Searching over Metapositions in Kriegspiel – p.8/34 PaoloC-Krieg, ICC Aprile 2003 ✤ ✴ ✢ ✤ ✳ ✂ ✲ ✤ ✱ ✰ ✜ ✯ ✤ ✮ ✛ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✤ ✴ ✢ ✤ ☎ ✳ ✂ ✬ ✲ ✤ ☎ ✱ ✄ ✰ ✜ ☎ ✯ ✫ ✤ ✮ ✛ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ 6.h4 ✁ ✴ ☎ ✳ ✬ ✲ ☎ ✱ ✄ ✰ ☎ ✯ ✫ ✮ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪
Searching over Metapositions in Kriegspiel – p.8/34 PaoloC-Krieg, ICC Aprile 2003 ✤ ✴ ✢ ✤ ✳ ✂ ✲ ✤ ✱ 6. ..Qe7 ✰ ✜ ✯ ✤ ✮ ✛ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✤ ✴ ✢ ✤ ☎ ✳ ✂ ✬ ✲ ✤ ☎ ✱ ✄ ✰ ✜ ☎ ✯ ✫ ✤ ✮ ✛ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✴ ☎ ✳ ✬ ✲ ☎ ✱ ✄ ✰ ☎ ✯ ✫ ✮ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪
Searching over Metapositions in Kriegspiel – p.8/34 PaoloC-Krieg, ICC Aprile 2003 ✤ ✴ ✢ ✤ ✳ ✂ ✲ ✤ ✱ ✰ ✜ ✯ ✤ ✮ ✛ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✤ ✴ ✢ ✤ ☎ ✳ ✂ ✬ ✲ ✤ ☎ ✱ ✄ ✰ ✜ ☎ ✯ ✫ ✤ ✮ ✛ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ 7.b4 ✁ ✴ ☎ ✳ ✬ ✲ ☎ ✱ ✄ ✰ ☎ ✯ ✫ ✮ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪
PaoloC-Krieg, ICC Aprile 2003 ✛ ✜ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✤ ✦ ☎ ☎ ☎ ✧ ★ ☎ ✩ ✄ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 7. ..h5 one pawn try ✛ ✜ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✤ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ☎ ✩ ✩ ✫ ✄ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
Searching over Metapositions in Kriegspiel – p.8/34 PaoloC-Krieg, ICC Aprile 2003 ✤ ✴ ✢ ✤ ✳ ✂ ✲ ✤ ✱ ✰ ✜ ✯ ✤ ✮ ✛ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✤ ✴ ✢ ✤ ☎ ✳ ✂ ✬ ✲ ✤ ☎ ✱ ✄ ✰ ✜ ☎ ✯ ✫ ✤ ✮ ✛ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ 8.a4 ✁ ✴ ☎ ✳ ✬ ✲ ☎ ✱ ✄ ✰ ☎ ✯ ✫ ✮ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪
PaoloC-Krieg, ICC Aprile 2003 ✛ ✜ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✤ ✧ ★ ✩ ✄ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 8. ..h × g4 pawn capture in g4 ✛ ✜ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✤ ✧ ✧ ★ ★ ✩ ✩ ✫ ✄ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✜ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ✄ ☎ ☎ ☎ ✧ ★ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 9.Q × g4 pawn capture in g4 ✛ ✜ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✄ ☎ ☎ ☎ ✧ ✧ ★ ★ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ✄ ☎ ☎ ☎ ✧ ★ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 9. ..Bd7 ✛ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✄ ☎ ☎ ☎ ✧ ✧ ★ ★ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 10.Qe2 ✛ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 10. ..a5 ✛ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 11.b × a5 pawn capture in a5 ✛ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 11. ..R × a5 pawn capture in a5 ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 12.Bd2 ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✜ ✣ ✛ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✫ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 12. ..Ra6 ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✛ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✫ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✜ ✣ ✛ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 13.Nc3 ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✛ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✜ ✣ ✛ ✤ ✤ ✥ ✦ ☎ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 13. ..Qe8 ✵ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✛ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
Searching over Metapositions in Kriegspiel – p.8/34 PaoloC-Krieg, ICC Aprile 2003 ✴ ✢ ✤ ✳ ✂ ✲ ✵ ✤ ✱ ✜ ✰ ✯ ✤ ✮ ✛ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✴ ✢ ✤ ✳ ✂ ✫ ✬ ✲ ✵ ✄ ✤ ☎ ✱ ✜ ✰ ☎ ✯ ✤ ✮ ✛ ☎ ✭ 14.Nf3 ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✁ ✴ ✳ ✫ ✬ ✲ ✄ ☎ ✱ ✰ ☎ ✯ ✮ ☎ ✭ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ✛ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 14. ..R × a4 pawn capture in a5 ✵ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✛ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✜ ✣ ✤ ✤ ✥ ✦ ✫ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 15.N × a4 piece capture in a4 ✵ ✢ ✚ ✚ ✂ ✤ ✜ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✫ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ✜ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 15. ..B × a4 piece capture in a4 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✜ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ✁ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 16.R × a4 piece capture in a4 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✁ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ✵ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 16. ..Q × a4 piece capture in a4 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✵ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ✵ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 17.Ba5 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✵ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 17. ..Qe8 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 18.Bc3 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 18. ..e5 one pawn try ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 19.d × e5 pawn capture in e5 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 19. ..d × e5 pawn capture in e5 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 20.f × e5 pawn capture in e5 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ✫ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 20. ..f × e5 pawn capture in e5 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✫ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✵ ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 21.N × e5 check by knight; pawn capture in e5 ✵ ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 21. ..Q × e5 piece capture in e5 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 22.B × e5 piece capture in e5 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 22. ..Na6 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 23.Bc3 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 23. ..Bb4 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 24.Kd2 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ★ ✄ ✩ ✬ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 24. ..Bf8 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✬ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✢ ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 25.Bh3 ✢ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 25. ..Ne7 ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✦ ☎ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 26.Rf1 check on file ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✤ ✥ ✢ ✦ ☎ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 26. ..Nf5 one pawn try ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ✢ ✦ ✦ ☎ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✤ ✥ ☎ ✦ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 27.e × f5 piece capture in f5 ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✤ ✥ ✥ ☎ ✦ ✦ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✥ ✤ ✦ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 27. ..g × f5 pawn capture in f5 ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✥ ✥ ✤ ✦ ✦ ☎ ✧ ✧ ✬ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✚ ✂ ✤ ✣ ✢ ✥ ✬ ✦ ☎ ✧ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 28.B × f5 pawn capture in f5 ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✥ ✥ ✬ ✦ ✦ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✚ ✂ ✤ ✣ ✢ ✥ ✬ ✦ ☎ ✧ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 28. ..Rg8 ✛ ✚ ✚ ✂ ✤ ✣ ✣ ✢ ✥ ✥ ✬ ✦ ✦ ☎ ✧ ✧ ★ ★ ✄ ✩ ✩ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 ✛ ✚ ✂ ✤ ✣ ✢ ✥ ✦ ☎ ✧ ✬ ★ ✄ ✩ ✁ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ 29.Bd3 # 1-0 checkmate ✛ ✛ ✚ ✚ ✂ ✂ ✤ ✤ ✣ ✣ ✢ ✢ ✥ ✥ ✦ ✦ ☎ ☎ ✧ ✧ ✬ ✬ ★ ★ ✄ ✄ ✩ ✩ ✁ ✁ ✪ ✪ ✲ ✲ ✮ ✰ ✴ ✮ ✰ ✴ ✭ ✯ ✱ ✭ ✯ ✱ ✳ ✳ Searching over Metapositions in Kriegspiel – p.8/34
PaoloC-Krieg, ICC Aprile 2003 Searching over Metapositions in Kriegspiel – p.8/34
Example of position (1/3) ✁ ✚ ✣ ✥ ✦ ✧ ★ ✩ ✪ ✮ ✰ ✲ ✴ ✭ ✯ ✱ ✳ • This diagram shows an example of position during a game of ending; � ✂ ✁ • the game tree has a branch of 11 moves, corresponding to the possible White’s moves, • plus the Black’s ones. Searching over Metapositions in Kriegspiel – p.9/34
Example of position (2/3) • Black’s choices compose the information sets for White, who does not know where Black has moved his pieces; • In the example seen before we have to deal with 33 information sets for White and 53 positions in total ✶ ✶ ✶ ✶ ✷ 8 ✶ ✶ ✶ ✸ 7 ✶ ✶ ✶ ✶ 6 ✶ ✶ ✶ ✹ 5 ✽ ✶ ✶ ✶ ✶ a8 ( × 5) 4 ✾ h7 ✶ ✶ ✶ ✶ 3 ✶ ✶ ✶ ✶ ✽ ✾ 2 f8 f7 ✶ ✶ ✶ ✶ 1 ✽ ✾ a b c d e f g h h8 h8 ✾ f8 g4 h4 f4 f5 h5 g4 g4 h4 f6 h6 h4 f6 h6 h4 f5 h5 h6 h4 f5 h5 f6 h4 h5 f5 f5 f5 f4 f4 h5 f4 h5 f4 f4 � �� � � �� � � �� � � �� � � �� � � �� � � �� � ✶ ✶ ✶ ✶ ✶ ✶ ✻ ✶ ✶ ✶ ✶ ✻ ✶ ✶ ✸ ✶ ✶ ✶ ✶ ✸ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷ ✷ ✷ ✷ ✷ 8 8 8 8 8 8 8 ✶ ✶ ✶ ✸ ✶ ✶ ✶ ✸ ✶ ✶ ✶ ✸ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✼ ✼ 7 7 7 7 7 7 7 ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✹ ✹ ✶ ✶ ✹ ✹ ✶ ✶ ✶ ✹ ✶ ✶ ✹ ✶ 6 6 6 6 6 6 6 ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ 5 5 5 5 5 5 5 ✶ ✶ ✹ ✹ ✶ ✶ ✶ ✹ ✶ ✶ ✹ ✶ ✶ ✶ ✹ ✹ ✶ ✶ ✹ ✹ ✶ ✶ ✹ ✹ ✶ ✶ ✹ ✹ ✺ ✺ ✺ 4 4 4 4 4 4 4 ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ 3 3 3 3 3 3 3 ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ 2 2 2 2 2 2 2 ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ 1 1 1 1 1 1 1 a b c d e f g h a b c d e f g h a b c d e f g h a b c d e f g h a b c d e f g h a b c d e f g h a b c d e f g h ( × 5) Searching over Metapositions in Kriegspiel – p.10/34
Example of position (3/3) Considering White moves f8: ✁ • Black’s possible moves are h5, h4 or g4; ✛ ✛ ✛ • White now faces two information sets of 2 and 1 elements respectively: • The former includes 21 possible moves, • the latter includes 19 moves • the game tree branches out with 2 · 21 + 19 = 61 further nodes We aim to obtain a tree without information sets with more than an element. Searching over Metapositions in Kriegspiel – p.11/34
Using metapositions We could join the moves made by Black which lead to the same information set: ✿ f8 Black ❀ ❀ g4 h5 ❀ h4 White White ❁ ❁ ❁ x7 x7 x5 ✿ ✿ ✿ x14 x14 x14 y x z Searching over Metapositions in Kriegspiel – p.12/34
Using metapositions We could join the moves made by Black which lead to the same information set: ❂ f8 Black ❃ ❃ ❃ g4 ∧ h4 h5 White White ❄ ❄ x7 x5 ❂ ❂ x14 x14 x ′ ≃ min ( x, y ) z Searching over Metapositions in Kriegspiel – p.12/34
Using metapositions We could join the moves made by Black which lead to the same information set: ❅ f8 Black ❆ ❆ ❆ g4 ∧ h4 h5 White White ❇ ❇ x7 x5 ❅ ❅ x14 x14 x ′ ≃ min ( x, y ) z Because of the complexity of a procedure which distinguishes between Black’s moves that lead to different information sets for White we define the game tree in a simpler but equivalent Searching over Metapositions in Kriegspiel – p.12/34
Metapositions • Metaposition : is a particular position which describes a set of positions. • Metamove : a move by Black player from one metaposition to another. • Pseudomove : a move by White player on a metaposition. ❊ ❊ ❊ ❊ ❉ ❈ ❊ ❊ ❊ ❊ ● ❋ ❊ ❊ ❊ ❊ ❍ ❊ ❊ ❊ ❏ ■ ❊ ❊ ❊ ❊ ❑ ❊ ❊ ❊ ❊ ▲ ❊ ❊ ❊ ❊ ◆ ▼ ❊ ❊ ❊ ❊ ❖ ◗ ❙ ❯ ❲ P ❘ ❚ ❱ Searching over Metapositions in Kriegspiel – p.13/34
Metapositions • Metaposition : is a particular position which describes a set of positions. • Metamove : a move by Black player from one metaposition to another. • Pseudomove : a move by White player on a metaposition. ❩ ❩ ❩ ❩ ❨ ❳ ❩ ❩ ❩ ❩ ❭ ❬ ❩ ❩ ❫ ❫ ❴ ❪ ❩ ❩ ❩ ❩ ❴ ❴ ❵ ❩ ❩ ❫ ❫ ❴ ❛ ❩ ❩ ❩ ❫ ❴ ❴ ❜ ❩ ❩ ❫ ❫ ❝ ❩ ❩ ❩ ❫ ❴ ❴ ❞ ❢ ❤ ❥ ❧ ❡ ❣ ✐ ❦ Searching over Metapositions in Kriegspiel – p.13/34
Metapositions • Metaposition : is a particular position which describes a set of positions. • Metamove : a move by Black player from one metaposition to another. • Pseudomove : a move by White player on a metaposition. ♥ ♥ ♥ ♥ ♠ ♥ ♥ ♥ ♥ ♣ ♦ ♥ ♥ r r s q ♥ ♥ ♥ ♥ s s t ♥ ♥ ♥ ♥ ✈ ✉ ♥ ♥ ♥ r s s ✇ ♥ ♥ r r ① ♥ ♥ ♥ r s s ② ④ ⑥ ⑧ ⑩ ③ ⑤ ⑦ ⑨ Searching over Metapositions in Kriegspiel – p.13/34
Cardinality of pseudomoves • The set P of pseudomoves has a cardinality equal to the maximum cardinality of legal moves for each position. | P | = max( | legal moves | ) • For the example seen before White always considers 21 pseudomoves, even if he is in the case with 19 legal moves, because he cannot distinguish between the two situations. Searching over Metapositions in Kriegspiel – p.14/34
The game tree ❷ ❷ ❸ ❷ ❶ ❷ ❷ ❷ ❺ ❹ ❷ ❷ ❷ ❷ ❻ ❷ ❷ ❷ ❷ ❽ ❼ ❷ ❷ ❷ ❿ ❽ ❾ ❷ ❷ ❷ ❷ ➀ ❷ ❷ ❷ ❷ ➁ ❷ ❷ ❷ ❷ ➂ ➄ ➆ ➈ ➊ ➃ ➅ ➇ ➉ ➋ x7 ➌ x14 pseudomoves Searching over Metapositions in Kriegspiel – p.15/34
The game tree ➎ ➎ ➏ ➎ ➍ ➎ ➎ ➎ ➑ ➐ ➎ ➎ ➎ ➎ ➒ ➎ ➎ ➎ ➎ ➔ ➓ ➎ ➎ ➎ ➣ ➔ → ➎ ➎ ➎ ➎ ↔ ➎ ➎ ➎ ➎ ↕ ➎ ➎ ➎ ➎ ➙ ➜ ➞ ➠ ➢ ➛ ➝ ➟ ➡ ➤ x7 ➥ x14 pseudomoves . . . . . . . S C I S C I y x z y ′ x ′ z ′ Searching over Metapositions in Kriegspiel – p.15/34
The game tree ➧ ➧ ➨ ➧ ➦ ➧ ➧ ➧ ➫ ➩ ➧ ➧ ➧ ➧ ➭ ➧ ➧ ➧ ➧ ➲ ➯ ➧ ➧ ➧ ➵ ➲ ➳ ➧ ➧ ➧ ➧ ➸ ➧ ➧ ➧ ➧ ➺ ➧ ➧ ➧ ➧ ➻ ➽ ➚ ➶ ➘ ➼ ➾ ➪ ➹ ➴ x7 ➷ x14 pseudomoves . . . . . . . S C I S C I x ′ ≥ x ≤ y ≤ z y ′ ≤ z ′ Searching over Metapositions in Kriegspiel – p.15/34
Exploiting the referee’s answers ➮ ➮ ➱ ➮ ➬ ➬ ➬ ➬ 8 ➮ ➮ ➮ ➮ ➬ ➬ ➬ e3 silent: we clean the squares around 7 ➱ ➱ ➱ ➮ ➬ ➬ ➬ 6 ➮ ✃ ➮ ➮ ➬ 5 ➱ ➱ ➱ ➮ the King and along row and column ➬ ➬ ➬ 4 ➱ ➱ ➱ ➱ ❐ 3 ➮ ➮ ➱ ➮ ➬ ➬ ➬ ➬ 2 ➮ ➮ ➮ ➮ ➬ ➬ ➬ 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
Exploiting the referee’s answers ❮ ❮ ❰ ❮ ❒ ❒ ❒ 8 ❮ ❮ ❰ ❮ ❒ ❒ ❒ e3 silent: we clean the squares around 7 ❰ ❰ ❰ ❮ ❒ ❒ 6 ❮ Ï ❰ ❮ ❒ 5 ❰ ❰ ❰ ❮ the King and along row and column ❒ ❒ 4 ❰ ❰ Ð ❰ 3 ❮ ❮ ❰ ❮ ❒ ❒ ❒ 2 ❮ ❮ ❰ ❮ ❒ ❒ ❒ 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
Exploiting the referee’s answers Ò Ò Ó Ò Ñ Ñ Ñ 8 Ò Ò Ó Ò Ñ Ñ Ñ e3 silent: we clean the squares around 7 Ó Ó Ó Ò Ñ Ñ 6 Ò Ô Ó Ò Ñ 5 Ó Ó Ó Ò the King and along row and column Ñ Ñ 4 Ó Ó Õ Ó 3 Ò Ò Ó Ò Ñ Ñ Ñ 2 Ò Ò Ó Ò Ñ Ñ Ñ 1 a b c d e f g h × × Ø × Ö Ö Ö Ö 8 × × × × Ö Ö Ö e3 check: we clean all the squares 7 Ø Ø Ø × Ö Ö Ö 6 × Ù × × Ö 5 Ø Ø Ø × Ö Ö Ö and we assume that the King is on the 4 Ø Ø Ø Ø Ú 3 × × Ø × Ö Ö Ö Ö 2 row or column × × × × Ö Ö Ö 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
Exploiting the referee’s answers Ü Ü Ý Ü Û Û Û 8 Ü Ü Ý Ü Û Û Û e3 silent: we clean the squares around 7 Ý Ý Ý Ü Û Û 6 Ü Þ Ý Ü Û 5 Ý Ý Ý Ü the King and along row and column Û Û 4 Ý Ý ß Ý 3 Ü Ü Ý Ü Û Û Û 2 Ü Ü Ý Ü Û Û Û 1 a b c d e f g h à à à à á 8 à à â à e3 check: we clean all the squares 7 à à à à á 6 à ã â à 5 à à à à á and we assume that the King is on the 4 à à ä à 3 à à à à á 2 row or column à à â à 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
Exploiting the referee’s answers æ æ ç æ å å å 8 æ æ ç æ å å å e3 silent: we clean the squares around 7 ç ç ç æ å å 6 æ è ç æ å 5 ç ç ç æ the King and along row and column å å 4 ç ç é ç 3 æ æ ç æ å å å 2 æ æ ç æ å å å 1 a b c d e f g h ê ê ê ê ë 8 ê ê ì ê e3 check: we clean all the squares 7 ê ê ê ê ë 6 ê í ì ê 5 ê ê ê ê ë and we assume that the King is on the 4 ê ê î ê 3 ê ê ê ê ë 2 row or column ê ê ì ê 1 a b c d e f g h ð ð ñ ð ï ï ï ï 8 ð ð ð ð ï ï ï 7 ñ ñ ñ ð ï ï ï 6 Kd5 illegal: we clean all the squares ð ò ð ð ï 5 ñ ñ ñ ð ï ï ï 4 only leaving those around the Black King% ñ ñ ñ ñ ó 3 ð ð ñ ð ï ï ï ï 2 ð ð ð ð ï ï ï 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
Exploiting the referee’s answers õ õ ö õ ô ô ô 8 õ õ ö õ ô ô ô e3 silent: we clean the squares around 7 ö ö ö õ ô ô 6 õ ÷ ö õ ô 5 ö ö ö õ the King and along row and column ô ô 4 ö ö ø ö 3 õ õ ö õ ô ô ô 2 õ õ ö õ ô ô ô 1 a b c d e f g h ù ù ù ù ú 8 ù ù û ù e3 check: we clean all the squares 7 ù ù ù ù ú 6 ù ü û ù 5 ù ù ù ù ú and we assume that the King is on the 4 ù ù ý ù 3 ù ù ù ù ú 2 row or column ù ù û ù 1 a b c d e f g h þ þ þ þ 8 þ þ þ þ d5 illegal: we clean all the squares 7 þ þ þ þ ÿ 6 þ � ✁ þ 5 þ þ þ þ ÿ leaving those around the 4 þ þ þ þ ✂ 3 þ þ þ þ 2 þ þ þ þ 1 a b c d e f g h Searching over Metapositions in Kriegspiel – p.16/34
The search algorithm • Moves that jeopardize important pieces are pruned; • recently-played moves are not played in direct sequence; • the search algorithm uses: • a static evaluation function : • to enable the algorithm to prune bad moves; • to enable the algorithm to choose one of the three referee’s answers; • a recursive evaluation function : • it is returned by the recursive call during the tree visit. • The value of a metaposition is obtained by adding the recursive value to the static value. Searching over Metapositions in Kriegspiel – p.17/34
Searching through metapositions ✄ ✄ ✄ ☎ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✆ ✆ ✆ ✄ ✝ ✝ ✝ ✝ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✆ ✆ ✆ ✄ ✝ ✝ ✝ ✝ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✆ ✆ ✆ ✄ ✝ ✝ ✝ ✆ ✆ ✆ ✄ ✝ ✝ ✝ ✞ Searching over Metapositions in Kriegspiel – p.18/34
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions ✠ ✟ ✟ ✟ ☞ ✡ ✡ ✡ ✟ ☛ ☛ ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✡ ✡ ✡ ✡ ☛ ☛ ☛ . . . ✟ ✟ ✟ ✟ ☞ g8 ✟ ✟ ✟ ✟ ✌ , − 526 ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✍ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ✟ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ✡ ✡ ✡ ✡ ☛ ☛ ☛
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions ✏ ✎ ✎ ✎ ✓ ✑ ✑ ✑ ✎ ✒ ✒ ✎ ✑ ✑ ✑ ✒ ✒ ✒ ✒ ✑ ✑ ✑ ✑ ✒ ✒ ✒ ✎ ✑ ✑ ✑ ✒ ✒ ✒ ✒ ✑ ✑ ✑ ✑ ✒ ✒ ✒ ✎ ✑ ✑ ✑ ✒ ✒ ✒ ✒ ✑ ✑ ✑ ✑ ✒ ✒ ✒ 272,272 . . . g2 ✎ ✎ ✎ ✎ ✓ g8 ✎ ✎ ✎ ✎ ✔ ✕ , − 526 ✎ ✑ ✑ ✑ ✒ ✒ ✒ ✒ ✕ . . . ✑ ✑ ✑ ✑ ✒ ✒ ✒ 313 ,313 ✎ ✑ ✑ ✑ ✒ ✒ ✒ ✒ g1 ✑ ✑ ✑ ✑ ✒ ✒ ✒ ✎ ✑ ✑ ✑ ✕ ✒ ✒ ✒ ✒ ✑ ✑ ✑ ✑ ✒ ✒ ✒
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions ✗ ✖ ✖ ✖ ✚ ✘ ✘ ✘ ✖ ✙ ✙ ✖ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✘ ✘ ✘ ✘ ✙ ✙ ✙ ✖ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✘ ✘ ✘ ✘ ✙ ✙ ✙ ✖ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✘ ✘ ✘ ✘ ✙ ✙ ✙ 272,272 . . . g2 ✖ ✖ ✖ ✖ ✚ − 213, − 526 g8 ✖ ✖ ✖ ✖ ✛ ✜ ✖ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✜ . . . ✘ ✘ ✘ ✘ ✙ ✙ ✙ 313 ,313 ✖ ✘ ✘ ✘ ✙ ✙ ✙ ✙ g1 ✘ ✘ ✘ ✘ ✙ ✙ ✙ ✖ ✘ ✘ ✘ ✜ ✙ ✙ ✙ ✙ ✘ ✘ ✘ ✘ ✙ ✙ ✙
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions ✢ ✢ ✢ ✣ ✦ ✤ ✤ ✤ ✢ ✥ ✥ ✥ ✤ ✤ ✤ ✢ ✥ ✥ ✥ ✥ ,262 ✤ ✤ ✤ ✤ ✥ ✥ ✥ h2 ✤ ✤ ✤ ✢ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥ ★ ✤ ✤ ✤ ✢ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥ ✣ ✢ ✢ ✢ ✦ . . . . . . ✤ ✤ ✤ ✢ ✥ ✥ ✢ ✤ ✤ ✤ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥ ✢ ✤ ✤ ✤ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥ ✢ ✤ ✤ ✤ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥ 272,272 . . . g2 ✢ ✢ ✢ ✢ ✦ − 213, − 526 g8 ✢ ✢ ✢ ✢ ✧ ★ ✢ ✤ ✤ ✤ ✥ ✥ ✥ ✥ ★ . . . ✤ ✤ ✤ ✤ ✥ ✥ ✥ 313 ,313 ✢ ✤ ✤ ✤ ✥ ✥ ✥ ✥ g1 ✤ ✤ ✤ ✤ ✥ ✥ ✥ ✢ ✤ ✤ ✤ ★ ✥ ✥ ✥ ✥ ✤ ✤ ✤ ✤ ✥ ✥ ✥
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions 272 ,272 g2 ✩ ✩ ✩ ✪ ✭ ✫ ✫ ✫ ✩ ✬ ✬ ✬ ✯ 534 ,262 ✫ ✫ ✫ ✩ ✬ ✬ ✬ ✬ . . . ✫ ✫ ✫ ✫ ✬ ✬ ✬ 263,263 h2 ✫ ✫ ✫ ✩ ✬ ✬ ✬ ✬ g2 ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✯ ✰ ✫ ✫ ✫ ✩ ✬ ✬ ✬ ✬ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✪ ✩ ✩ ✩ ✭ . . . . . . ✫ ✫ ✫ ✩ ✬ ✬ ✩ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✩ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✩ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✫ ✫ ✫ ✫ ✬ ✬ ✬ 272,272 . . . g2 ✩ ✩ ✩ ✩ ✭ − 213, − 526 g8 ✩ ✩ ✩ ✩ ✮ ✯ ✩ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✯ . . . ✫ ✫ ✫ ✫ ✬ ✬ ✬ 313 ,313 ✩ ✫ ✫ ✫ ✬ ✬ ✬ ✬ g1 ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✩ ✫ ✫ ✫ ✯ ✬ ✬ ✬ ✬ ✫ ✫ ✫ ✫ ✬ ✬ ✬
Searching over Metapositions in Kriegspiel – p.18/34 Searching through metapositions 272 g2 ✱ ✱ ✱ ✲ ✵ ✳ ✳ ✳ ✱ ✴ ✴ ✴ ✷ ✳ ✳ ✳ ✱ ✴ ✴ ✴ ✴ . . . ✳ ✳ ✳ ✳ 272 ✴ ✴ ✴ h2 ✳ ✳ ✳ ✱ ✴ ✴ ✴ ✴ g2 263 ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✷ ✸ ✳ ✳ ✳ ✱ ✴ ✴ ✴ ✴ ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✲ ✱ ✱ ✱ ✵ . . . . . . ✳ ✳ ✳ ✱ ✴ ✴ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✳ ✳ ✳ ✳ ✴ ✴ ✴ . . . 272 g2 ✱ ✱ ✱ ✱ ✵ g8 ✱ ✱ ✱ ✱ ✶ ✷ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✷ . . . 313 ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ g1 313 ✳ ✳ ✳ ✳ ✴ ✴ ✴ ✷ ✱ ✳ ✳ ✳ ✴ ✴ ✴ ✴ ✳ ✳ ✳ ✳ ✴ ✴ ✴
The evaluation function • contains the notion of progress which leads the player towards victory; • is a linear weighted sum of features EVAL ( m ) = w 1 f 1 ( m ) + w 2 f 2 ( m ) + ... + w n f n ( m ) for example: w 1 = − 1 , f 1 ( s ) = number of black kings ; • returns the move which immediately gives Checkmate , if it exists; • prunes moves that risk leading to a stalemate; • is different according to each single ending. Searching over Metapositions in Kriegspiel – p.19/34
The rook ending ( ) (1/3) � ✂ ✁ The evaluation function for this ending considers n = 6 different features. 1. it avoids jeopardizing the Rook: w 1 = − 1000 and f 1 is a boolean function which is true if white Rook is under attack; 2. it brings the two Kings closer: w 2 = − 1 and f 2 returns the distance (number of squares) between the two Kings; 3. it reduces the number of black Kings on the quadrants of the board as seen from the Rook and it favors having the black Kings grouped together in as few quadrants as � 4 possible: w 3 = − 1 and f 3 = c i =1 q i where c ∈ { 1 , 2 , 3 , 4 } is a constant which counts the quadrants that contains a black King and q i counts the number of possible black Kings on i th quadrant; Searching over Metapositions in Kriegspiel – p.20/34
The rook ending ( ) (2/3) � ✂ ✁ 4. it avoids the black King to go between white Rook and white King: w 5 = − 500 and f 5 is a boolean function which returns true if the black King is inside the rectangle formed by white King and white Rook on two opposite corners; 5. it keeps White pieces close to each other: w 5 = +1 and f 5 is a boolean function which returns true if the Rook is adjacent to the King; 6. it pushes the black King toward the corner of the board: � 63 w 6 = +1 and f 6 = i =0 v [ i ] , where v is a numerical 64-element vector, shown in the following slide, that returns a grade for each squares which possibly holds the black King or returns 0 otherwise. Searching over Metapositions in Kriegspiel – p.21/34
The rook ending ( ) (3/3) � ✂ ✁ The simple numerical matrix v [] used in the 6 th features: 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 − 2 − 4 − 4 − 2 0 0 0 0 − 4 − 4 − 4 − 4 0 0 0 0 − 4 − 4 − 4 − 4 0 0 0 0 − 2 − 4 − 4 − 2 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 Searching over Metapositions in Kriegspiel – p.22/34
Histogram of ending � ✂ ✁ This histogram represents the number of moves needed to win each game, starting from metapositions with greatest uncertainty. Searching over Metapositions in Kriegspiel – p.23/34
The queen ending ( ) � ✄ ✂ 7. it avoids metapositions where Queen risks to be captured: w 7 = − 100 and f 7 is a boolean function that returns true if Queen is under attack; 8. it penalizes those metapositions with a big number of black Kings: w 8 = − 1 and f 8 is equal to the number of black Kings on White’s reference board; 9. it reduces the number of black Kings on the areas traced by the Queen’s diagonals: w 9 = − 1 and f 9 = evalRhombArea ( S ) where (0) evalRhombArea ( S ) = c · ( a 0 + a 1 + a 2 + a 3 ) and c ∈ { 1 , 2 , 3 , 4 } is a constant which counts the areas that possibly contains a black King and a i ( i = 0 , ..., 3 ) counts the number of possible black Kings on i th area. Searching over Metapositions in Kriegspiel – p.24/34
Description of evalRhombArea() ✺ ✻ ✺ ✻ ✻ ✻ ✻ ✻ ✹ ✹ ✹ ✹ 8 8 ✻ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✹ ✹ ✹ a 0 7 7 ✻ ✻ ✻ ✺ ✻ ✻ ✻ ✻ ✹ ✹ ✹ ✹ 6 6 ✺ ✻ ✻ ✺ ✻ ✻ ✻ ✻ ✹ ✹ ✹ 5 5 ✻ ✼ ✻ ✻ ✻ ✼ ✻ ✻ a 3 a 1 4 4 ✺ ✻ ✻ ✺ ✻ ✻ ✻ ✻ ✹ ✹ ✹ 3 3 ✻ ✻ ✻ ✺ ✻ ✻ ✻ ✻ ✹ ✹ ✹ 2 2 a 2 ✻ ✺ ✽ ✻ ✻ ✻ ✽ ✻ ✹ ✹ 1 1 a b c d e f g h a b c d e f g h • Miniature on the right shows the 4 different areas if the queen is on d4; • miniature on the left represents a metaposition with greatest uncertainty: in this case the function returns 4 · (12 + 12 + 3 + 6) = 132 . Searching over Metapositions in Kriegspiel – p.25/34
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