Searching over Metapositions in Kriegspiel Andrea Bolognesi and - - PowerPoint PPT Presentation
Searching over Metapositions in Kriegspiel Andrea Bolognesi and Paolo Ciancarini {abologne,ciancarini}@cs.unibo.it Dipartimento di Scienze dellInformazione University of Bologna - Italy Searching over Metapositions in Kriegspiel p.1/34
Andrea Bolognesi and Paolo Ciancarini
{abologne,ciancarini}@cs.unibo.it
Dipartimento di Scienze dell’Informazione University of Bologna - Italy
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I II R
✆ ✆ ✆ ✝ ✞ ✆ ✆ ✆ ✟ ✠ ✆ ✆ ✆ ✝ ✡ ✆ ✆ ✆ ✟ ☛ ✆ ✆ ✆ ✝ ☞ ✆ ✆ ✆ ✟ ✌ ✆ ✆ ✆ ✝ ✍ ✆ ✆ ✆ ✟ ✎ ✏ ✑ ✆ ✆ ✝ ✞ ✒ ✓ ✆ ✆ ✟ ✠ ✔ ✑ ✆ ✆ ✝ ✡ ✕ ✓ ✆ ✆ ✟ ☛ ✖ ✑ ✆ ✆ ✝ ☞ ✗ ✓ ✆ ✆ ✟ ✌ ✘ ✑ ✆ ✆ ✝ ✍ ✙ ✓ ✆ ✆ ✟ ✎ ✏ ✑ ✆ ✆ ✆ ✒ ✓ ✆ ✆ ✆ ✔ ✑ ✆ ✆ ✆ ✕ ✓ ✆ ✆ ✆ ✖ ✑ ✆ ✆ ✆ ✗ ✓ ✆ ✆ ✆ ✘ ✑ ✆ ✆ ✆ ✙ ✓ ✆ ✆ ✆Searching over Metapositions in Kriegspiel – p.4/34
Tipografia Senese, Siena,1980.
Mathematical Gardener, pag 28-36. Prindle, Weber & Smith, 1981.
Theoretical Computer Science, 96:389-403, 1992.
UCLA, 1995.
Uncertainty: A Rational Approach to Kriegspiel. In J. van den Herik and J. Uiterwijk editors, Advances in Computer Chess 8, 1997.
a Transposition Table. ICCA Journal, 23(4):218-229, 2000.
Lloyd Shapley. 64-Chess Review, #10, pages 58-59, 1999.
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to exploit current computer chess knowledge and technology
progressing to victory knowing nothing about the opponent’s position
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SKIP
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8
✶ ✶ ✶ ✷ ✶7
✶ ✶ ✶ ✸6
✶ ✶ ✶ ✶5
✶ ✶ ✶ ✹4
✶ ✶ ✶ ✶3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✷ ✶ ✶ ✶ ✶7
✶ ✶ ✶ ✸6
✶ ✶ ✶ ✶5
✶ ✶ ✶ ✺ ✶ ✺4
✶ ✶ ✹ ✺ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h
(×5)
8
✶ ✶ ✻ ✶7
✶ ✶ ✶ ✸6
✶ ✶ ✶ ✶5
✶ ✶ ✶ ✶ ✺4
✶ ✶ ✶ ✺ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✶ ✶ ✶ ✻7
✶ ✶ ✶ ✸6
✶ ✶ ✶ ✶5
✶ ✶ ✶ ✺ ✶4
✶ ✶ ✹ ✺ ✶3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✶ ✶ ✸ ✷ ✶7
✶ ✶ ✶ ✶6
✶ ✶ ✹ ✹5
✶ ✶ ✶ ✺ ✶ ✺4
✶ ✶ ✹ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✶ ✶ ✶ ✷ ✸7
✶ ✶ ✶ ✶6
✶ ✶ ✹ ✹5
✶ ✶ ✶ ✺ ✶ ✺4
✶ ✶ ✹ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✶ ✶ ✶ ✷ ✶7
✶ ✶ ✶ ✼ ✶6
✶ ✶ ✶ ✹5
✶ ✶ ✶ ✺ ✶ ✺4
✶ ✶ ✹ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h 8
✶ ✶ ✶ ✷ ✶7
✶ ✶ ✶ ✶ ✼6
✶ ✶ ✹ ✶5
✶ ✶ ✶ ✺ ✶ ✺4
✶ ✶ ✹ ✹3
✶ ✶ ✶ ✶2
✶ ✶ ✶ ✶1
✶ ✶ ✶ ✶a b c d e f g h
✽a8 (×5)
✽f8
✽h8
✾f8
✾h8
✾f7
✾h7
h4 h5
f4 g4
h5 f4 h4 f6 h6
h5 f4 h4 f6 h6
h4 f5 h5 h6
h4 f5 h5 f6
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f8 Black White White
❀g4
❀h5
❀h4
❁x7
✿x14 x
❁x7
✿x14 y
❁x5
✿x14 z
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f8 Black White White
❃g4∧
❃h4
❃h5
❄x7
❂x14 x′ ≃ min(x, y)
❄x5
❂x14 z
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f8 Black White White
❆g4∧
❆h4
❆h5
❇x7
❅x14 x′ ≃ min(x, y)
❇x5
❅x14 z
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x7
➌x14 pseudomoves
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x7 . . . . . . .
➥x14 pseudomoves S C I S C I x y z x′ y′ z′
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x7 . . . . . . .
➷x14 pseudomoves S C I S C I x ≤ y ≤ z x′ ≥ y′ ≤ z′
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8
➬ ➮ ➬ ➮ ➬ ➱ ➬ ➮7
➮ ➬ ➮ ➬ ➮ ➮ ➬6
➬ ➱ ➱ ➬ ➱ ➬ ➮5
➮ ✃ ➮ ➮ ➬4
➬ ➱ ➱ ➬ ➱ ➬ ➮3
➱ ➱ ➱ ❐ ➱2
➬ ➮ ➬ ➮ ➬ ➱ ➬ ➮1
➮ ➬ ➮ ➬ ➮ ➮ ➬a b c d e f g h
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8
❒ ❮ ❒ ❮ ❰ ❒ ❮7
❮ ❒ ❮ ❒ ❰ ❮ ❒6
❒ ❰ ❰ ❰ ❒ ❮5
❮ Ï ❰ ❮ ❒4
❒ ❰ ❰ ❰ ❒ ❮3
❰ ❰ Ð ❰2
❒ ❮ ❒ ❮ ❰ ❒ ❮1
❮ ❒ ❮ ❒ ❰ ❮ ❒a b c d e f g h
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8
Ñ Ò Ñ Ò Ó Ñ Ò7
Ò Ñ Ò Ñ Ó Ò Ñ6
Ñ Ó Ó Ó Ñ Ò5
Ò Ô Ó Ò Ñ4
Ñ Ó Ó Ó Ñ Ò3
Ó Ó Õ Ó2
Ñ Ò Ñ Ò Ó Ñ Ò1
Ò Ñ Ò Ñ Ó Ò Ña b c d e f g h
8
Ö × Ö × Ö Ø Ö ×7
× Ö × Ö × × Ö6
Ö Ø Ø Ö Ø Ö ×5
× Ù × × Ö4
Ö Ø Ø Ö Ø Ö ×3
Ø Ø Ø Ú Ø2
Ö × Ö × Ö Ø Ö ×1
× Ö × Ö × × Öa b c d e f g h
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8
Û Ü Û Ü Ý Û Ü7
Ü Û Ü Û Ý Ü Û6
Û Ý Ý Ý Û Ü5
Ü Þ Ý Ü Û4
Û Ý Ý Ý Û Ü3
Ý Ý ß Ý2
Û Ü Û Ü Ý Û Ü1
Ü Û Ü Û Ý Ü Ûa b c d e f g h
8
à à á à à7
à à â à6
à à á à à5
à ã â à4
à à á à à3
à à ä à2
à à á à à1
à à â àa b c d e f g h
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8
å æ å æ ç å æ7
æ å æ å ç æ å6
å ç ç ç å æ5
æ è ç æ å4
å ç ç ç å æ3
ç ç é ç2
å æ å æ ç å æ1
æ å æ å ç æ åa b c d e f g h
8
ê ê ë ê ê7
ê ê ì ê6
ê ê ë ê ê5
ê í ì ê4
ê ê ë ê ê3
ê ê î ê2
ê ê ë ê ê1
ê ê ì êa b c d e f g h
8
ï ð ï ð ï ñ ï ð7
ð ï ð ï ð ð ï6
ï ñ ñ ï ñ ï ð5
ð ò ð ð ï4
ï ñ ñ ï ñ ï ð3
ñ ñ ñ ó ñ2
ï ð ï ð ï ñ ï ð1
ð ï ð ï ð ð ïa b c d e f g h
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8
ô õ ô õ ö ô õ7
õ ô õ ô ö õ ô6
ô ö ö ö ô õ5
õ ÷ ö õ ô4
ô ö ö ö ô õ3
ö ö ø ö2
ô õ ô õ ö ô õ1
õ ô õ ô ö õ ôa b c d e f g h
8
ù ù ú ù ù7
ù ù û ù6
ù ù ú ù ù5
ù ü û ù4
ù ù ú ù ù3
ù ù ý ù2
ù ù ú ù ù1
ù ù û ùa b c d e f g h
8
þ þ þ þ7
þ þ þ þ6
þ þ ÿ þ þ5
þ4
þ þ ÿ þ þ3
þ þ þ ✂ þ2
þ þ þ þ1
þ þ þ þa b c d e f g h
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,−526
✍g8 . . .
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313,313 . . . 272,272
✕g1
✕g2
✕g8 . . . ,−526
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313,313 . . . 272,272
✜g1
✜g2
✜g8 . . . −213,−526
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,262
✢ ✢ ✢ ✧ ✢ ✤ ✥ ✤ ✥ ✤ ✥ ✢ ✥ ✤ ✥ ✤ ✥ ✤ ✢ ✤ ✥ ✤ ✥ ✤ ✥ ✢ ✥ ✤ ✥ ✤ ✥ ✤ ✢ ✤ ✥ ✤ ✥ ✤ ✥ ✢ ✥ ✤ ✥ ✤ ✥ ✤ ✢ ✤ ✥ ✤ ✥ ✤ ✥ ✢ ✦313,313 . . . 272,272
★g1
★g2
★h2
★g8 . . . . . . . . . −213,−526
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313,313 . . . 272,272
✯g1
✯g2 263,263 . . . 272,272
✰g2
✯g2
✯h2
✯g8 . . . . . . . . . −213,−526 534,262
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313 . . . 272
✷g1
✷g2 263 . . . 272
✸g2
✷g2
✷h2
✷g8 . . . . . . . . . 313 272
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4
i=1 qi where
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63
i=0 v[i], where v is a numerical
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1 1 1 1 1 1 −2 −4 −4 −2 −4 −4 −4 −4 −4 −4 −4 −4 −2 −4 −4 −2 1 1 1 1 1 1
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8
✹ ✺ ✹ ✻ ✹ ✺ ✹ ✻7
✻ ✹ ✺ ✺ ✹ ✻ ✹6
✹ ✻ ✹ ✻ ✹ ✻ ✹ ✺5
✺ ✹ ✻ ✻ ✹ ✺ ✹4
✻ ✼ ✻ ✻3
✺ ✹ ✻ ✻ ✹ ✺ ✹2
✹ ✻ ✹ ✻ ✻ ✹ ✺1
✻ ✹ ✺ ✽ ✻ ✹a b c d e f g h 8
✻ ✻ ✻ ✻7
✻ ✻ ✻ ✻6
✻ ✻ ✻ ✻5
✻ ✻ ✻ ✻4
✻ ✼ ✻ ✻3
✻ ✻ ✻ ✻2
✻ ✻ ✻ ✻1
✻ ✻ ✽ ✻a b c d e f g h
a0 a1 a2 a3
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63
i=0 b[i], where b is a numerical
−10 −50 −100 −100 −50 −10 −10 −10 −40 −40 −40 −40 −10 −10 −50 −40 −40 −40 −40 −40 −40 −50 −100 −40 −40 −50 −50 −40 −40 −100 −100 −40 −40 −50 −50 −40 −40 −100 −50 −40 −40 −40 −40 −40 −40 −50 −10 −10 −40 −40 −40 −40 −10 −10 −10 −50 −100 −100 −50 −10
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6
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8
✾ ✾ ✾ ✾7
✾ ✿ ✾ ❀ ✾6
✿ ❁ ❂ ✾ ❀ ✾5
✾ ❁ ✾ ❂ ✾4
✾ ❂ ✾ ❁ ✾3
✾ ❀ ✾ ❂ ❁ ✿2
✾ ❀ ✾ ✿ ✾1
✾ ✾ ✾ ✾a b c d e f g h 8
❃ ✾ ✾ ✾ ✾7
❄ ✿ ✾ ✾6
✾ ✾ ✾ ✾5
✾ ✾ ✾ ✾4
✾ ❂ ✾ ✾ ✾3
✾ ✾ ✾ ✾2
✾ ✾ ✾ ✾1
✾ ✾ ✾ ✾a b c d e f g h 8
❃ ✾ ✾ ✾ ✾7
✾ ✿ ✾ ✾6
✾ ✾ ✾ ✾5
✾ ❁ ✾ ✾4
✾ ❂ ✾ ✾ ✾3
✾ ✾ ✾ ✾2
✾ ✾ ✾ ✾1
✾ ✾ ✾ ✾a b c d e f g h 8
❃ ✾ ✾ ✾ ✾7
✾ ✿ ✾ ✾6
✾ ✾ ✾ ✾5
✾ ❁ ❂ ✾ ✾4
✾ ✾ ✾ ✾3
✾ ✾ ✾ ✾2
✾ ✾ ✾ ✾1
✾ ✾ ✾ ✾a b c d e f g h
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