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二人単貧民の定理を Coqで証明する試み
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Katsuki Ohto
August 31, 2020
Research
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200
二人単貧民の定理を Coqで証明する試み
単貧民をテーマに、Coqでゲームを扱うことについて検討する。
Katsuki Ohto
August 31, 2020
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Transcript
ೋਓ୯ශຽͷఆཧΛ COQͰূ໌͢ΔࢼΈ େ উݾʢϑϦʔΤϯδχΞʣ 2020.8.31 ୈ4ճຊ߹ͤήʔϜཧݚڀूձ
ൃදɿCOQ୯ශຽ ➤ తɿ ఆཧূ໌ࢧԉܥCoqʹͯ ػցʹΑΔอূ͖ͷ ೋਓ୯ශຽͷূ໌Λॻ͘ ➤ ݁Ռɿ ݁·Ͱͷূ໌ͷྲྀΕ ࡉ͔͍ิΛશͯূ໌͢Δͷ
ਐߦதʢ͕͜͜ΩπΠʣ ➤ CoqʹΑΔήʔϜݕূͷ ՄೳੑΛհ
ೋਓ୯ශຽ
େ߽ܥήʔϜͷݚڀ ➤ ୩ɾখ (2017, 2018) ೋਓ୯ශຽͷউऀΛઢܗ࣌ؒͰٻΊΔΞϧΰϦζϜͷఏҊ ➤ ೋਓ୯ශຽʢ, 2007ʣͱ… τϥϯϓήʔϜେ߽ͷ؆ུԽ
ɾೋਓ ɾಛघϧʔϧͳ͠ ɾҰຕग़͠ͷΈ ➤ શใήʔϜ
ೋਓ୯ශຽͷఆࣜԽ ➤ ୯ශຽήʔϜͷҙͷঢ়ଶҎԼͷࡾͭͰදݱͰ͖Δ ɾख൪ଆͷखࡳɹ ɾඇख൪ଆͷखࡳ ɾͷࡳͷڧ͞ɹ ➤ ྫ
ʢಉ͡ڧ͞ͷࡳΛෳ࣋ͭ͜ͱ͋Δʣ X ¯ X r X = {1,3}, ¯ X = {2,2,4}, r = 0
ೋਓ୯ශຽͷఆࣜԽ ➤ ྫ ➤ ઌख͕̏Λग़ͨ͠߹ɹɹɹɹɹ {1} {2, 2, 4}
3 ޙख͕̐Λग़͢ɹɹɹɹɹɹɹɹ {1} {2, 2} 4 ઌखग़ͤͳ͍ͷͰύεɹɹɹɹ {1} {2, 2} 0 ޙख͕̎Λग़͢ɹɹɹɹɹɹɹɹ {1} {2} 2 ઌखύεɹɹɹɹɹɹɹɹɹɹ {1} {2} 0 ޙख͕̎Λग़ͯ͠ޙखউͪɹɹɹ {1} {} 2 X = {1,3}, ¯ X = {2,2,4}, r = 0
ೋਓ୯ශຽͷఆࣜԽ ➤ ྫ ➤ ઌख͕ Λग़͠ ˠ ޙख͕ Λग़ͨ͠߹ ˠ
ઌख͕ Λग़ͯ͠ઌखউͪ ➤ ઌख͕ Λग़͠ ˠ ޙख͕ Λग़ͨ͠߹ ˠ ઌखύεɺޙख͕ Λग़͢ ˠ ઌख͕ Λग़ͯ͠ઌखউͪ ➤ Ҏ্ͷ݁Ռ͔Βɹઌखউͪɹͷہ໘ X = {1,3}, ¯ X = {2,2,4}, r = 0 1 2 3 1 4 2 3
ೋਓ୯ශຽͷղ๏ ➤ ୩ɾখ (2018) ➤ ҎԼͷΞϧΰϦζϜʹΑΓखࡳͷϚονϯάΛܭࢉ 1 ڧ͞ 3 2
2 4 0 ख൪ଆ ඇख൪ଆ ࣗͷࡳ͔Βɺ ͦΕΑΓऑ͍૬खͷࡳʹ ϚονϯάลΛҾ͘ ख൪ଆ ࡳରʹ
ೋਓ୯ශຽͷղ๏ ➤ ୩ɾখ (2018) ➤ ҎԼͷΞϧΰϦζϜʹΑΓखࡳͷϚονϯάΛܭࢉ 1 ڧ͞ 3 2
2 4 0 ख൪ଆ ඇख൪ଆ ૬खͷ ࠷ऑࡳʹ Ҿ͚ͳ͍ʂ ࣗͷࡳ͔Βɺ ͦΕΑΓऑ͍૬खͷࡳʹ ϚονϯάลΛҾ͘
ೋਓ୯ශຽͷղ๏ ➤ ୩ɾখ (2018) ➤ ҎԼͷΞϧΰϦζϜʹΑΓखࡳͷϚονϯάΛܭࢉ 1 ڧ͞ 3 2
2 4 0 ख൪ଆ ඇख൪ଆ 2 1 ∨
ೋਓ୯ශຽͷղ๏ ➤ ہ໘ ʹ͓͍ͯ खࡳͦΕͧΕͷ࠷খࡳ ຕΛআ͍ͨखࡳΛ ➤ ऑ͍ࡳͷϚονϯάͷΛܭࢉ͢Δؔ ➤ ख൪ଆඞউ
⁶ (X, ¯ X, r) 1 X− , ¯ X− μ μ(X, ¯ X− + {r}) > μ( ¯ X, X− )
୯ශຽͷূ໌
ೋਓ୯ශຽͷΞϧΰϦζϜͷূ໌ ➤ ήʔϜʹଐ͢ΔԿΒ͔ͷʹର͢Δؼೲ๏Ͱূ໌Ͱ͖Δ ͜͜Ͱɺ྆ऀͷखࡳຕͷ߹ܭͱ͢Δ 1 ڧ͞ 3 2 2 4
0 ख൪ଆ ඇख൪ଆ 2 1 ∨
ೋਓ୯ශຽͷΞϧΰϦζϜͷূ໌ ➤ ূ໌͍ͨ͠ఆཧΛ࣍ͷ (a) (b) ʹׂ (a) ͳΒख൪ଆඞউ (b) ͳΒඇख൪ଆඞউ
➤ جૅεςοϓ खࡳ̍ຕͣͭʢ߹ܭ̎ຕʣͷͱ͖ (a) (b) ཱ͕ ➤ ؼೲεςοϓ खࡳ߹ܭ͕ ຕͷͱ͖ɺ ɾ(a)ہ໘ͳΒҰखͰউͪ or ૬खͷ(b)ہ໘ʹભҠ͢Δख͕͋Δ ɾ(b)ہ໘ͳΒ૬खͷ(a)ہ໘ʹඞͣભҠ͢Δ ͜ͱ͕ࣔͤͨͱ͢Δʢ͕͍͜͜͠ͷ͕ͩʣ μ(X, ¯ X− + {r}) > μ( ¯ X, X− ) μ(X, ¯ X− + {r}) ≤ μ( ¯ X, X− ) k
ೋਓ୯ශຽͷΞϧΰϦζϜͷূ໌ ➤ ূ໌͍ͨ͠ఆཧΛ࣍ͷ (a) (b) ʹׂ (a) ͳΒख൪ଆඞউ (b) ͳΒඇख൪ଆඞউ
➤ ؼೲεςοϓ खࡳ߹ܭ͕ ຕͷͱ͖ɺ ɾ(i) (a)ہ໘Ͱग़ͤΔࡳ͕͋Εؼೲ๏ͷԾఆͷ(b)Λద༻ ɾ(ii) (b)ہ໘Ͱग़ͤΔࡳ͕͋Εؼೲ๏ͷԾఆͷ(a)ɺ ɹɹɹग़ͤΔࡳ͕ͳ͚Ε (i) Λద༻ ɾ(i) (a)ہ໘Ͱग़ͤΔࡳ͕ͳ͚Ε (ii) Λద༻ Αͬͯखࡳ߹ܭ ຕͰ(a)(b)ཱ͕ μ(X, ¯ X− + {r}) > μ( ¯ X, X− ) μ(X, ¯ X− + {r}) ≤ μ( ¯ X, X− ) k + 1 k + 1
ఆཧূ໌ࢧԉܥ COQ
COQʹΑΔূ໌ͷྲྀΕ ➤ ఆཧূ໌ࢧԉܥ ূ໌Λهड़͢ΔͨΊͷϓϩάϥϛϯάݴޠ γεςϜ͕डཧͨ͠ূ໌ʢίʔυʣͨ͠ূ໌ͱͯ͠ Ҏ߱ͷূ໌ʹར༻Ͱ͖Δ ➤ CoqAgdaͳͲ͕༗໊ ➤ ʢຊൃදͷൣғͰʣ͍ΘΏΔʮࣗಈূ໌ʯͰͳ͍
➤ ྫɿࣗવ ʹରͯ͠ Λূ໌ͯ͠ΈΔ n n = 0 + n n = n + 0
None
None
ήʔϜ ON COQ
COQͰήʔϜΛѻ͏ख๏ ➤ ήʔϜ(ͷΈͳΒ༷ͣʑͳରʹʹڞ௨Ͱ͋Δ͕) CoqͰෳࡶͳରΛఆٛ͢Δ̎௨ΓͷΓํ ɾ࠶ؼؔʹΑΔఆٛ ɹ ɾؼೲతͳఆٛ ɹ ➤
ͲͪΒ͋Γ
࠶ؼؔʹΑΔήʔϜͷఆٛ ➤ ४උ
࠶ؼؔʹΑΔήʔϜͷఆٛ ➤ ࠶ؼؔͰશ୳ࡧΛఆٛ
࠶ؼؔʹΑΔήʔϜͷఆٛ ➤ Compute Ͱશ୳ࡧΛ࣮ߦ
࠶ؼؔʹΑΔήʔϜͷఆٛ ➤ ख੍ݶͳ͠ˠ࠶ؼؔͷఀࢭੑೳ͕CoqʹΘΒͣ×
ؼೲతͳήʔϜͷউഊఆٛ ➤ ҰྫɿউഊΛҾʹ໋Λฦؔ͢Λఆٛ͢Δ
ؼೲతͳήʔϜͷউഊఆٛ ➤ ۩ମྫʹର͢Δউഊͷܾఆ…ܭࢉͰ͖ͣɺূ໌Ͱߦ͏
COQͰήʔϜΛѻ͏ख๏ ➤ CoqͰήʔϜΛѻ͏߹ ɾ࠶ؼؔʹΑΔఆٛ ɹॴɿײʹ͍ۙఆٛɺܭࢉͰ͖Δ ɹॴɿఆٛ͢Δ࣌Ͱ੍ݶ͕େ͖͍ ɹ ɾؼೲతͳఆٛ ɹॴɿఆٛͰ͖Δൣғ͕͍ɺతͳূ໌
ɹॴɿײతͳήʔϜͱͷဃ