圏論とコンピュータサイエンス / Category Theory and Theoretical Computer Science

Transcript

1. ݍ࿦ͱίϯϐϡʔλαΠΤϯε
   ࢁా༔೭հ
   yunosuke(at)ueda.info.waseda.ac.jp
   ૣҴాେֶେֶӃ جװཧ޻ֶݚڀՊ
   ৘ใཧ޻ɾ৘ใ௨৴ઐ߈ ্ాݚڀࣨ
   2020/3/9
   

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2. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ࣗݾ঺հ
   ݚڀ
   • ݴޠ (HydLa) ͷҙຯ࿦
   ʢݩʣझຯ
   • ਺ֶ
   • ڝϓϩ
   2 / 23
   

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3. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ֓ཁ
   ͜ͷൃදͰ࿩͢͜ͱ
   • ݍͷఆٛͱ؆୯ͳྫ
   • ݍ࿦ͷߟ͑ํ͕৽͍͠ࢹ఺Λ༩͑ͯ͘ΕΔ͜ͱ
   • ݍ࿦͕ CS ෼໺ʹͲ͏Ԡ༻͞Ε͍ͯΔ͔
   ஫ҙ
   ͨͿΜ෼͔Βͳ͍ྫ͕ग़ͯ͘Δͱࢥ͍·͢
   ஌Βͳ͍෼໺ͷ۩ମྫ͸దٓεϧʔ͍ͯͩ͘͠͞
   ͨͩ͠ҎԼͷ஌͍ͬͯΔ΋ͷͱͯ͠ਐΊ·͢
   • ू߹ɼࣸ૾ͷجૅ஌ࣝʢશࣹɼ୯ࣹɼ௚ੵɼ௚࿨ʣ
   3 / 23
   

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4. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ݍ࿦ͱ͸
   ݍ࿦ͱ͸ԿͰ͋Δ͔
   • 20 ੈلத൫ʹͰ͖ͨ৽͍͠਺ֶ
   • ௒ந৅తͰ͓΋͠Ζ͍
   • ਺ֶʹݶΒͳ͍ཧ࿦෼໺ͷڞ௨ݴޠ
   ݍ࿦ͱ͸ԿͰ͸ͳ͍ͷ͔
   • ϓϩάϥϜΛॻ্͘Ͱ໾ʹཱͭ΋ͷ
   4 / 23
   

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5. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ਺ֶͷ෮श 1
   ○
   ू߹ͱࣸ૾
   ू߹ͱࣸ૾ʹରͯ͠ҎԼ͕੒Γཱͭ
   • ೚ҙͷू߹ X ʹରͯ͠߃౳ࣸ૾ idX
   ͕ଘࡏ
   • ೚ҙͷࣸ૾ f : X → Y, g : Y → Z ʹରͯ͠
   ߹੒ࣸ૾ g ◦ f : X → Z ͕ଘࡏ
   άϥϑͱ४ಉܕ
   άϥϑͱͦͷ४ಉܕʹରͯ͠ҎԼ͕੒Γཱͭ
   • ೚ҙͷάϥϑ G ʹରͯ͠߃౳ࣸ૾ idG
   ͸४ಉܕ
   • ೚ҙͷ४ಉܕ f : G1
   → G2
   , g : G2
   → G3
   ʹରͯ͠
   ߹੒ࣸ૾ g ◦ f : G1
   → G3
   ͸४ಉܕ
   5 / 23
   

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6. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ਺ֶͷ෮श 2
   ○
   ܈ͱ४ಉܕ
   ܈ͱͦͷ४ಉܕʹରͯ͠ҎԼ͕੒Γཱͭ
   • ೚ҙͷ܈ G ʹରͯ͠߃౳ࣸ૾ idG
   ͸४ಉܕ
   • ೚ҙͷ४ಉܕ f : G1
   → G2
   , g : G2
   → G3
   ʹରͯ͠
   ߹੒ࣸ૾ g ◦ f : G1
   → G3
   ͸४ಉܕ
   ൒ॱংू߹ͱ୯ௐࣸ૾
   ൒ॱংू߹ͱ୯ௐʢ૿Ճʣࣸ૾ʹରͯ͠ҎԼ͕੒Γཱͭ
   • ೚ҙͷ൒ॱংू߹ X ʹରͯ͠߃౳ࣸ૾ idX
   ͸୯ௐ
   • ೚ҙͷ୯ௐࣸ૾ f : X → Y, g : Y → Z ʹରͯ͠
   ߹੒ࣸ૾ g ◦ f : X → Z ͸୯ௐ
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7. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ݍͷؾ࣋ͪ
   Ҏ্ͷྫͰ͸Ϟϊͱ໼ҹʹରͯ࣍͠ͷ 2 ͕ͭ੒Γཱ͍ͬͯͨ
   • ೚ҙͷϞϊʹରͯ͠߃౳తͳ໼ҹ͕ଘࡏ
   • ߦ͖ઌͱग़ൃ఺͕ಉ͡໼ҹʹରͯ͠߹੒ͨ͠໼ҹ͕ଘࡏ
   ݍ͸͜ͷ͋Γ;Εͨಛ௃Ͱఆٛ͞ΕΔ
   7 / 23
   

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8. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ݍͷఆٛ
   ఆٛ [1]
   ݍ͸࣍ͷཁૉ͔Β੒Γཱͭ
   • ର৅ɿA, B, C, ...
   • ࣹɿf, g, h, ...
   • ೚ҙͷࣹ f ʹରͯ͠ର৅ dom(f), cod(f) ͕͋Δ
   A = dom(f), B = cod(f) Ͱ͋Δ f Λ f : A → B ͱॻ͘
   • ࣹ f : A → B, g : B → C s.t. cod(f) = dom(g) ʹରͯ͠
   ࣹ g ◦ f : A → C ͕͋Γɼ
   ೚ҙͷࣹ f : A → B, g : B → C, h : C → D ʹରͯ͠
   h ◦ (g ◦ f) = (h ◦ g) ◦ f ͕੒Γཱͭ
   • ೚ҙͷର৅ A ʹରࣹ͠ idA
   : A → A ͕͋Γɼ
   ೚ҙͷࣹ f : A → B ʹର͠ f ◦ idA
   = f = idB
   ◦ f ͕੒Γཱͭ
   8 / 23
   

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9. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
   ஫ҙ
   ର৅ͱͯ͠ू߹ɺࣹͱͯࣸ͠૾ΛऔͬͨݍΛ Sets ͱݺͿ
   ݍͷྫ͸ʢߏ଄෇͖ʣू߹ͱʢߏ଄Λอͭʣࣸ૾͚ͩͰ͸ͳ͍
   ൒ॱংͦͷ΋ͷ
   ൒ॱংू߹ P Λߟ͑Δ
   ର৅ͱͯ͠ཁૉΛͱΓɼॱংؔ܎͕͋Δͱ͖ʹࣹ͕͋Δͱ͢Δ
   • ೚ҙͷཁૉ x ʹର͠ x ≤ x
   • x ≤ y, y ≤ z ͳΒ x ≤ z
   ݍͰͳ͍ྫ
   • ൒ॱংू߹ͱ୯ௐݮগࣸ૾
   ∵ ୯ௐݮগࣸ૾ͷ߹੒͸୯ௐ૿Ճࣸ૾
   • ϚάϚ M ͷཁૉͱͦͷԋࢉ
   ∵ ϚάϚͷԋࢉ͸୯Ґݩ͕ͳ݁͘߹తͰ΋ͳ͍
   9 / 23
   

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10. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    CS ͬΆ͍ྫ
    ໋୊ͱূ໌
    • ೚ҙͷ໋୊ φ ʹରͯ͠ φ ⊢ φ
    • φ ⊢ ψ, ψ ⊢ ρ ͳΒ φ ⊢ ρ
    ϓϩάϥϜʹ͓͚Δܕͱؔ਺
    ϓϩάϥϛϯάݴޠ L Λߟ͑Δ
    • L ͷ೚ҙͷܕ T ʹର͠Ҿ਺Λͦͷ··ฦؔ͢਺͕ॻ͚Δ
    • ؔ਺ f : T1
    → T2
    , g : T2
    → T3
    ʹର͠ g(f(t)) Λฦؔ͢਺
    10 / 23
    

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11. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ԋश
    ͜ΕΒ͕ݍʹͳ͍ͬͯΔ͜ͱΛ֬ೝͯ͠Έ͍ͯͩ͘͞
    ର৅ ࣹ
    ϕΫτϧۭؒ ઢܗࣸ૾
    ࣮਺ R ͷΈ ࿈ଓؔ਺
    ू߹ 2 ߲ؔ܎
    ϞϊΠυ ϞϊΠυ४ಉܕ
    ू߹ X ͷཁૉ =
    ܈ G ͷཁૉ ܈ԋࢉ
    11 / 23
    

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12. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ݍ࿦ͷಛ௃
    ݍ࿦Ͱ͸ର৅΍ࣹͷಛ௃෇͚ΛࣹͷΈΛ༻͍ͯߦ͏
    ू߹࿦ͱҟͳΓର৅ͷத਎͸ݟͳ͍
    ʢཁૉΛݟͳͯ͘΋ಉ౳ͷදݱྗ͕͋Δʣ
    ཁૉΛݟͳ͍͜ͱʹΑͬͯ
    • ఆ͕ٛ؆ܿʹͳΔ
    • ΑΓந৅తʹͳΔ
    • ࠓ·Ͱ෼͔Βͳ͔ͬͨ͜ͱ͕෼͔Δ
    12 / 23
    

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13. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    Ϟϊࣹ
    ࣹͷಛ௃෇͚ͷྫͱͯ͠୯ࣹੑͱશࣹੑΛऔΓ্͛Δ
    ఆٛ
    ࣹ f : A → B ͕ϞϊࣹͰ͋Δͱ͸
    ೚ҙͷࣹ g, h : C → A ʹରͯ͠ fg = fh ͳΒ g = h Ͱ͋Δ͜ͱ
    C
    g //
    h
    // A f // B
    Sets ʹ͓͍ͯϞϊࣹͰ͋Δ͜ͱ͸୯ࣹͰ͋Δ͜ͱͱಉ஋ʢԋशʣ
    g, h ͕ A ͷ೚ҙͷݩΛද͍ͯ͠Δͱࢥ͑͹ࣗવ
    Ͱ͸શࣹ͸ʁ
    13 / 23
    

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14. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    Τϐࣹ
    ఆٛ
    ࣹ f : A → B ͕ΤϐࣹͰ͋Δͱ͸
    ೚ҙͷࣹ i, j : B → C ʹରͯ͠ if = jf ͳΒ i = j Ͱ͋Δ͜ͱ
    A f // B i //
    j
    // C
    Sets ʹ͓͍ͯΤϐࣹͰ͋Δ͜ͱ͸શࣹͰ͋Δ͜ͱͱಉ஋ʢԋशʣ
    if = jf ͔Β i = j ͕ݴ͑ΔͨΊʹ͸
    f Λ߹੒ͨ͜͠ͱͰ৘ใ͕ࣦΘΕͯ͸͍͚ͳ͍ͱߟ͑Δͱࣗવ
    ͞ΒʹϞϊࣹͱΤϐࣹ͸૒ରʹͳ͍ͬͯΔ
    14 / 23
    

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15. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ૒ର
    ݍ࿦ʹ͓͍ͯ 2 ͭͷ֓೦͕૒ରͰ͋Δͱ͸ɺ
    ยํͷࣹͷ޲͖Λશͯٯ޲͖ʹ͢Δͱ΋͏ยํʹͳΔ͜ͱ
    ࢼ͠ʹΤϐࣹͷਤΛͻͬ͘Γฦͯ͠ΈΔͱ
    Ϟϊࣹͷਤ C
    g //
    h
    // A f // B
    ͻͬ͘Γฦͨ͠Τϐࣹͷਤ C i //
    j
    // B f // A
    ͭ·Γ୯ࣹੑͱશࣹੑ͸ͪΐ͏Ͳ൓ରͷ֓೦Ͱ͋Δ͜ͱ͕෼͔ͬͨ
    15 / 23
    

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16. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ੵ
    ର৅ͷಛ௃෇͚ͷྫͱͯ͠ੵͱ༨ੵΛऔΓ্͛Δ
    ෳ਺ͷࣹΛߟ͑Δ࣌ɼਤࣜͱ͍͏ɽ
    ੵ
    ਤࣜ A p1
    ←− P p2
    −→ B ͕ A, B ͷੵͰ͋Δͱ͸
    ೚ҙͷର৅ Z ͱਤࣜ A z1
    ←− Z z2
    −→ B ʹରͯ͠
    pi
    ◦ u = zi
    Λຬͨ͢Ұҙͳ u : Z → P ͕ଘࡏ͢Δ͜ͱ
    Z
    z1
    
    u
    
    z2
    
    A P
    p1
    oo
    p2
    // B
    Sets ʹ͓͍ͯ P ͕ੵͰ͋Δ͜ͱ͸௚ੵͰ͋Δ͜ͱͱಉ஋ʢԋशʣ
    (P = A × Bɼpi
    ͕ࣹӨɼu = ⟨z1
    , z1
    ⟩)
    16 / 23
    

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17. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ༨ੵ
    ݍ࿦ʹ͓͍ͯ͸૒ରͷ֓೦ΛΑ͘༨ʙͱݺͿ
    ༨ੵ
    ਤࣜ A q1
    −→ Q q2
    ←− B ͕ A, B ͷ༨ੵͰ͋Δͱ͸
    ೚ҙͷର৅ Z ͱਤࣜ A z1
    −→ Z z2
    ←− B ʹରͯ͠
    u ◦ qi
    = zi
    Λຬͨ͢Ұҙͳ u : Q → Z ͕ଘࡏ͢Δ͜ͱ
    Z
    A
    z1
    ??
    q1
    // Q
    u
    OO
    B
    z2
    __
    q2
    oo
    Sets ʹ͓͍ͯ Q ͕༨ੵͰ͋Δ͜ͱ͸௚࿨Ͱ͋Δ͜ͱͱಉ஋ʢԋशʣ
    ௚ੵͱ௚࿨͸ͪΐ͏Ͳ൓ରͷ֓೦Ͱ͋Δ͜ͱ͕෼͔ͬͨ
    ͦΕͧΕ͸Α͘
    ∏
    ͱ
    ⨿
    Ͱॻ͕͘૒ରͳͷͰࣗવ
    17 / 23
    

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18. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ܕແ͠ϥϜμܭࢉ
    ܕແ͠ϥϜμܭࢉͰ࣍ͷΑ͏ͳϥϜμࣜΛߟ͑Δ
    λx.xx
    ܕແ͠ϥϜμܭࢉͷϞσϧΛ୳ͯ͠ΈΔ
    1 x ͸ؔ਺ͷ͸ͣ
    2 A → B ͱ͍͏ؔ਺ͩͱࢥͬͯΈΔ
    3 x ͸ͦͷҾ਺ͱͯ͠༩͑ΒΕΔͷͰू߹ A ͷཁૉͰ͋Δ
    4 BA = A ͸ໃ६͢ΔͷͰ BA ∼
    = A ͱͯ͠ΈΔ
    5 ͦΕͰ΋ೱ౓Λߟ͑Δͱ |A| = |B| = 1 ͱ͍͏ࣗ໌ͳղͷΈ
    6 ܕແ͠ϥϜμܭࢉͷϞσϧ͸ͳ͍ʁ
    ʢ༨༟͕͋ͬͨΒͲ͕͓͔͍͔͜͠ߟ͔͑ͯΒ࣍ʹߦ͍ͬͯͩ͘͞ʣ
    18 / 23
    

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19. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ΋͔ͯ͠͠ Sets Ͱ
    ߟ͍͑ͯ·ͤΜ͔ʁ
    19 / 23
    

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20. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ී௨͸ |A| ͕େ͖͘ͳΕ͹ |A| < |BA| ͳͷͰղʹ͸ͳΒͳ͍
    ∵ Sets Ͱߟ͍͑ͯΔ͔Βʂ
    ܕແ͠ϥϜμܭࢉͷඇࣗ໌ͳϞσϧͷߏ੒๏
    1 cpoʢ׬උ൒ॱংू߹ʣͷݍͰߟ͑Δ
    2 উखͳ cpo D ʹରͯ͠ɺD0
    = D, Dn+1
    = DDn
    n
    ͱ͢Δ
    3 Dn
    ͸ Dn+1
    ʹຒΊࠐΊɺ·ͨ D∞
    ͸ cpo Ͱ͋Δ͜ͱ͕ࣔͤΔ
    4 ͦͯ͠ D∞
    ∼
    = DD∞
    ∞
    ΋ࣔͤΔ
    5 D ͸ D∞
    ʹຒΊࠐΊͯ
    D Λେ͖͘औΕ͹ D∞
    ΋޷͖ͳ͚ͩେ͖͘Ͱ͖Δ
    ͜ͷߏ੒๏͸͋Δ৚݅Λຬͨ͢ݍʹҰൠԽͰ͖ΔͨΊ
    ܕແ͠ϥϜμܭࢉͷඇࣗ໌ͳϞσϧ͕ແ਺ʹ͋Δ͜ͱ͕෼͔Δ [2]
    20 / 23
    

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21. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ܕ෇͖ϥϜμܭࢉ
    ܕ෇͖ϥϜμܭࢉͷ΄͏͸ͦΕ΄Ͳ೉͘͠ͳ͍
    Sets ͸΋ͪΖΜϞσϧʹͳ͍ͬͯΔ
    Ұൠʹ͋Δݍ͕ϥϜμܭࢉͷϞσϧʹͳΔʹ͸
    ݍ࿦తͳҙຯͰͷႈ͕͋Ε͹Α͍
    ·ͨ௚ੵܕ͕͋ΔͳΒશͯͷ༗ݶੵ͕͋Ε͹ྑ͍͠
    ௚࿨ܕ͕͋ΔͳΒશͯͷ༗ݶ༨ੵ͕͋Ε͹ྑ͍
    ܕ෇͖ϥϜμܭࢉʹ͓͍ͯ΋Ϟσϧʹඞཁͳ৚͕݅෼͔ͬͨ
    21 / 23
    

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22. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ͦͷଞ
    ϥϜμܭࢉҎ֎ͷԠ༻ͱͯ͠͸
    Ϟφυ Haskell, Scala Ͱ࢖ΘΕͯΔ
    ݍ࿦͸ؔ਺ܕϓϩάϥϛϯάͷ໾ʹ͸ཱ͔ͭ΋
    τϙε ߴ֊௚؍ओٛ࿦ཧͷϞσϧʹͳ͍ͬͯΔΒ͍͠ [3]
    ༨୅਺ ঢ়ଶભҠܥ΍ແݶσʔλߏ଄Λ௒؆୯ʹॻ͚Δ [4]
    ͳͲ͕͋Δ
    22 / 23
    

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23. ֓ཁ ಋೖ ݍ࿦ͷಛ௃ CS ΁ͷԠ༻
    ࢀߟจݙ
    [1] Steve Awodey.
    ݍ࿦ ݪஶୈ 2 ൛.
    ڞཱग़൛, 2015.
    [2] ԣ಺׮จ.
    ϓϩάϥϜҙຯ࿦.
    ڞཱग़൛, 1994.
    [3] ਗ਼ਫٛ෉.
    ݍ࿦ʹΑΔ࿦ཧֶ ߴ֊࿦ཧͱτϙε.
    ౦ژେֶग़൛ձ, 2007.
    [4] J. Rutten.
    Universal coalgebra: a theory of systems.
    1996.
    23 / 23
    

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