5分でわかる Curry–Howard 同型対応

5分で伝えることが無茶だとわかりました

https://connpass.com/event/54292/

Transcript

1. ෼ Ͱ Θ ͔ Δ
   C u r r y – H o w a rd ಉ ܕ ର Ԡ
   2 0 1 7 - 0 5 - 2 8 ୈ 6 ճ O U CC LT ձ
   

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2. ͓ લ ͸ ୭ ͩ
   susisu / @susisu2413
   • JavaScript
   • Haskell
   •
   きんいろモザ
   イク
   

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3. ࿩͢͜ͱ
   ୯७ܕ෇͖ϥϜμܭࢉ
   ࣗવԋ៷ʢ໋୊࿦ཧʣ
   Curry–HowardಉܕରԠ
   
   System F
   ࿩ͤͳ͍͜ͱʢ࣌ؒతʹʣ
   w ϥϜμܭࢉͷৄࡉ
   w ड़ޠ࿦ཧ
   w ݹయ࿦ཧͱ௚؍ओٛ࿦ཧ
   

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4. ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
   

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5. ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
   w ϥϜμܭࢉʜ୯७ͳϓϩάϥϛϯάݴޠΈ͍ͨͳͷ
   w ม਺
   w ؔ਺
   w ؔ਺ద༻ͷͭͷΈ
   w ؔ਺ܕϓϩάϥϛϯάͷཧ࿦తͳج൫
   w ୯७ܕ෇͖ϥϜμܭࢉ͸ͦΕʹܕΛ͚ͭͨ΋ͷͷ಺ɺ
   ࠷΋؆୯ͳ΋ͷ
   

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6. ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
   t ::= -- ߲
   x -- ม਺
   λx: T. t -- λந৅ʢؔ਺ʣ
   t t -- ؔ਺ద༻
   T ::= -- ܕ
   X -- ܕม਺
   T → T -- ؔ਺ͷܕ
   

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7. ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
   w ܕ෇͚نଇ͸࣍ͷΑ͏ͳܗͰॻ͘
   ʮલఏ͕͢΂ͯ੒Γཱͭͱ͖ɺ݁࿦΋੒Γཱͭʯ
   w લఏ΍݁࿦͸࣍ͷΑ͏ͳܗࣜ
   ʮจ຺ΓͷԼͰɺ߲ t͸Tͱ͍͏ܕΛ΋ͭʯ
   ৈଥ (0 ڎΤࣣ)
   ٗ໼
   ` t : T
   

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8. ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
   x
   :
   A
   2
   `
   x
   :
   A
   , x
   :
   A
   `
   t
   :
   B
   `
   x
   :
   A. t
   :
   A
   !
   B
   ` t1 : A ! B ` t2 : A
   ` t1 t2 : B
   

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9. ࣗ વ ԋ ៷
   

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10. ࣗ વ ԋ ៷
    w ࿦ཧֶͷܗࣜମܥͷͻͱͭ
    w ࿦ཧతʹਖ਼໋͍͠୊ʢఆཧʣ
    จ຺ʹΑΒͣ߃ʹਅͳ໋୊
    w ܗࣜମܥ͸ͦͷΑ͏ͳఆཧΛಋͨ͘Ίͷ΋ͷ
    w ͳΜ͔ࣗવͬΆ͍ʢਓ͕ؒߦ͏ਪ࿦ʹࣅ͍ͯΔʣ
    w ͜ͷεϥΠυͰ͸໋୊࿦ཧͷൣғΈߟ͑·͢
    

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11. ࣗ વ ԋ ៷
    P ::= -- ໋୊
    X -- ໋୊ม਺
    P → P -- ࿦ཧแؚʮͳΒ͹ʯ
    P ∧ P -- ࿦ཧੵʮ͔ͭʯ
    P ∨ P -- ࿦ཧ࿨ʮ·ͨ͸ʯ
    ¬P -- ൱ఆ
    

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12. ࣗ વ ԋ ៷
    w ಋग़نଇ͸࣍ͷΑ͏ͳܗͰॻ͘
    ʮલఏ͕͢΂ͯ੒Γཱͭͱ͖ɺ݁࿦΋੒Γཱͭʯ
    w લఏ΍݁࿦͸࣍ͷΑ͏ͳܗࣜ
    ʮจ຺ΓͷԼͰɺP͸߃ʹਅʯ
    ৈଥ (0 ڎΤࣣ)
    ٗ໼
    ` P
    

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13. ࣗ વ ԋ ៷
    A 2
    ` A
    , A ` B
    ` A ! B
    ` A ! B ` A
    ` B
    

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14. ࣗ વ ԋ ៷
    A 2
    ` A
    , A ` B
    ` A ! B
    ` A ! B ` A
    ` B
    x
    :
    A
    2
    `
    x
    :
    A
    , x
    :
    A
    `
    t
    :
    B
    `
    x
    :
    A. t
    :
    A
    !
    B
    ` t1 : A ! B ` t2 : A
    ` t1 t2 : B
    ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    

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15. ࣗ વ ԋ ៷
    A 2
    ` A
    , A ` B
    ` A ! B
    ` A ! B ` A
    ` B
    x
    :
    A
    2
    `
    x
    :
    A
    , x
    :
    A
    `
    t
    :
    B
    `
    x
    :
    A. t
    :
    A
    !
    B
    ` t1 : A ! B ` t2 : A
    ` t1 t2 : B
    ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    ໋୊㱻ܕ
    

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16. ࣗ વ ԋ ៷
    A 2
    ` A
    , A ` B
    ` A ! B
    ` A ! B ` A
    ` B
    x
    :
    A
    2
    `
    x
    :
    A
    , x
    :
    A
    `
    t
    :
    B
    `
    x
    :
    A. t
    :
    A
    !
    B
    ` t1 : A ! B ` t2 : A
    ` t1 t2 : B
    ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    Curry–Howard
    ಉܕରԠ
    

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17. ࣗ વ ԋ ៷
    A 2
    ` A
    , A ` B
    ` A ! B
    ` A ! B ` A
    ` B
    x
    :
    A
    2
    `
    x
    :
    A
    , x
    :
    A
    `
    t
    :
    B
    `
    x
    :
    A. t
    :
    A
    !
    B
    ` t1 : A ! B ` t2 : A
    ` t1 t2 : B
    ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    ͱΜ͔ͭDJ
    

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18. C u r r y – H o w a rd ಉ ܕ ର Ԡ
    ଞͷ࿦ཧԋࢉࢠʹؔͯ͠ରԠΛߟ͑Δ͜ͱ΋Ͱ͖Δ
    ࣗ વ ԋ ៷ ୯ ७ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    ໋ ୊ ܕ
    ఆ ཧ ߲ ͕ ଘ ࡏ ͢ Δ ܕ
    ࿦ ཧ แ ؚ → ؔ ਺ ͷ ܕ →
    

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19. C u r r y – H o w a rd ಉ ܕ ର Ԡ
    ϥϜμܭࢉͰɺ͋Δܕͷ߲͕ଘࡏ͢Δ͜ͱΛࣔ͢
    ରԠ͢Δ໋୊ʢఆཧʣͷূ໌
    → ূ໌ࢧԉγεςϜʢ$PRͳͲʣ
    ϥϜμܭࢉͷ߲ʢϓϩάϥϜʣΛॻ͘
    ίϯϐϡʔλʔ͕ܕΛ֬ೝ
    ఆཧ͕ূ໌Ͱ͖ͨ
    

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20. Sy s t e m F
    

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21. Sy s t e m F
    w ֊ͷܕ෇͖ϥϜμܭࢉ
    ୯७ܕ෇͖ϥϜμܭࢉܕʹؔ͢Δந৅
    w ໋୊࿦ཧͷൣғΛશͯΧόʔͰ͖Δ
    ܕʹؔ͢Δந৅Ͱ࿦ཧԋࢉࢠΛΤϯίʔυՄೳ
    

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22. Sy s t e m F
    t ::= -- ߲
    x -- ม਺
    λx: T. t -- λந৅ʢؔ਺ʣ
    t t -- ؔ਺ద༻
    ΛX. t -- ܕʹؔ͢Δந৅ʢؔ਺ʣ
    t [T] -- ܕ΁ͷద༻
    T ::= -- ܕ
    X -- ܕม਺
    T → T -- ௨ৗͷؔ਺ͷܕ
    ∀X. T -- શশྔԽ͞Εͨܕ
    

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23. Sy s t e m F
    , X ` t : A
    ` ⇤X. t : 8X. A
    ` t : 8X. A
    ` t [B] : [X 7! B]A
    

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24. Sy s t e m F
    ֤छ࿦ཧԋࢉࢠͷΤϯίʔυ͸ҎԼͷͱ͓Γ
    A ^ B ⌘ 8X. ((A ! B ! X) ! X) ! X
    A _ B ⌘ 8X. ((A ! X) ! (B ! X) ! X) ! X
    ? ⌘ 8X. X
    ¬A ⌘ A ! ?
    9X. A ⌘ 8Y. (8X. A ! Y ) ! Y
    

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25. Sy s t e m F
    w ূ໌ࢧԉγεςϜͬΆ͍ͷΛ࣮૷ͯ͠Έͨ
    IUUQTHJUIVCDPNTVTJTVTZTUFNGKTJNQM
    w σϞʢ͕࣌ؒ͋Ε͹ʣʢͨͿΜ࣌ؒͳ͍ʣ
    

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26. · ͱ Ί
    

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27. · ͱ Ί
    w Curry–HowardಉܕରԠ͸࿦ཧֶͱϓϩάϥϜͷؒͷ
    ରԠؔ܎
    w System FΛ࣮૷͢Δͷָ͔ͬͨ͠ʢंྠͷ࠶ൃ໌ʣ
    ࣗ વ ԋ ៷ ܕ ෇ ͖ ϥϜ μ ܭ ࢉ
    ໋ ୊ ܕ
    ఆ ཧ ߲ ͕ ଘ ࡏ ͢ Δ ܕ
    

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28. λ→
    λ2
    λω
    λ ω
    λP
    λPω
    λP2
    λC
    

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29. λ→
    λ2
    λω
    λ ω
    λP
    λPω
    λP2
    λC
    ୯७ܕ෇͖ϥϜμܭࢉ
    System F
    System Fω
    (Haskell)
    CoC
    (Coq)
    

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30. λ→
    λ2
    λω
    λ ω
    λP
    λPω
    λP2
    λC
    ୯७ܕ෇͖ϥϜμܭࢉ
    System F
    System Fω
    (Haskell)
    CoC
    (Coq)
    :zenzenyowai:
    

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31. ࢀ ߟ จ ݙ
    • ஜ೾େֶͷܭࢉ࿦ཧֶͷߨٛࢿྉ
    http://www.cs.tsukuba.ac.jp/~kam/complogic/
    •
    ޒेཛྷ ३, ϓϩάϥϛϯάݴޠͷجૅ֓೦, αΠΤϯε
    ࣾ, 2011
    • B. C. Pierce, Types and Programming Languages,
    The MIT press, 2002
    ʢ೔ຊޠ༁΋͋Δʣ
    

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