Game Playing Pruning

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Game Playing ?-? Pruning

• Introduction to Artificial Intelligence
• Hadi Moradi
• moradi_at_usc.edu

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2. ?-? pruning search cutoff

• Pruning eliminating a branch of the search tree
  from consideration without exhaustive examination
  of each node
• ?-? pruning the basic idea is to prune portions
  of the search tree that cannot improve the
  utility value of the max or min node, by just
  considering the values of nodes seen so far.
• Does it work?
• Yes, it roughly cuts the branching factor from b
  to ?b resulting in double as far look-ahead than
  pure minimax

3
?-? pruning example
? 6
MAX
MIN
6
6
12
8
4
?-? pruning example
? 6
MAX
MIN
6
? 2
6
12
8
2
5
?-? pruning example
? 6
MAX
MIN
6
? 2
? 5
6
12
8
2
5
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?-? pruning example
? 6
MAX
Selected move
MIN
? 5
6
? 2
6
12
8
2
5
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?-? pruning general principle
Player
m
?
6
Opponent
If ? gt v then MAX will chose m so prune tree
under n Similar for ? for MIN
Player
n
v
Opponent
2
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Properties of ?-?
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The ?-? algorithm
//the leaf node (terminal state)
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The ?-? algorithm (cont.)
//the leaf node (terminal state)
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Remember Minimax Recursive implementation
Time complexity O(bm)Space complexity O(bm)
( DFSDoes not keep all nodes in memory.)
Complete Yes, for finite state-spaceOptimal
Yes
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More on the ?-? algorithm start from Minimax
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The ?-? algorithm
Note These are both Local variables. At
the Start of the algorithm, We initialize them
to ? -? and ? ?
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A Walk Through Example
? Best choice so far for MAX ? Best choice so
far for MIN
MAX
? -? ? ?
MIN

MAX
5 10 4 2
8 7
15
In Max-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• V -?

Max-Value loops over these
MIN

MAX
5 10 4 2
8 7
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In Max-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

? -? ß ?
MIN

MAX
5 10 4 2
8 7
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

• ? -?
• ?
• v ?

MIN

MAX
5 10 4 2
8 7
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• ?
• v ?

MIN

Min-Value loops over these
MAX
5 10 4 2
8 7
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? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• ?
• v ?

MIN

• ? -?
• ?

MAX
5 10 4 2
8 7
Utility(state) 5
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• ?
• v 5

MIN

MAX
5 10 4 2
8 7
21
In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX
? -? ? ?

• ? -?
• 5
• v 5

MIN

MAX
5 10 4 2
8 7
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX
? -? ? ?

• ? -?
• 5
• v 5

MIN

• ? -?
• 5
• v 5

MAX
5 10 4 2
8 7
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• ? Best choice so far for MAX
• Best choice so far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• 5
• v 5

MIN

• ? -?
• 5

MAX
5 10 4 2
8 7
Utility(state) 10
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX
? -? ? ?

• ? -?
• 5
• v 5 lt10

MIN

MAX
5 10 4 2
8 7
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• ? Best choice so far for MAX
• Best choice so far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• 5
• v 5

MIN

• ? -?
• 5

MAX
5 10 4 2
8 7
Utility(state) 4
26
In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• 5
• v 5 gt4

MIN

MAX
5 10 4 2
8 7
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In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• 5
• v 4

MIN

MAX
5 10 4 2
8 7
28
In Min-Value
? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v -?

MAX

• ? -?
• 4
• v 4

MIN

MAX
5 10 4 2
8 7
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? Best choice so far for MAX ? Best choice so
far for MIN

• ? -?
• ?
• v 4

MAX
MIN

MAX
5 10 4 2
8 7
30
? Best choice so far for MAX ? Best choice so
far for MIN

• ? 4
• ?
• v 4

MAX
MIN

MAX
5 10 4 2
8 7
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In Max-Value
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

• ? 4
• ?
• v 4

MIN

MAX
5 10 4 2
8 7
32
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

• ? 4
• ?
• v ?

MIN

MAX
5 10 4 2
8 7
33
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

MIN

MAX
5 10 4 2
8 7

• ? 4
• ?
• v ?

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? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

MIN

MAX
5 10 4 2
8 7

• ? 4
• ?

35
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

• ? 4
• ?
• v 2

MIN

MAX
5 10 4 2
8 7
36
? Best choice so far for MAX ? Best choice so
far for MIN
MAX

• ? -?
• ?
• v -?

• ? 4
• ?
• v 2

MIN

MAX
5 10 4 2
8 7
37
? Best choice so far for MAX ? Best choice so
far for MIN

• ? 4
• ?
• v 4 gt2

MAX
MIN

MAX
5 10 4 2
8 7
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Another way to understand the algorithm

• From http//yoda.cis.temple.edu8080/UGAIWWW/lect
  ures95/search/alpha-beta.html
• For a given node N,
• ? is the value of N to MAX
• ? is the value of N to MIN

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Alpha/Beta pruning - example (4)
A move (max)
B move (min)
A move (max)
8 7 3 9 1 6 2 4
1 1 3 5 3 9 2 6
5 2 1 2 3 9 7 2
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Alpha/Beta pruning - example (4)
A move (max)
B move (min)
A move (max)
8 7 3 9 1 6 2 4
1 1 3 5 3 9 2 6
5 2 1 2 3 9 7 2
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State-of-the-art for deterministic games
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Chess As An Example

• Chess basics
• 8x8 board, 16 pieces per side, average branching
  factor of 38
• Rating system based on competition
• 500 --- beginner/legal
• 1200 --- good weekend warrier
• 2000 --- world championship level
• 2500 --- grand master
• time limited moves
• important aspects position, material

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Sketch of Chess History

• First discussed by Shannon, Sci. American, 1950
• Initially, two approaches
• human-like
• brute force search
• 1966 MacHack ---1100 --- average tournament
  player
• 1970s
• discovery that 1 ply 200 rating points
• hash tables
• quiescence search

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Sketch of Chess History

• Chess 4.x reaches 2000 (expert level), 1979
• Belle 2200, 1983
• special purpose hardware
• 1986 --- Cray Blitz and Hitech 100,000 to 120,000
  position/sec using special purpose hardware

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Deep Blue History

• 1985, Hsu build BLSI move generator (using DARPA
  funding!)
• Anantharaman combined with chess program leading
  to 50K moves/second algorithm
• 1986 CCC ---- no luck, but notice blind forced
  move was a problem
• singular extension if you see a position where
  only a single move determines the line, follow it
• 1987 --- new algorithm won (Chiptest) 400-500K
  position/sec

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IBM checks in

• Deep thought
• 250 chips (2M pos/sec /// 6-7M pos/soc)
• Evaluation hardware
• piece placement
• pawn placement
• passed pawn eval
• file configurations
• 120 parameters to tune
• Tuning done to masters games
• hill climbing and linear fits
• 1989 --- rating of 2480 Kasparov beats

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IBM Checks In

• Deep Blue is the next generation
• parallel version of deep thought
• 200 M pos/sec ? 60B positions in the 3 minutes
  allotted for move
• DB 1 32 Rs/6000s with 6 chess proc/node
• DB 2 faster 32 nodes w 8 chess proc/node (256
  proc)
• message passing architecture
• search as much as 20-30 levels deep using sing.
  extension
• In 1997, Kasparov beaten
• Kasparov changed strategy in earlier games
• As much a psychological as mental victory
• http//www.research.ibm.com/deepblue/

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Nondeterministic games
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Algorithm for nondeterministic games
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Remember Minimax algorithm
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Nondeterministic games the element of chance
expectimax and expectimin, expected values over
all possible outcomes
?
CHANCE
0.5
0.5
?
3
?
8
8
17
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Evaluation functions Exact values DO matter
Order-preserving transformation do not
necessarily behave the same!
NOTE It is not the same as MINIMAX algorithm
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State-of-the-art for nondeterministic games
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Summary
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Exercise Game Playing
Consider the following game tree in which the
evaluation function values are shown below each
leaf node. Assume that the root node corresponds
to the maximizing player. Assume the search
always visits children left-to-right.

• (a) Compute the backed-up values computed by the
  minimax algorithm. Show your answer by writing
  values at the appropriate nodes in the above
  tree.
• (b) Compute the backed-up values computed by the
  alpha-beta algorithm. What nodes will not be
  examined by the alpha-beta pruning algorithm?
• (c) What move should Max choose once the values
  have been backed-up all the way?

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Exercise Game Playing
? -? ? ?
Max-Value loops over these
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Exercise Game Playing
? -? ? ?
Min-Value loops over these
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Exercise Game Playing
? -? ? ?
Max-Value loops over these
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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Exercise Game Playing
? -? ? ?
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