COSC 4350 and 5350 Artificial Intelligence

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COSC 4350 and 5350Artificial Intelligence

• Kasparov vs. Machine Reflection on the Essence
  of Intelligence via Game Playing (Part 2)
• Dr. Lappoon R. Tang

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Outline

• Enhancing Minimax search with Alpha Beta pruning
• Kasparov vs. Deep Blue 2 what lessons have we
  learned?
• Open questions

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Readings

• Section 6.3
• Section 6.4

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Last time

• But you may ask what about the chess game? Can
  we use minimax to play chess as Deep Blue used
  it?
• Good question
• The game tree of chess has an average branching
  factor of 30 or more
• Usually, a game can have 30 ply or more (i.e. a
  tree depth of 30) 2.05 x 1044 nodes!
• Can minimax work on such a huge search space?
  Lets find out more about it next time

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Review of Minimax search
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Alpha beta pruning

• A complete minimax search requires exploring
  every nodes until meeting the terminal state
• Too inefficient
• Idea An algorithm that tells which parts of the
  game tree can be ignored because computing the
  minimax values of the nodes in those parts will
  not affect the final choice made by MAX

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Alpha Beta Pruning Effectiveness

• Effectiveness depends on node ordering
• Worst Case ...
• No advantage due to useless node ordering. That
  is, complexity O(bd).
• Best Case ...
• Happens when the score of the leftmost node is
  smaller than the best so far the O(bd)
  complexity of MiniMax becomes O(bd/2 ).
• Dont worry about the Math, the important point
  is that
• Since bd/2 ?bd, this is the same as having a
  branching factor of ?b instead of b, thereby
  doubling the allowable tree depth to explore
• For example Chess
• Branching factor goes from 35 to 6.
• Allows for a much deeper search given the same
  amount of time.
• Expected Case ...
• Empirical studies indicate and expected
  complexity of O(b3d/4).

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Dealing with Limited Time

• In real games, there is usually a time limit T on
  making a move
• How do we take this into account?

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Dealing with Limited Time

• Could we set a conservative depth-limit that
  guarantees we will find a move in time
• Example Confine Minimax to explore up to a tree
  depth of 10
• BUT there are problems

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Dealing with Limited Time

• Problem 1 What if we havent reached the
  terminal nodes to get their utility values?
• Problem 2 Even if some of the terminal nodes
  have been reached, how do we select the minimax
  value for a level if some nodes at the level do
  not have minimax values?

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Evaluation function to the rescue (instead of
using utility function)

• We can solve both problems by combining
  alpha-beta pruning with a state evaluation
  function
• Idea Replace the terminal-test with a cut-off
  test stops when a certain depth limit T has
  been reached
• Instead of returning utility values from terminal
  nodes, one just computes the expected value of a
  non-terminal node (by using a heuristic function)
• The purpose of a heuristic function is to
  estimate the quality of a resulting board
  configuration so that the best move can be made

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Cut-off Test

• IF level(game_state) T, then return
  value_of(game_state)
• Idea For every game state
• Check if its level in the game tree T
• If so, we stop and treat the state as if it was a
  terminal game state
• Evaluate its expected utility score given the
  state description and return it back

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Cut-off Test (contd)
Depth limited Minimax search
We would like to do Minimax on this full game
tree but ... ?
we dont have time, so we will explore it to
some manageable depth.
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Heuristic Evaluation Functions

• Often called static evaluation heuristics.
• Evaluate board without knowing where exactly it
  will lead to
• Use it to estimate the probability of winning
  from that node
• Example a chess game state in which the queen
  has been captured but not the opponents queen
  is not likely to lead to a winning state
• Important qualities
• Must agree with the utility function f at the
  terminal states (i.e. h(end-game) f(end-game))
• Must not take long, to compute.
• Should be accurate enough.

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What should the heuristic function return?
Expected Value vs. Material Value

• Approach 1 Expected (utility) value of a state
  probability-weighted average of the different
  possible utility values a state can have
• Example (0.72 1) (0.20 -1) (0.08 0)
  0.52 (52)
• Problem Estimation of probabilities is difficult
• Approach 2 Material value of a state weighted
  linear combination of the different features of a
  state
• Example amaterialBalance ßcenterControl ?
  where material balance Value of white pieces
  - Value of black pieces (pawn 1, rook 5,
  queen 9, etc).
• Problems 1) features are assumed to be
  independent of each other 2) can be time
  consuming

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The Horizon Effect

• Sometimes disaster lurks just beyond search depth
• computer captures queen, but a few moves later
  the opponent checkmates (i.e., wins)
• The computer has a limited horizon it cannotsee
  that this significant event could happen
• How does one avoid catastrophic losses due to
  short-sightedness?
• Quiescence search
• Secondary search
• Basically these are searches that look ahead a
  bit to see if a disaster is just beneath the
  horizon

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Computers and Grand Master Chess

• Deep Blue 2 (IBM)
• Parallel processor, 32 node cluster
• Each node has 8 dedicated VLSI chess chips
• Can search 250 million configurations/second an
  average human being can search at most 30
  moves/second?
• Uses minimax, alpha-beta pruning, sophisticated
  heuristics
• Currently it can search up to 14 plies (i.e. 7
  pairs of moves)
• Can avoid horizon by searching as deep as 40
  plies
• Uses book moves

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Computers and Grand Master Chess

• Kasparov vs. Deep Blue 2, May 1997
• 6 game full-regulation chess match sponsored by
  the Association of Computing Machinery (ACM)
• Kasparov lost the match 2 wins 1 tie to 3 wins
  1 tie
• This was a historic achievement the first time a
  computer became the best chess player on the
  planet

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Kasparov vs. Deep Blue 2 Lessons learned

• Minimax search and alpha beta pruning are not
  intelligent by themselves they are just
  mathematical calculations
• There is arguably no observable intelligence at a
  microscopic level
• However, an extremely large amount of seemingly
  unintelligent steps organized to achieve a well
  defined purpose can produce intelligent behavior
  at a macroscopic level
• Deep Blue 2 can see more board configurations in
  one second than all the moves a person can see in
  his life time
• Quantity becomes quality G. Kasparov

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How far can we go with such an approach?

• Checkers
• Current world champion is Chinook (a computer
  system ?)
• Othello
• computers can easily beat the world experts
• Go
• branching factor b 360 (very large!)
• 2 million prize for any system that can beat a
  world expert

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Open questions

• Currently, brute force search a big machine can
  amount to a sufficient level of computational
  intelligence that beats human intelligence on
  some non-trivial board games (e.g. chess and
  checkers)
• Such an approach still wont work for more
  complicated games with a crazy branching factor
  (e.g. Go)
• Open question Can we further combine human
  intelligence and raw computational power to build
  a game playing machine that plays like a God?
• It can beat any system that relies purely on
  computational intelligence
• It can also beat any human expert