Dr' Lewis Stiller

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Computer Endgame Analysis

• Dr. Lewis Stiller

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HEURISTICS VERSUS PERFECTION

• Heuristics Tools to simplify real-life problems
• Weather prediction
• Game-playing
• Stock-market analysis
• Theorem proving
• Language understanding
• Perfection Only attainable in fairly small
  domains.

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Overview

• History
• Method
• Results
• Conclusion

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History
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Historical Human

• Al-Adli - 9th century
• Carrera - 1617
• Philidor - 1749
• Lolli - 1763
• Horwitz and Kling - 1851
• Crosskill - 1864
• Molien - 1893
• Amelung ca. 1901
• Troitsky - 1934

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AL-ADLI 9TH CENTURY A.D.
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Historical Computer

• Quevedo - 1890, 1915
• Zermelo - 1913
• Bellman - 1965
• Strohlein - 1970
• Komissarchik and Futer - 1974
• Herik, Herschberg - 1980s and 90s
• Thompson - 1986
• Michie and Bratko - 1987
• Stiller - 1989
• Stiller - 1991

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QUEVEDOS ROBOT
Torres believes that the limit has by no means
been reached on what automatic machinery will do,
and in substantiation of his opinions presents
his automatic chess-playing machine Scientific
American Suppl. 1915
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DEUTSCHE SCHACHZEITUNG Approx. 1902
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REPRESENTATIONS (1897)
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Read a hundred novels in a language, and you will
know that language
THE MOLIENS
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TOMSK MANUSCRIPT
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Method
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Endgame computation
Set of chess pieces
Endgame algorithm
All winning positions using those pieces
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Algorithm overview
White wins in 1
White wins in 2
White wins in 3
White wins in 4
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Backup
Complement
White to play and win within j moves
Black unmove
White cannot win within j moves
Complement
Black can avoid loss for at least j moves
White unmove
Black loses within j moves
White wins within j1 moves
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HYPERBOARD
(6,3)
(2,3)
(2,3,6,3)
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HYPERBOARD
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OPERATOR
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HYPERBOARD OPERATOR
(4,3,6,3)
(5,3,6,3)
(3,3,6,3)
(2,3,6,3)
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HYPERBOARD UNMOVE ROOK RIGHT
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Unmove-rook-right
Set of positions
Marbles in hyperboard
Holes in hyperboard
Tilt hyperboard
Get points rolled over
New set of positions
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Unmove rook up
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Unmoving pieces

• Rook Tilt up, right, down, left
• Bishop Tilt diagonally
• Queen Rook Bishop
• Knight Ignore holes
• King Tilt only for one time unit

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ORBIT UNDER D4
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D4 ACTION ON HYPERBOARD 8-ELEMENT ORBIT
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QUOTIENT SPACE
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Unpromotion
KQ v. K
KP v. K
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Implementation

• Reduced problem to manipulation of boolean arrays

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3-Dimensional Hypercube
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4-Dimensional Hypercube
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5-Dimensional Hypercube
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16-Dimensional Hypercube
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Results
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Stiller results

• Five-piece games
• Previous 2 weeks (Thompson)
• 2 minutes
• Six-piece games

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Win versus Mate metric

• WTM and mate in 5
• WTM and win in 5
• Win Mate or winning capture or winning pawn move
• If WTM and win in k then WTM and mate in kgtk

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WTM and win in 223
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WTM and Win in 243
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NUMBER OF MOVES REQUIRED TO WIN CERTAIN
ENDGAMES
Maximum value of the max-to-win over all
positions with White to move and win in the given
endgame.
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6-PIECE MAX-TO-WINS
Maximum value of the max-to-win over all
positions with White to move and win in the given
endgame.
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KARPOV-KASPAROV TILBURG,1991
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Mutual Zugzwangs

• White to play and draw Black to play and lose
• Leads to elegant studies

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Whites turn White drawsBlacks turn White wins
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Whites turn White drawsBlacks turn White wins
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Study by N. Elkies
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Results
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Conclusion
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Conclusions

• Chess history Theodor Molin founded two fields
  at about the same time group representation
  theory and numerical analysis of chess endgames
  (before spending the rest of his life in
  Siberia).
• Deep and surprising results can lurk in the
  simplest settings.

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ACKNOWLEDGMENTS

• N. Elkies, S. Kasif, H. Berliner, D. Waltz, B.
  Wendroff.
• Computing facilities were provided by the
  Advanced Computing Laboratory of the Los Alamos
  National Laboratory, Los Alamos NM 87545. This
  work was also supported in part by the Army
  Research Office under Contract Number
  DAALO3-C-0038 with the University of Minnesota
  Army High Performance Computing Research Center
  (AHPCRC) and the DoD Shared Resource Center at
  AHPCRC. The Author was partly supported under a
  National Defence Science and Engineering Graduate
  Fellowship, U.S. Army Grant DAAL03-929G-0345.
• Alpha Server Computing Facilities provided by
  Digital Equipment Corporation the Author thanks
  D. Hunter and T. Mann of Digital Equipment
  Corporation.