,9 B C 4,7 D 8 DFS with depth bound = 4

22 Iterative Deepening Search Depth Bound = 4 A 1,2,5,8 3,6,9 B C 4,7,10 D 8 DFS with depth bound = 4

23 Iterative Deepening Search Depth Bound = 4 A 1,2,5,8 3,6,9 B C 4,7,10 D 11 DFS with depth bound = 4

24 Iterative Deepening Search Depth Bound = 4 A 1,2,5,8 3,6,9 B C 4,7,10 Iterative Deepening finds solution 3 steps from start rather than 2! D 11 DFS with depth bound = 4

25 Iterative Deepening Search Algorithm given in class: Finds optimal solution for search trees (complete and optimal) Finds a solution, but not necessarily an optimal one for search graphs (complete but not optimal) Homework 1 Question 2: Draw what iterative deepening does

26 Other Search Methods Informed Search

27 Local Beam Search Like hill-climbing, only maintain k states, k>1 BeamSearch(states,ops): { states k} successors {s s = apply(s,o) where s states and o ops}; If goal(s ) for any s successors then return that s { implementation might interleave successor generation b } { and goal testing so as to stop when first solution is found } Else If successors > k then throw away all but best k of successors; BeamSearch(successors,ops) (If you let k=1 you get hill-climbing)

28 Genetic Algorithm Motivated by evoluationary biology Similar to Local Beam Search: Maintains k states called the population f(s) is typically called the fitness function Often greater values mean better Assumes states have structure, typically that they are n-tuples Genetic programming: states are programs Common operators: Crossover: Pick two states with good f values, and generate a new state whose elements are each randomly chosen from one or the other of the two states Mutation: With low probability mutate a state by a small amount

29 Genetic Algorithm Example: 8-queens Start with a population of k boards with 8 queens randomly placed in each Fitness(s) = # pairs of attacking queens (lower is better) Create a new population of k boards from the existing population by a combination of: Crossover: Select two boards s 1 and s 2 from population Bias board selection towards better boards (better board greater probability of selection) Select random i, i {2,,7} Create new board taking i random queens from s 1 and the remaining queens from s 2 Mutation: take a board and with low probability randomly move each queen

30 Other Directions (Not Covering) Continuous spaces / gradient-based methods Newton s method Linear programming (Until machine learning) Non-deterministic actions Partially observable states Online search Dynamic programming Searching AND/OR trees

31 Adversarial Search

32 Adversarial Search (Game Tree Search)

33 Let s Play a Game Each player takes one of the digits 1-9 and removes it First player to wind up with 3 of its numbers adding up to 15 wins If all numbers are used up it s a tie

34 Let s Play a Game Example Player 1: Player 2:

35 Let s Play a Game Example Player 1: 3 Player 2:

36 Let s Play a Game Example Player 1: 3 Player 2: 8

37 Let s Play a Game Example Player 1: 3 7 Player 2: 8

38 Let s Play a Game Example Player 1: 3 7 Player 2: 8

39 Let s Play a Game Example Player 1: 3 7 Player 2: 8 5

40 Let s Play a Game Example Player 1: 3 7 Player 2: 8 5

41 Let s Play a Game Example Player 1: Player 2: 8 5

42 Let s Play a Game Example Player 1: Player 2: 8 5

43 Let s Play a Game Example Player 1: Player 2: 8 5 6

44 Let s Play a Game Example Player 1: Player 2: 8 5 6

45 Let s Play a Game Example Player 1: Player 2: 8 5 6

46 Let s Play a Game Example Player 1: Player 2:

47 Let s Play a Game

48 Magic Square Tic Tac Toe

49 Minimax Value of a Game My Turn

50 Minimax Value of a Game My Turn Terminal Nodes

51 Minimax Value of a Game My Turn I Win Opponent Wins Terminal Nodes

52 Minimax Value of a Game I win = + I lose = My Turn Terminal Nodes

53 Minimax Value of a Game I win = + I lose = (Tie = 0) My Turn Terminal Nodes

54 Minimax Value of a Game I win = + I lose = (Tie = 0) (could choose +1 / 0 / -1, 1 / ½ / 0, or any appropriate scale) + My Turn - - Terminal Nodes

55 Minimax Value of a Game I win = + I lose = My Turn Terminal Nodes

56 Minimax Value of a Game I win = + I lose = My Turn Terminal Nodes

57 Minimax Value of a Game I win = + I lose = My Turn Terminal Nodes

58 Minimax Value of a Game I win = + I lose = My Opponent s Turn Terminal Nodes

59 Minimax Value of a Game I win = + I lose = My Opponent s Turn Terminal Nodes

60 Minimax Value of a Game Current state: s Available operators: ops Value of a state: V(s) My turn: Value of state s = V(s) = max Best move = = argmax o ops o ops Opponent s turn: Value of state s = V(s) = min {V apply s, o } {V apply s, o } o ops {V apply s, o } Best move = = argmin{v apply s, o } o ops

61 Minimax Value of a Game Current state: s Available operators: ops Value of a state: V(s) My turn: Value of state s = V(s) = max Best move = = argmax o ops o ops Opponent s turn: Value of state s = V(s) = min {V apply s, o } {V apply s, o } o ops {V apply s, o } Best move = = argmin{v apply s, o } o ops

62 Minimax Value of a Game Current state: s Available operators: ops Value of a state: V(s) My turn: Value of state s = V(s) = max Best move = = argmax o ops o ops {V apply s, o } {V apply s, o } Opponent s turn: Assumes opponent always does best move Value of state s = V(s) = min o ops {V apply s, o } Best move = = argmin{v apply s, o } o ops

63 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

64 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

65 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

66 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

67 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

68 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

69 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply

70 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

71 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

72 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

73 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

74 Minimax Value of a Game My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

75 Minimax Value of a Game GAME TREE My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors Can be continued arbitrarily deeply My Turn V(s) = Max of successors My Opponent s Turn V(s) = Min of successors

76 Pictorial Depiction of Full Tic Tac Toe Game Tree From "Fractal images of formal systems" by Paul St Denis and Patrick Grim, Journal of philosophical logic 26.2 (1997):

77 (Partial) Game Tree for Tic Tac Toe 0 +

78 Minimax Value of a Game Current state: s Available operators: ops Value of a state: V(s) My turn: Value of state s = V(s) = max Best move = = argmax o ops o ops Opponent s turn: Value of state s = V(s) = min {V apply s, o } {V apply s, o } o ops {V apply s, o } Best move = = argmin{v apply s, o } o ops

79 Minimax Algorithm Initial call: If I go first: minimax(initial-state,ops) If opponent goes first: maximin(initial-state,ops)

80 Minimax Algorithm minimax(s,ops): if terminal(s) then return V(s) else val - ; foreach o ops val maximin(apply(s,o),ops); if val > val then val val ; bestop o; return val

81 Minimax Algorithm minimax(s,ops): if terminal(s) then return V(s) else val - ; foreach o ops val maximin(apply(s,o),ops); if val > val then val val ; bestop o; return val maximin(s,ops): if terminal(s) then return V(s) else val + ; foreach o ops val minimax(apply(s,o),ops); if val < val then val val ; bestop o; return val

82 Minimax Search Multiagent environment Often just two agents Common setting is a game Competitive Each agent is self-interested Non-cooperative non-collaborative (except through self-interest) Assume opponent always optimizes what s best for the opponent Zero sum Deterministic (example: not backgammon dice!) Perfect information (example: poker hidden cards!)

83 Minimax Algorithm minimax(s,ops): if terminal(s) then return V(s) else val - ; foreach o ops val maximin(apply(s,o),ops); if val > val then val val ; bestop o; return val maximin(s,ops): if terminal(s) then return V(s) else val + ; foreach o ops val minimax(apply(s,o),ops); if val < val then val val ; bestop o; return val

84 Minimax Algorithm (Complete Search) minimax(s,ops): if terminal(s) then return V(s) else val - ; foreach o ops val maximin(apply(s,o),ops); if val > val then val val ; bestop o; return val maximin(s,ops): if terminal(s) then return V(s) else val + ; foreach o ops val minimax(apply(s,o),ops); if val < val then val val ; bestop o; return val

85 Minimax Algorithm (Complete Search) minimax(s,ops): if terminal(s) then return V(s) else val - ; foreach o ops val maximin(apply(s,o),ops); if val > val then val val ; bestop o; return val maximin(s,ops): if terminal(s) then return V(s) else val + ; foreach o ops val minimax(apply(s,o),ops); if val < val then val val ; bestop o; return val