Programming a Computer for Playing Chess

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Programming a Computer for Playing Chess

• By Claude E. Shannon

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Programming a Computer for Playing Chess

• Published in Philosophical Magazine, in 1950.
• Considered as an important milestone in computer
  chess programming

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About the paper

• Written while Shannon was working in Bell
  Telephone Laboratories
• One of Shannon famous hobbies as well as
  cycling and juggling
• Surveys the problem of constructing a computer
  routine for a modern computer that will play chess

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First chess machine

• First machine built in 1769
• Constructed by the Hungarian engineer von
  Kempelen for queen Maria Theresia.
• Had outstanding playing strength
• Mechanical device supplied by a human chess
  master hidden inside.
• It was a FAKE

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El Ajedrecista (chess player)- Playing a
Rook-King Endgame

• Invented in 1914 by Leonardo Torres y Quevedo
• Played rook and king against a king
• Machine forced checkmate no matter however its
  opponent played
• Explicit set of rules - relatively simple idea

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Turings paper machine

• Was written soon after WWII
• Instruction that would enable a machine to play
  chess
• no machines that could execute the
  instructions...
• Turing acts as a human CPU
• More than ½ hour per move !
• Howeverlooses to one of his colleagues

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Turings paper machine

• Here's the historic game Turing's paper machine
  Alick Glennie, Manchester 1952
• 1.e4 e5 2.Nc3 Nf6 3.d4 Bb4 4.Nf3 d6 5.Bd2 Nc6
  6.d5 Nd4 7.h4 Bg4 8.a4 Nxf3 9.gxf3 Bh5 10.Bb5
  c6 11.dxc6 0-0 12.cxb7 Rb8 13.Ba6 Qa5 14.Qe2 Nd7
  15.Rg1 Nc5 16.Rg5 Bg6 17.Bb5 Nxb7 18.0-0-0 Nc5
  19.Bc6 Rfc8 20.Bd5 Bxc3 21.Bxc3 Qxa4 22.Kd2?
  22.h5 would have trapped the bishop 22...Ne6
  23.Rg4 Nd4? 23...Rxb2! 24.Bxb2 Rxc2 24.Qd3 Nb5
  25.Bb3 Qa6 26.Bc4 Bh5 27.Rg3 Qa4 28.Bxb5 Qxb5
  29.Qxd6 Rd8 0-1.

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First win for a computer

• MANIAC I was delivered in 1950 for developing
  atomic weapons
• Before calculating detonation parameters, was
  tested with a limited chess program
• Simplified 6x6 board without bishops
• Twelve minutes to search up to four ply (2 moves)
• In Los Alamos, 1956 it wins in 23 moves against
  a young lady who learnt the game a week before
  the game.
• First win of a machine against a human player

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Why Chess?

• Sharply defined problem both in operations
  (moves) and ultimate goal (checkmate)
• Neither trivial nor too difficult for
  satisfactory solution
• Its discrete structure fits well modern computers

Chess is generally considered to require
thinking for a skilful play. A solution to this
problem will force us to admit that machine
thinks, or further restrict the concept of
thinking
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Definition of a Chess Position

1. A statement of the positions of all pieces on the
   board.
2. A statement of which side, White or Black, has
   the move.
3. A statement as to whether the king and rooks have
   moved.
4. A statement of, say, the last move. This will
   determine whether a possible en passant capture
   is legal
5. A statement of the number of moves made since the
   last pawn move or capture.

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Game Types

• Games can be deterministic or non-deterministic
• probabilistic, stochastic
• and have perfect information or not
• Accessible The world state is fully
  accessible to the program or agent (nothing
  hidden)

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50-move drawing rule

• The F.I.D.E. Laws of Chess
• 10.12
• the game is drawn when a player having the move
  claims a draw and demonstrates that at least the
  last? 50 consecutive moves have been made by
  each side without the capture of any piece and
  without the movement of any pawn. This number of
  50 moves can be increased for certain positions,
  provided that this increase in number and these
  positions have been clearly announced by the
  organisers before the event starts.
• The most extreme case yet known of a position
  which might take more than 50 moves to win is
  king, rook and bishop against king and two
  knights, which can run for 223 moves between
  captures!

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Von Neumann and Morgenstern, 1944

• In chess there is no element of chance, and since
  each opponent has prefect information at each
  move
• For any given position, either
• A won position for White. That is, White can
  force a win, however Black defends.
• A draw position. White can force at least a draw,
  however Black plays, and likewise Black can force
  at least a draw, however White plays. If both
  sides play correctly the game will end in a draw.
• A won position for Black. Black can force a win,
  however White plays.

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Von Neumann and Morgenstern, 1944

• Nature of existence theorem
• No practical method is known to determine which
  of the three is correct
• If would be determined, probably would lose
  interest in chess

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Altering the rules

• A slight change of the rules, guarantees at least
  a draw for white.
• Suggested change A player may choose to pass.

Proof If white has a winning move make
it. Otherwise choose to pass. By the symmetry
Black is faced with the same position white just
had Since white had no winning move before,
Black has none now. Hence, black at best can
draw
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Evaluation function - f(P)

• Sometimes a simple evaluation function can be
  applied to any position P to determine whether it
  is a won/lost/drawn position
• Example The game of Nim
• If number of ones in each column binary
  representation) is even, the player about to move
  looses

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Evaluation function - f(P)

• It is possible, in principle, to play a perfect
  game.
• However
• In typical chess position there are 30-40 legal
  moves.
• One move for white, and one for black yields 103
  possibilities
• Typical game lasts 40 moves to resignation of
  one party
• Need to check 10120 variations infeasible
• It is easy to play a trivial game not
  interesting.

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Approximating Evaluation Functions

• The relative values of queen, rook, bishop,
  knight and pawn are about 9, 5, 3, 3,1,
  respectively.
• Mobility - Rooks should be placed on open files.
• Backward, isolated and doubled pawns are weak.
• An exposed king is a weakness (until the end
  game).

200(K-K') 9(Q-Q') 5(R-R') 3(B-B'N-N')
(P-P') - 0.5(D-D'S-S'I-I') 0.1(M-M') ...
f(P)
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Two Players Games

• One Search Tree for both Players
• Even layers Max Player move
• Odd Layers Min Player move
• The state evaluated according to evaluation
  function.

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MinMax search strategy

• Generate the whole game tree. (Or up to a
  constant depth)
• Evaluate Terminal states (Leafs)
• propagate Min-Max values up from leafs
• Search for MAX best next move, so that no matter
  what MIN does MAX will be better off

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22
-1
23
1
-1
24
1
-2
-1
25
1
1
-2
-1
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MinMax Complexity

• For branching factor b and depth search d the
  complexity is O(bd)

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Cuting Off Search

• We want to prune the tree stop exploring
  subtrees with values that will not influence the
  final MinMax root decision

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Alpha Beta Simple Example
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Alpha-Beta search cutoff rules

• Keep track and update two values so far
• alpha is the value of best choice in the MAX path
  
• beta is the value of best choice in the MIN path
• Rule do not expand node n when beta lt
  alpha
• for MAX node return beta
• for MIN node return alpha

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Alpha-Beta Prunning

• function Max-Value(state,game,alpha,beta) returns
  
• minmax value of state
• if Cutoff-Test(state) then return Eval(state)
• for each s in Successor(state) do
• alpha Max(alpha,Min-Value(s, game,alpha,beta)
• if beta lt alpha then return beta
• end return alpha
• function Min-Value(state,game,alpha,beta) returns
  
• minmax value of state
• if Cutoff-Test(state) then return Eval(state)
• for each s in Successor(state) do
• beta Min(beta,Max-Value(s, game,alpha,beta)
• if beta lt alpha then return alpha
• end return beta

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Complexity

• In the worst case, no cut-off. The complexity is
  O(bd). In practice, O(bd/2), with branching
  factor of b1/2 instead of b.

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Type A Strategy

• Consider all positions down the min-Max, up to a
  fixed depth d.
• For each position in depth d, use some evaluation
  function to estimate the position
• Possible to implement using only 3000 bits !
• Simple and ineffective

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Type A Strategy - Backdraws

• Type A Strategy is Slow Weak
• 109 evaluations needed after 3 moves (6 plies)
• Evaluates positions at unreliable states
• Rather limited thinking
• World champion can construct (at best) 15-20
  moves deep

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Human Strategies

• Studied experimentally by De Groot, 1946.
• showed various typical positions to chess masters
• In certain case the chess master examined sixteen
  variations, ranging in depth from 1/2 (one Black
  move) to 4-1/2 (five Black and four White) moves.
• The total number of positions considered was 44.

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Type B Strategy

• Examine forceful variations out as far as
  possible and evaluate only at reasonable
  positions, where some quasi-stability
  (quiescence) has been established.
• Select the variations to be explored by some
  process so that the machine does not waste its
  time in totally pointless variations.

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Quiescence Positions

• Define a function g(P) of a position which
  determines whether approximate stability exists
• g(P) 1 ,
• g(P) 0 , Otherwise.
• Variations can be explored until g(P)0
• Making sure we check at least 2 moves and at most
  10 moves

• any piece is attacked by a piece of lower value
  or by more pieces than defenses, or
• any check exists on a square controlled by the
  opponent

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h(P,M) Is this move essential?

• For position P, and move M evalutate decide
  whether it is worth exploring
• High value
• Check
• Captures
• Attack on major pieces
• Medium value
• Developing moves
• Defensive moves
• Low value - other moves
• should NOT eliminate moves that merely look bad
  at first sight (piece en prise)

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Further Developments..

• Opening Book
• Statistical variation for position P,
• create some distribution and play by it
• Positional vs. Combination Player
• Endgame Solver
• Iterative Deepening

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Recent Achivements
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References

• Claude E. Shannon , Programming a Computer
  Playing Chess, Philosophical Magazine, Ser. 7,
  Vol 41, No. 312 March 1950
• The F.I.D.E. Laws of Chess, http//www.chessvarian
  ts.com/fidelaws.html
• S. Russel, P. Norvig, Artificial Intelligence A
  Modern Approach, Prentice Hall, 1995.
• Lecture notes of Workshop in Reinforcement
  Learnning 2003/4 by Prof. Yishay Mansour, TAU
• ChessBase, A short history of computer chess,
  http//www.chessbase.com/columns/column.asp?pid10
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