Alpha Beta pruning in AI

1. 1
UNIT: 5
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

3. GAME PLAYING: ALPHA BETA PRUNING

4. Alpha-BetaPruning= MinimaxExcept
• This game search strategy is a modification to Minimax game
search that avoids exploring nodes that are not useful in the
search.
• It gives the same results as Minimax but avoids exploring
some nodes.
• In the previous example, the path explored using Minimax was
A-B-D-J.
• Also the Alpha-Beta Pruning path will be A-B-D-J but
without exploring all nodes as in Minimax.

5. Alpha-Beta PruningMotivation
Never explore values that are not useful.
• =Min(Max(1, 2, 5), Max(6, x, y), Max(1, 3, 4))
• =Min(5, Max(6, x, y), 4)
• =Min(Max(6, x, y), 4)
• =4

6. Alpha-BetaGameSearch

7. Alpha-BetaGameSearch

8. Alpha-BetaGameSearch

9. Alpha-BetaGameSearch

10. Alpha-BetaGameSearch

11. Alpha-BetaGameSearch
5

12. Alpha-BetaGameSearch
5
5

13. Alpha-BetaGameSearch
5
5

14. Alpha-BetaGameSearch
5
5

15. Alpha-BetaGameSearch
5
5

16. Alpha-BetaGameSearch
7
5
5

17. Alpha-BetaGameSearch
7
5
5
Min(5, max(7, x,
y))
=5

18. Alpha-BetaGameSearch
7
5
5

19. Alpha-BetaGameSearch
7
5
5

20. Alpha-BetaGameSearch
7
5
5

21. Alpha-BetaGameSearch
7
5
5
5

22. Alpha-BetaGameSearch
7
5
5
5

23. Alpha-BetaGameSearch
7
5
5
5

24. Alpha-BetaGameSearch
7
5
5
5

25. Alpha-BetaGameSearch
7
5
5
5

26. Alpha-BetaGameSearch
7
5
5
5
4

27. Alpha-BetaGameSearch
7
5
5
5
4
4
Max(5, min(4, x, y,
z))
=5

28. Alpha-BetaGameSearch
7
5
5
5
4
4

29. Alpha-BetaGameSearch
7
5
5
5
4
4

30. Alpha-BetaGameSearch
7
5
5
5
4
4
Same results as
Minimax with fewer
nodes explored.
5 Unexplored
Branches.

34. Introduction
34
• Alpha-beta pruning is a way of finding the optimal minimax solution while
avoiding searching subtrees of moves which won't be selected. In the
search tree for a two-player game, there are two kinds of nodes, nodes
representing your moves and nodes representing your opponent'smoves.
• Alpha-beta pruning gets its name from two parameters.
– They describe bounds on the values that appear anywhere along the
path under consideration:
• α = the value of the best (i.e., highest value) choice found so far
along the path for MAX
• β = the value of the best (i.e., lowest value) choice found so far
along the path for MIN

35. Alpha Beta Pruning
35
• Alpha-beta pruning gets its name from two bounds that are passed along
during the calculation, which restrict the set of possible solutions based on
the portion of the search tree that has already been seen. Specifically,
• Beta is the minimum upper bound of possiblesolutions
• Alpha is the maximum lower bound of possiblesolutions
• Thus, when any new node is being considered as a possible path to the
solution, it can only work if:
where N is the current estimate of the value of the node

36. Example
36

37. Initial Assumption for Alpha and Beta
37

38. Example
38

39. Example
39

40. Example
40
• And one more time

41. Example
41

42. Example
42

43. Example
43

44. Example
44

45. Example
45
• We generate the next child and pass the bounds along

46. Example
46

47. Example
47

48. Example
48

49. Example
49

50. Example
50

51. Example
51

52. Example
52
• ... it's first child min node ...

53. Example
53

54. Example
54

55. Example
55

56. Example
56

57. Example
57

58. Example
58

59. Example
59

60. Example
60
• ... and its first child ...

61. Example
61

62. Example
62

63. Example
63

64. Example
64

65. Example
65

66. Example
66

67. Example
67

68. Example
68

69. Example
69

70. Example
70

71. 71
• Once again, we're at a point where alpha and beta are tied, so we prune.
Note that a real solution doesn't just indicate a number, but what move led
to that number.
• If you were to run minimax on the list version presented at the start of the
example, your minimax would return a value of 3 and 6 terminal nodes
would have been examined

72. SYLLABUS: UNIT - 5
Advanced Topics: Game Playing: Minimax search procedure-
Adding alpha-beta cutoffs. Expert System: Representation-
Expert System shells-Knowledge Acquisition. Robotics:
Hardware-Robotic Perception-Planning-Application domains

73. GAME PLAYING: MINMAX Search Procedure

74. What is MINIMAX ?
Minimax is a kind of backtracking algorithm that is used in decision
making and game theory to find the optimal move for a player,
assuming that your opponent also plays optimally. It is widely used in
two player turn-based games such as Tic-Tac-Toe, Backgammon,
Mancala, Chess, etc

75. GAME COMPONENTS
• Initial State
• Successor Function
• Terminal Test
• Utility Function

76. GAME

77. Game tree

78. Minimax GameSearch

79. Minimax Game
Search Two Players take
turns: Max and
Min

80. Minimax GameSearch
Two Players take
turns: Max and
Min
Max : Maximizes
Score. Min :
Minimizes Score.
MAX

81. Minimax GameSearch
Two Players take
turns: Max and
Min
MAX
MIN

82. Minimax GameSearch
Two Players take
turns: Max and
Min
Max : Maximizes
Score. Min :
Minimizes Score.
MAX
MIN

83. Minimax GameSearch
Two Players take
turns: Max and
Min
Max : Maximizes
Score. Min :
Minimizes Score.
Special Case.
Max is an
expert. Min is
a beginner.
MAX
MIN

84. Minimax Game
Search A

85. Minimax GameSearch
Which node to
follow?
No heuristic
values.
A
B C

86. Minimax GameSearch
Which node to
follow?
No heuristic
values.
A
B C

87. Minimax GameSearch
A
B C
Which node to follow?
No heuristic values.
Hot to find heuristic
values for other
nodes?

88. Minimax GameSearch
A
B C
Which node to follow?
No heuristic values.
Hot to find heuristic
values for other
nodes?
Use children
heuristics to
calculate parent
heuristic.

89. Minimax GameSearch
A
B C
Which node to follow?
No heuristic values.
Hot to find heuristic
values for other
nodes?
Use children
heuristics to
calculate parent
heuristic.
Minimax Game
Search Steps

90. Minimax GameSearch
Which node to
follow?
No heuristic
values.
A
B C
Hot to find heuristic
values for other
nodes?
Use children
heuristics to
calculate parent
heuristic.
Minimax Game Search Steps
Calculate Heuristics

91. Minimax GameSearch
Which node to
follow?
No heuristic
values.
A
B C
Hot to find heuristic
values for other
nodes?
Use children
heuristics to
calculate parent
heuristic.

92. Phase 1 :Heuristic ValueCalculation
Depth-FirstSearch
A

93. Phase 1 :Heuristic ValueCalculation
A
B

94. Phase 1 :Heuristic ValueCalculation
A
B C

95. Phase 1 :Heuristic ValueCalculation
A
B C
B C

96. Phase 1 :Heuristic ValueCalculation
A
B
B
C

97. Phase 1 :Heuristic ValueCalculation
A
B
B
C
D E
B C

98. Phase 1 :Heuristic ValueCalculation
A
B
B
C
D E
B C
D E

99. Phase 1 :Heuristic ValueCalculation
A
B
B
D
C
D E
B C
D E

100. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
B C
D E

101. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
B C
D E
H I
J

102. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
B C
D E
H I J

103. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J

104. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J
Max

105. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J
Max
5

106. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J
Ma
x
5
5

107. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J
Ma
x
5
5
5

108. Phase 1 :Heuristic ValueCalculation
A
B
B
D
H I
C
D E
J
3 -2 5
B C
D E
H I J
Ma
x
5
5
5
5

109. Phase 1 :Heuristic ValueCalculation
A
B
B
C
D E
B C
D E
5
5

110. Phase 1 :Heuristic ValueCalculation
A
B
B
E
C
D E
B C
D E
5
5

111. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
B C
D E
5
5

112. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
B C
D E
K L M
5
5

113. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
5
5

114. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Max
5
5

115. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Max
7
5
5

116. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
5
5

117. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
7
5
5

118. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
7
Min
5
7
5

119. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
7
Mi
n
5
5
5
7
5

120. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
7
Mi
n
5
5
5
5
7
5

121. Phase 1 :Heuristic ValueCalculation
A
B
B
E
K L
C
D E
M
7 0 3
B C
D E
K L M
Ma
x
7
7
7
Mi
n
5
5
5
5
7
5
5

122. Phase 1 :Heuristic ValueCalculation
A
B C
B C
5
7
5
5

123. Phase 1 :Heuristic ValueCalculation
A
B
C
C
B C
5
7
5
5

124. Phase 1 :Heuristic ValueCalculation
A
B
C
C
F G
B C
5
7
5
5

125. Phase 1 :Heuristic ValueCalculation
A
B
C
C
F G
B C
5
F G
7
5
5

126. Phase 1 :Heuristic ValueCalculation
A
B
C
F
C
F G
B C
5
F G
7
5
5

127. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
B C
5
F G
7
5
5

128. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
B C
N O P
5
F G
7
5
5

129. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
5
F G
7
5
5

130. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
Max
5
F G
7
5
5

131. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
Max
4
5
F G
7
5
5

132. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
Ma
x
4
5
4
F G
7
5
5

133. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
Ma
x
4
5
4
F G
4
7
5
5

134. Phase 1 :Heuristic ValueCalculation
A
B
C
F
N O
C
F G
P
0 -5 4
B C
N O P
Ma
x
4
5
4
F G
4
4
7
5
5

135. Phase 1 :Heuristic ValueCalculation
A
B
C
C
F G
B C
5
F G
4
4
7
5
5

136. Phase 1 :Heuristic ValueCalculation
A
B
C
G
C
F G
B C
5
F G
4
4
7
5
5

137. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
B C
5
F G
4
4
7
5
5

138. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
B C
Q R S
5
F G
4
4
7
5
5

139. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
5
F G
4
4
7
5
5

140. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Max
5
F G
4
4
7
5
5

141. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Max
8
5
F G
4
4
7
5
5

142. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4
4
7
5
5

143. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
4
7
5
5

144. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Min
8
4
7
5
5

145. Phase 1 :Heuristic ValueCalculation
Depth-First
Search A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
8
4
7
5
5

146. Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
4
8
4
7
5
5
4

147. Max
Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
4
8
4
7
5
5
4

148. Max
Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
4
5
8
4
4
7
5
5

149. Ma
x
Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
4
5
5
8
4
4
7
5
5

150. Ma
x
Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
-6 8 2
B C
Q R S
Ma
x
8
5
8
F G
4 8
Mi
n
4
4
4
5
5
8
4
5
4
7
5
5

151. Ma
x
Phase 1 :Heuristic ValueCalculation
A
B
C
G
Q R
C
F G
S
8
B C
Q R S
Ma
x
5
8
F G
4 8
Mi
n
4
4
4
5
5
8
4
5
4
7
5
5
-6 8 2
If both players play optimally then Max will win by a score 5.

152. Game TreeAfterHeuristic ValueCalculation
8
4
5
4
7
5
5

153. Phase 2 :GameSearch
8
4
5
4
7
5
5

154. Phase 2 :GameSearch
8
4
5
4
7
5
5
A

155. Phase 2 :GameSearch
8
4
5
4
7
5
5
B
Max
A

156. Phase 2 :GameSearch
8
4
5
4
7
5
5
B
D
Max
A
Min

157. Phase 2 :GameSearch
8
4
5
4
7
5
5
B
D
J
Max
Min
Max
A

158. Minimax Game Search
Drawback
• Expandsall the
tree
while not all
expanded nodes are
useful.

159. Minimax Game Search
Drawback
• Expandsall the
tree
while not all
expanded nodes are
useful.
• In this example, just
few nodes of the
whole tree was
useful in reaching
the goal.

160. Alpha-Beta
Pruning

161. Alpha-BetaPruning= Minimax
Except
• This game search strategy is a modification to Minimax game
search that avoids exploring nodes that are not useful in the
search.
• It gives the same results as Minimax but avoids exploring
some nodes.
• In the previous example, the path explored using Minimax was
A-B-D- J.
• Also the Alpha-Beta Pruning path will be A-B-D-J but
without exploring all nodes as in Minimax.

162. Alpha-BetaGameSearch

163. Alpha-BetaGameSearch

164. Alpha-BetaGameSearch

165. Alpha-BetaGameSearch

166. Alpha-BetaGameSearch

167. Alpha-BetaGameSearch
5

168. Alpha-BetaGameSearch
5
5

169. Alpha-BetaGameSearch
5
5

170. Alpha-BetaGameSearch
5
5

171. Alpha-BetaGameSearch
5
5

172. Alpha-BetaGameSearch
7
5
5

173. Alpha-BetaGameSearch
7
5
5
Min(5, max(7, x,
y))
=5

174. Alpha-BetaGameSearch
7
5
5

175. Alpha-BetaGameSearch
7
5
5

176. Alpha-BetaGameSearch
7
5
5

177. Alpha-BetaGameSearch
7
5
5
5

178. Alpha-BetaGameSearch
7
5
5
5

179. Alpha-BetaGameSearch
7
5
5
5

180. Alpha-BetaGameSearch
7
5
5
5

181. Alpha-BetaGameSearch
7
5
5
5

182. Alpha-BetaGameSearch
7
5
5
5
4

183. Alpha-BetaGameSearch
7
5
5
5
4
4
Max(5, min(4, x, y,
z))
=5

184. Alpha-BetaGameSearch
7
5
5
5
4
4

185. Alpha-BetaGameSearch
7
5
5
5
4
4

186. Alpha-BetaGameSearch
7
5
5
5
4
4
Same results as
Minimax with fewer
nodes explored.
5 Unexplored
Branches.