Check to Checkmate !

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Check to Checkmate !

• Group 1
• Ashutosh
• Pushkar
• Ameya
• Sudhir
• From

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Motivation

• Game playing was one of the first tasks
  undertaken in AI
• Study of games brings us closer to
• Machines capable of logical deduction
• Machines for making strategic decisions
• Analyze the limitations of machines to human
  thought process
• Games are an idealization of worlds
• World state is fully accessible
• Actions outcomes are well-defined

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Outline of the presentation

• Evaluation Functions
• Algorithms
• Deep Blue
• Conclusions from Deep Blue
• Conclusion
• References

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Chess

• Neither too simple
• Nor too difficult for satisfactory solution
• Requires thinking for a skilled player
• Designing a chess playing program
• Perfect chess playing INTRACTABLE
• Legal Chess TRIVIAL
• Play tolerably good game SKILLFULLY

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Evaluation Function
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Evaluation Function

• Utility function
• Whole game tree is explored
• computationally expensive task !!
• Estimates the expected utility of a state
• Evaluation functions
• cut off the exploration depth by estimating
  whether a state will lead to a win or loss

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Evaluation Function (cont.)

• A good evaluation function should
• not take too long
• Preserve ordering of the terminal states
  otherwise it will lead to bad decision making
• Consider strategic moves that lead to long term
  advantages

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Evaluation Function (cont.)

• Typically includes
• Material Advantage (difference in total material
  of both sides)
• f (P) 200(k k) 9(q q) 5(r r)
• 3(b b) (p p) g(P) h(P)
• Positions of pieces
• Rook on open file
• double rooks
• rook on seventh rank etc. and their relative
  positions.
• Pawn Formation
• Mobility
• Evaluation function is an attempt to write a
  mathematical formula for intelligence ?

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Algorithms
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Games as Search Problems

• Initial states
• Where game starts
• Initial position in chess
• Successor function
• List of all legal moves from current position
• Terminal State
• Where the game is concluded
• Utility function
• Numeric value for all terminal states

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Minimax Strategy
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Minimax Strategy

• Optimal strategy
• Assumption opponent plays his best possible
  move.
• An option is picked which
• Minimizes damage done by opponent
• Does most damage to the opponent
• Idea
• For each node find minimum minimax value
• Choose the node with maximum of such values
• This will ensure best value against most damage
  done by opponent

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Strategy of Minimax

• Opponent tries to reduce utility functions value
• For any move made by opponent in reply of
  computers move, choose minimum reduced value by
  opponent
• Find the move with maximum such value

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Analysis

• Algorithm is complete for complete tree only
• Not best strategy against irrational opponent
• According to definition
• Time complexity O(bm)
• b max. no. of possible moves
• m max. depth of tree
• In chess even in average case, b 35 and
• m 100 gt time exceeds practical limits
• of states grows exponentially as per number of
  moves played

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a-ß Pruning
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a-ß Pruning

• The problem of minimax search
• of state to examine exponential in number of
  moves
• Returns same moves as minimax does
• Prunes away branches that cant influence final
  decision.
• a the value of the best (highest) choice so far
  in search of MAX
• ß the value of the best (lowest) choice so far
  in search of MIN
• Order of considering successor

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Algorithm for a-ß Pruning

• Current highest ß is found and assigned as a
• ß is current lowest for as from that move
• For next possible node, while finding ß, if some
  a is found lower than current highest ß
• It will only give lesser value of final ß
• S0, other as are not found for that node
• After calculating ß for this node, a is replaced
  by max(a,ß for this node)
• In this way after all possible set of moves final
  value of a is found

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a-ß Pruning (2)

• If m is better than n for Player, we will never
  get to n in play
• and just prune it.

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Analysis of a-ß Pruning
Improvement over minimax algorithm

• Does not affect final results
• Worthwhile to examine that successor first which
  is likely to be best
• Time complexity O(b(m/2))
• Effective branching factor vb
• i.e. 6 rather than 35
• In case of random ordering
• Total number of nodes examined is of the order
  O(b(3m/4))

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Transposition table

• Dynamic programming
• Multiple paths to the same position
• Savings through memorization
• Use a hash table of evaluated positions

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Iterative Deepening

• Sometime chess is played under a strict time
• Depth of search depend on time
• Use of Breadth first Search
• Advantage program know which move was
• best at previous level

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Horizon Effect

• Problem with fixed depth search
• Positive Horizon Effect
• Negative horizon effect

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Quiescence Search

• Search till quiet position
• Quiet Position
• Doesnt affect the current position so much
• Example no capture of any piece, no check, no
  pawn promotions/threats

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State of the art DEEP BLUE
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Defeats Gary Kasparov

• Won a match in 1997
• Brute force computing power
• Massive, parallel architecture
• Special purpose hardware for chess
• Parameters of the evaluation function
• Learnt by studying many master games
• Different evaluation function for different
  positions
• Utilized heavily loaded endgame databases

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Humans vs. Computers
Humans Computers
Lower Computational Speed Higher
Errors Possible Error Free
Tend to be instinctive No instincts
Imaginative None
High Learning Capabilities Limited
Inductive Not Inductive
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Some intricacies of a chess playing system

• Should not play the same sequence of moves again
• A player wins a match against the computer
• Starts playing the same sequence of moves
• Hence, a statistical element is required
• Opponent can learn the algorithm used by computer
• Hence, again the need for a statistical element
• Different game play during different phases
• Start Game
• Mid Game
• End Game

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Conclusion

• Computer chess as a search problem
• Good enough decisions
• Simulation of skill by knowledge
• Limitations of computers to humans
• Future work
• Better evaluation functions through learning
• Need for different AI techniques to play chess

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References

• Claude E. Shannon Programming a Computer for
  Playing Chess, Philosophical Magazine, Ser.7,
  Vol. 41, No. 314, March 1950.
• S.Russel P. Norvig Artificial Intelligence A
  Modern Approach 2/E, Prentice Hall, ISBN-10
  0137903952
• Wikipedia

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Thank You