プログラマのためのモナド(圏論) / #cat4pg

プログラマのための圏論勉強会 - connpass #cat4pg (Oct 16, 2018)
https://cat4pg.connpass.com/event/100728/

YouTube:
https://www.youtube.com/watch?v=fhuAlDPgi2Q

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■ 前回

圏論とSwiftへの応用 / iOSDC Japan 2018 - Speaker Deck
https://speakerdeck.com/inamiy/iosdc-japan-1

iOSDC Japan 2018 「圏論とSwiftへの応用」発表スライドメモ - Qiita
https://qiita.com/inamiy/items/3e0c10d5eaf234b41c3d

iOSDC Japan 2018 で発表した「圏論とSwiftへの応用」の補足 - Qiita
https://qiita.com/inamiy/items/c4e85f22273e98b8db26

Transcript

1. !ϓϩάϥϚͷͨΊͷ"
   Ϟφυ ݍ࿦ 
   ϓϩάϥϚͷͨΊͷݍ࿦ษڧձ
   DBUQH
   Ҵݟ
   ହ޺
   !JOBNJZ
2. Ҵݟହ޺
   !JOBNJZ
   "CFNB57
 J04ΤϯδχΞ
   4XJGU
   
   )BTLFMM
   
   &MN
   ݍ࿦ྺ
   ೥
   J04%$
   ొஃ
5. ϒϩάهࣄ w ݍ࿦ͷΦεεϝษڧ๏ʢϓϩάϥϚ޲͚ʣ
   
   2JJUB
   w J04%$
   +BQBO
   
   ʮݍ࿦ͱ4XJGU΁ͷԠ༻ʯൃදεϥΠυϝϞ
   
   2JJUB
   w J04%$
   +BQBO
   
   Ͱൃදͨ͠ʮݍ࿦ͱ4XJGU΁ͷԠ༻ʯͷิ଍
   
   2JJUB
6. !ϓϩάϥϚͷͨΊͷ"
   Ϟφυೖ໳
7. ϓϩάϥϚͷͨΊͷϞφυೖ໳ ശͰߟ͑Δ'VODUPSɺ"QQMJDBUJWFͦͯ͠.POBE
   
   2JJUB
8. !ϓϩάϥϚͷͨΊͷ"
   Ϟφυ ݍ࿦
9. ͳͥϓϩάϥϚ͸
   Ϟφυʹऒ͔ΕΔͷ͔ʁ
11. ؔ਺ܕϓϩάϥϛϯά
    αʔόʔαΠυ։ൃ
13. ΫϥΠΞϯτ։ൃ
    ΦϒδΣΫτࢤ޲ʴؔ਺ܕ
14. ΫϥΠΞϯτ։ൃݴޠͷॆ࣮ʢྫɿ4XJGUʣ w0QUJPOBM
    DIBJOJOH
    •view?.superview?.frame ==
 view.flatMap { $0.superview }.map { $0.frame } w1SPNJTF
    
    'VUVSF
    •sendRequest(req).then { nextRequest($0) } wؔ਺ܕϦΞΫςΟϒϓϩάϥϛϯά
    wObservable.flatMap
    (merge, concat, first, latest, race, …)
15. // RxSwiftΛ࢖ͬͨɺΠϯΫϦϝϯλϧαʔνͷྫ searchBar.rx.text .debounce(0.1, scheduler: backgroundScheduler) .flatMapLatest { API.fetchData(query: $0) } .map { $0.results.map(SectionModel.init) } .observeOn(mainScheduler) .bind(to: tableView.rx.items( cellIdentifier: "Cell", configureCell: { index, model, cell in cell.textLabel?.text = model } ) .disposed(by: disposeBag)
16. // RxSwiftΛ࢖ͬͨɺΠϯΫϦϝϯλϧαʔνͷྫ searchBar.rx.text .debounce(0.1, scheduler: backgroundScheduler) .flatMapLatest { API.fetchData(query: $0) } .map { $0.results.map(SectionModel.init) } .observeOn(mainScheduler) .bind(to: tableView.rx.items( cellIdentifier: "Cell", configureCell: { index, model, cell in cell.textLabel?.text = model } ) .disposed(by: disposeBag)
17. Ϟφυ
    flatMap / then / >>=
18. ͦͷଞͷϞφυ׆༻ࣄྫ w)BTLFMM
    EPߏจ
    
    4DBMB
    GPS಺แදه
    %4-
    w&JUIFSʢΤϥʔॲཧʣ
    w45
    
    4UBUF
    ঢ়ଶૢ࡞
    w*0ʢ෭࡞༻ʣ
    w$PODVSSFOUʢฒྻॲཧʣ
    w1BSTFDʢߏจղੳʣ
19. Ϟφυ
    
    ߏ଄Խఆཧͷͭͷجຊ੍ޚߏ଄Λ΋ͭ  ॱ࣍
    ˺
    "QQMJDBUJWF 
 ϓϩάϥϜʹه͞Εͨॱʹɺஞ࣍ॲཧΛߦͳ͍ͬͯ͘
     ൓෮
    ˺
    'VODUPS 
 Ұఆͷ৚͕݅ຬͨ͞Ε͍ͯΔؒॲཧΛ܁Γฦ͢
     ෼ذ
    ˺
    .POBE 
 ͋Δ৚͕݅੒ཱ͢ΔͳΒॲཧ"Λɺͦ͏Ͱͳ͚Ε͹ॲཧ#Λߦ͏
21. A monad is just a monoid in the category of endofunctors, 
 what's the problem? ෆ׬શʹ͓ͯ͠Αͦਖ਼͘͠ͳ͍ϓϩάϥϛϯάݴޠখ࢙
    
    ݩωλ͸ɺ.BD
    -BOFʮݍ࿦ͷجૅʯ
22. Ϟφυ͸୯ͳΔࣗݾؔख ͷݍʹ͓͚ΔϞϊΠυର ৅ͩΑɻԿ͔໰୊Ͱ΋?
24. Ϟφυ͸୯ͳΔࣗݾؔख ͷݍʹ͓͚ΔϞϊΠυର ৅ͩΑɻԿ͔໰୊Ͱ΋?
25. 
    ࣗݾؔखͷݍ
    
    ϞϊΠυର৅
26. ؔख Functor
27. ݍ ݍ
28. ݍ ݍ
29. ݍ ݍ
30. ܕͷݍ String [String] Int [Int] Bool [Bool] Array ܕͷݍ f g map(f ) map(g) g ∘ f map(g ∘ f ) map(g) ∘ map(f ) =
31. A B C ݍ F f g g ∘ f ݍ F(f) F(g) F(A) F(B) F(C) F(g ∘ f) F(g) ∘ F(f) =
32. ؔखͷ߹੒ Functor Composition
33. A B C ݍ f g g ∘ f ݍ F(f) F(g) F(A) F(B) F(C) F(g ∘ f) F(g) ∘ F(f) = F
34. ݍ ݍ F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) =
35. খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat
36. খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat খ͍͞ݍ G G(F( f )) G(F(g)) G(F(A)) G(F(B)) G(F(C)) G(F(g ∘ f )) G(F(g)) ∘ G(F( f )) =
37. খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat খ͍͞ݍ G G(F( f )) G(F(g)) G(F(A)) G(F(B)) G(F(C)) G(F(g ∘ f )) G(F(g)) ∘ G(F( f )) = G ∘ F
38. A B C ݍ G ∘ F f g g ∘ f ݍ G(F(f)) G(F(g)) G(F(A)) G(F(B)) G(F(g ∘ f)) G(F(g)) ∘ G(F(f)) = G(F(C))
39. (খ͍͞ݍͷݍ) F C D Cat
40. (খ͍͞ݍͷݍ) F G C D E Cat G ∘ F ؔखͷ߹੒
41. (খ͍͞ݍͷݍ) F G C D E Cat G ∘ F I I I ߃౳ؔख ؔखͷ߹੒
42. (খ͍͞ݍͷݍ) Cat F G C D E G ∘ F E′ H H ∘ G (H ∘ G) ∘ F H ∘ (G ∘ F) ྆ऀ͸౳͍͠ (݁߹཯Λຬͨ͢)
43. ؔखݍ Functor Category
44. (খ͍͞ݍͷݍ) Cat F C D
45. (খ͍͞ݍͷݍ) Cat C D F H G
46. (খ͍͞ݍͷݍ) Cat C D F H G α β
47. A B F f H F(A) F(B) F(f) α C D G(A) G(B) G(f) H(A) H(B) H(f) β G αA αB βA βB
48. A B F f H F(A) F(B) F(f) α C D G(A) G(B) G(f) H(A) H(B) H(f) β G αA αB βA βB β ∘ α βA ∘ αA βB ∘ αB ࣗવม׵ͷ ਨ௚߹੒
 (݁߹཯Λຬͨ͢)
49. (খ͍͞ݍͷݍ) Cat C D F H G α β ؔखݍ DC
50. (ؔखݍ) α β F G H DC
51. α β F G H β ∘ α id id id ߃౳ࣗવม׵ ࣗવม׵ͷ(ਨ௚)߹੒ (݁߹཯Λຬͨ͢) (ؔखݍ) DC
52. ࣗݾؔखͷݍ Category of Endofunctors
53. (খ͍͞ݍͷݍ) Cat C D F H G
54. (খ͍͞ݍͷݍ) Cat C C F H G
55. (খ͍͞ݍͷݍ) Cat C F G H
56. (খ͍͞ݍͷݍ) Cat C G ∘ F F G H I
57. (খ͍͞ݍͷݍ) Cat C G ∘ F F G H I ࣗݾؔखͷݍ CC α β
58. α β F G H (ؔखݍ) DC
59. α β F G H (ࣗݾؔखͷݍ) CC I G ∘ F ߃౳ؔख (୯Ґݩ) ؔखͷ߹੒ (ೋ߲ԋࢉ)
60. ϞϊΠυ Monoid
61. ϞϊΠυ
    
    ೋ߲ԋࢉ
    
    ୯Ґݩͷଘࡏ
62. (Int, +, 0) a + 0 = a 0 + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ
63. (Int, *, 1) a * 1 = a 1 * a = a (a * b) * c = a * (b * c) ϞϊΠυͷྫ
64. (String, +, ””) a + ”” = a ”” + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ
65. (Array, +, []) a + [] = a [] + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ
66. (Bool, &&, true) a && true = a true && a = a (a && b) && c = a && (b && c) ϞϊΠυͷྫ
67. (Bool, ||, false) a || false = a false || a = a (a || b) || c = a || (b || c) ϞϊΠυͷྫ
68. (Int, max, minBound) max(a, minBound) = a max(minBound, a) = a max(max(a, b), c)
 = max(a, max(b, c)) ϞϊΠυͷྫ
69. (Int, min, maxBound) min(a, maxBound) = a min(maxBound, a) = a min(min(a, b), c)
 = min(a, min(b, c)) ϞϊΠυͷྫ
70. ϞϊΠυ w ू߹
    
    
    
    
    ɺೋ߲ԋࢉ
    
    
    υοτ ɺ୯Ґݩ
    
    
    
    
    
    
    
    
    
    
    
    ͷͭ૊
    
    
    
    
    
    
    
    
    
    
    
    
 ͰɺҎԼͷ৚݅Λຬͨ͢ɿ
    w ݁߹཯
    
    
    ೚ҙͷ
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    ʹ͍ͭͯ
    w ୯Ґݩ
    
    
    ೚ҙͷ
    
    
    
    
    
    
    
    
    
    
    
    ʹ͍ͭͯ (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) e ⋅ a = a ⋅ e = e a, b, c ∈ M a ∈ M M ⋅ e ∈ M (M, ⋅ ,e) (M, ⋅ ,e)
71. // Swift ʹ͓͚ΔϞϊΠυ protocol Monoid { static func <> (lhs: Self, rhs: Self) -> Self static var empty: Self } -- Haskell ʹ͓͚ΔϞϊΠυ class Monoid a where (<>) :: a -> a -> a mempty :: a
72. ݍ࿦ʹ͓͚ΔϞϊΠυ Monoid in Category Theory
73. Int 1 … 3 (+): (Int,Int) -> Int ΛࣹͰදݱ͍ͨ͠l 0 2 ୯Ґݩ
74. Int 0 1 2 … 3 ୯Ґݩ ϖΞΛ࡞ͬͯɾɾɾ ೋ߲ԋࢉʁ ͜ͷॻ͖ํ͸ݍ࿦Ͱ͸ͳ͍ ΛࣹͰදݱ͍ͨ͠l (+): (Int,Int) -> Int
75. σΧϧτੵ
    ੵू߹ (0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 (Int,Int) ͱ΋ॻ͘ Int × Int Int 0 1 2 … 3 Int 0 1 2 … 3 ϖΞ λϓϧ (+)
76. (0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 (+) ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ಺෦Ͱࣹ(ߏ଄)Λ ߟ͑ͳ͍ (Int,Int) Int 0 1 2 … 3 Int 0 1 2 … 3 ϖΞ λϓϧ σΧϧτੵ
    ੵू߹ σΧϧτੵ Int × Int
77. (+) : (Int, Int) -> Int ΧϦʔԽ
    ͢Δͱɺ
    (+) : Int -> (Int -> Int) ϖΞΛ࡞ͬͯɺIntʹม׵ ϖΞΛ࡞Βͣɺ IntΛʮࣗݾؔ਺ʯʹม׵ ͱ͜ΖͰɾɾɾ
78. ୯Ұର৅ݍ Int (0+) (1+) (2+) Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+)
79. ୯Ұର৅ݍ Int (0+) (1+) (2+) Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+) ߃౳ม׵ ࣹͳͷͰɺ ߹੒Մೳ
80. 0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) (+)ͷ෦෼ద༻
    ࣗݾؔ਺Խ Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+)
81. Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) { f in f(0) } \f -> f 0 0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) ୯ҐݩΛద༻
    ஋ʹ໭͢ PS
82. Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) 0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) ͷ෦෼ద༻
    ࣗݾؔ਺Խ ୯ҐݩΛద༻
    ஋ʹ໭͢ ୯Ґݩ ೋ߲ԋࢉ ୯Ґݩ ೋ߲ԋࢉ ࣗݾؔ਺͸ ϞϊΠυʂ
83. 0 + 1 + 2 ͷ෦෼ద༻
    ؔ਺Խ ୯ҐݩΛద༻
    ஋ʹ໭͢ (0+) ∘ (1+) ∘ (2+) ୯Ґݩ ೋ߲ԋࢉ έΠϦʔදݱ
    ೚ҙͷϞϊΠυ͸ɺͦͷࣗݾ४ಉܕ
    ˺ࣗݾؔ਺ ͷϞϊΠυͷ෦෼ϞϊΠυ Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+)
84. ϞϊΠυର৅ Monoid Object
85. σΧϧτੵ
    ੵू߹ (0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 Int 0 1 2 … 3 Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ (Int,Int) ϖΞ ಺෦Ͱࣹ(ߏ଄)Λ ߟ͑ͳ͍ λϓϧ (+)
86. (Int,Int) Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ (+)
87. Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ () ୯ҐݩΛܾΊΔࣹΛ ʮ֎ʯͰఆٛ͢Δ(ҰൠԽݩ) Ұݩू߹ (Int,Int) 0 (+) Unit
88. ϞϊΠυର৅ (M, μ, η) μ η 1 ୯Ґࣹ ೋ߲ԋࢉ M × M M
89. Int 0 1 2 … 3 Unit () (Int,Int) ⟨ Int, Int ⟩ ϖΞ 0 (+) = Int ⊗ Int ⊗ = (, ) (Int × IntͷҰൠԽ)
90. (Int × IntͷҰൠԽ) Int 0 1 2 … 3 () ()  ϖΞ(ܕͰ͸ͳ͍) λϓϧܕ ʮϖΞΛ࡞Δʯ͜ͱͱʮϖΞΛܕͱ Έͳ͢(λϓϧ)ʯ͜ͱΛ෼཭͢Δ ςϯιϧੵ (ޙड़) (Int,Int) Int ⊗ Int = ⟨ Int, Int ⟩ ϖΞ ⊗ = (, )
91. ϞϊΠυର৅ (M, μ, η) μ η I M ⊗ M M ୯Ґࣹ ೋ߲ԋࢉ
92. (σΧϧτੵͷҰൠԽ) ⊗ ୯Ґର৅ (Ұݩू߹ͷҰൠԽ) μ η I M ⊗ M M ⊗ ςϯιϧੵ
93. ϞϊΠμϧݍ Monoidal Category
94. ςϯιϧੵ ⊗ : C × C → C M I M ⊗ M ⊗ ୯Ґର৅ (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ)
95. M I M ⊗ M M ⊗ I M ⊗ M′ (M ⊗ M′) ⊗ M′′ طଘͷର৅ͷϖΞ͔Β ৽ͨͳର৅Λ࡞Γग़ͤΔ ςϯιϧੵ ⊗ : C × C → C (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ⊗
96. ςϯιϧੵ ⟨M ⊗ M′, M′′⟩ ੵͷݍ I M M ⊗ M ⟨M, M⟩ C × C M ⊗ M′ ⟨M, M′⟩ ⋯ ⊗ (M ⊗ M′) ⊗ M′′ ݍͷϖΞ ⊗ ⋯ (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ૒ؔख
97. ϞϊΠμϧݍ wςϯιϧੵ
    
    w୯Ґର৅
    w݁߹ࢠ
    wࠨ୯Ґࢠ
    wӈ୯Ґࢠ
    
    ⊗ : C × C → C I ∈ Ob(C) αA,B,C : (A ⊗ B) ⊗ C ≅ ⟶ A ⊗ (B ⊗ C) λA : I ⊗ A ≅ ⟶ A ρA : A ⊗ I ≅ ⟶ A ౳߸ͷͱ͖ɺ ετϦΫτϞϊΠμϧݍ (C, ⊗ ,I, α, λ, ρ)
98. ϞϊΠμϧݍ (C, ⊗ ,I, α, λ, ρ) ݁߹ࢠ ݁߹ॱংͷೖΕସ͑Մೳ ୯Ґࢠ ୯Ґݩ Λ௥ՃɾআڈՄೳ I
99. μ η I M ⊗ M M ϞϊΠυʹʮϞϊΠμϧݍʯʹ͓͚ΔʮϞϊΠυର৅ʯ (ϞϊΠμϧݍ) (C, ⊗ ,I, α, λ, ρ) C
100. ࠶ߟɿࣗݾؔखͷݍ Revisiting Category of Endofunctors
101. α β F G H I G ∘ F ߃౳ؔख (୯Ґݩ) ؔखͷ߹੒ (ೋ߲ԋࢉ) (ࣗݾؔखͷݍ) CC
102. ؔखͷ߹੒ (ೋ߲ԋࢉ) (ࣗݾؔखͷݍ) CC ߃౳ؔख (୯Ґݩ) F I F ∘ F ∘
103. M ⊗ M ςϯιϧੵ ⊗ : C × C → C ୯Ґର৅ M I M ⊗ M (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ⊗
104. ؔखͷ߹੒
     ˠ
     ςϯιϧੵ
     ߃౳ؔख
     ˠ
     ୯Ґର৅
     ؔखͷ߹੒ʹ͍ͭͯ
     ݁߹཯ɾ୯Ґ཯͕੒Γཱͭ
105. ࣗݾؔखͷݍ
     
     ʹ
     ετϦΫτϞϊΠμϧݍ
106. Ϟφυ Monad
107. F(C) D ݍ F C ݍ U(D) ࣹͷू߹ U ≅
108. F(C) D ݍ F U C U(D) ݍ F(U(D)) U(F(C)) f f F(f) U(f) counitD ηC ϵD unitC = =
109. F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC ϵD
110. F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC ϵD U(F(U(F(C)))) F(U(F(C)))
111. F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC U(F(U(F(C)))) F(U(F(C))) ϵD ϵF(C)
112. F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC U(F(U(F(C)))) F(U(F(C))) ϵD ϵF(C) UϵF(C)
113. C ݍ T(C) ηC T2(C) T = UF UϵF(C)
114. C ݍ ηC T T ∘ T I T(C) T2(C) UϵF(C)
115. C ݍ ηC T T ∘ T μ η I T(C) T2(C) UϵF = ࣗݾؔखͷݍ UϵF(C)
116. μ η I M ⊗ M M ϞϊΠυʹʮϞϊΠμϧݍʯʹ͓͚ΔʮϞϊΠυର৅ʯ (ϞϊΠμϧݍ) (C, ⊗ ,I, α, λ, ρ) C
117. T T ∘ T μ η Ϟφυʹʮࣗݾؔखͷݍ(ετϦΫτϞϊΠμϧݍ)ʯ ʹ͓͚ΔʮϞϊΠυର৅ʯ (ࣗݾؔखͷݍ) CC (CC, ∘ ,I) I
118. Ϟφυ wࣗݾؔख
     wࣗવม׵
     wࣗવม׵ T : C → C μ : T ⊗ T → T η : I → T μ ∘ Tμ = μ ∘ μT μ ∘ Tη = μ ∘ ηT = 1T Ϟφυଇ ݁߹཯ ୯Ґ཯ (T, μ, η)
119. Ϟφυ (T, μ, η) μ ∘ Tμ = μ ∘ μT μ ∘ Tη = μ ∘ ηT = 1T ݁߹཯ ୯Ґ཯
120. ʹࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅ (T, μ, η) Ϟφυ
121. )BTLFMMʹ͓͚ΔϞφυ class Functor f where fmap :: (a -> b) -> f a -> f b class Functor f => Applicative f where pure :: a -> f a (<*>) :: f (a -> b) -> f a -> f b class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure (ࣗݾ)ؔख ΞϓϦΧςΟϒؔख Ϟφυ
122. join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure Ϟφυ
123. join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure ࣗવม׵ η : I → T
124. join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure μ : T ⊗ T → T ࣗવม׵
125. join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure flatMap
126. ϓϩάϥϛϯάͷੈքͰ͸ɺ
     join
     ΑΓ΋
     
     flatMap(>>=)ͷํ͕ॏཁ
127. )BTLFMMʹ͓͚Δ
     flatMap == (>>=) (>>=) :: m a -> (a -> m b) -> m b ͜ͷܕͷҙຯ͸Կʁ
128. ΫϥΠεϦ߹੒ͱ
     ΫϥΠεϦɾτϦϓϧ Kleisli Composition & Kleisli Triple
129. C B A C ࣹͷ߹੒͕Մೳ
130. T(B) C B A T(C) C Ϟφυ͕ίυϝΠϯ(ऴҬ)ͷ ࣹ͸߹੒Ͱ͖ͳ͍
131. C B A C B A T(B) T(C) ⋯ C CT ΫϥΠεϦݍ ⋯ ⋯ f g gT fT ΫϥΠεϦࣹ (A -> T(B)ͷܗͷࣹ)
132. C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT ྆ऀ ͷࣹͷू߹ Λ
     ରԠ͍ͤͨ͞
133. f g C B A C B A T(B) T(C) ⋯ ⋯ C CT ⋯ gT fT gT ∘ fT ରԠͰ͖Δͱɺ
     ݍ
     
     
     
     
     
     ͷࣹͷ߹੒
     A -> C
     ͕
     ݍ ͷΫϥΠεϦࣹͷ߹੒ A -> T(C)ʹରԠ͢Δ
     
     
     
     
     CT C
134. F G C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT
     
     
     
     
     ͱ
     
     
     
     
     
     
     ͷ
     ਵ൐ؔखΛߟ͑Δ CT C
135. C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT F G
     ຒΊࠐΈؔख
     ର৅ͱࣹΛ
     ͦͷ··ࣸ͢ fT ≠ F(f) ஫ҙɿ
136. f g C B A F G C B A T(B) T(C) ⋯ ⋯ C CT T(A) ⋯ gT fT G(fT ) G(gT ) ֦ுؔख
     
     
     
     
     ͷର৅
     
     
     
     Λ
     
     
     
     
     ͷର৅
     
     
     
     
     
     
     
     
     ʹࣸ͢ A CT C T(A) G(A) = T(A) G(fT ) ≠ T(f) ஫ҙɿ
137. f g C B A F G C B A T(B) T(C) ⋯ C CT T(A) F gT fT G(fT ) G(gT ) T2(C) T2(B) T(f ) T(g) μB ηB ηC μA ηA
     
     
     
     
     ͸ϞφυͳͷͰ
     ͜ΕΒͷࣹ͕ଘࡏ͢Δ T μC
138. g C B A F G C B ⋯ C CT T(A) F gT fT G(fT ) G(gT ) T2(B) gT ∘ fT T(f ) μB ηB μA ηA f A T(B) T(C) T2(C) T(g) μC μC ∘ T(g) ∘ f ηC
     
     
     
     ͱ
     
     
     
     ͷΫϥΠεϦ߹੒ f g
139. ΫϥΠεϦ߹੒ C A gT ∘ fT f g B A T(B) T(C) μC ∘ T(g) ∘ f g < = < f = def < = < /PUF
     ΫϥΠεϦ߹੒͸
     ݁߹཯ɾ୯Ґ཯Λຬͨ͢
140. wΫϥΠεϦ߹੒
     ࢀߟ
     ؔ਺߹੒ ࠨ୯Ґ཯ return <=< f ≡ f ӈ୯Ґ཯ f <=< return ≡ f ݁߹཯ (h <=< g) <=< f ≡ h <=< (g <=< f) Ϟφυଇ
     ΫϥΠεϦ߹੒ (<=<) :: (b -> m c) -> (a -> m b) -> (a -> m c) (.) :: (b -> c) -> (a -> b) -> (a -> c) ࠨ୯Ґ཯ id . f ≡ f ӈ୯Ґ཯ f . id ≡ f ݁߹཯ (h . g) . f ≡ h . (g . f)
141. )BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒ (<=<) :: Monad m => (a -> m b) -> m a -> m b g <=< f = join . fmap g . f μC ∘ T(g) ∘ f g < = < f = def
142. )BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒
     ࣮ࡍͷίʔυ -- | Left-to-right composition of Kleisli arrows. (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c) f >=> g = \x -> f x >>= g -- | Right-to-left composition of Kleisli arrows. -- > (.) :: (b -> c) -> (a -> b) -> a -> c -- > (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \x -> g =<< f x —- (<=<) = flip (>=>)
143. -- | Left-to-right composition of Kleisli arrows. (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c) f >=> g = \x -> f x >>= g -- | Right-to-left composition of Kleisli arrows. -- > (.) :: (b -> c) -> (a -> b) -> a -> c -- > (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \x -> g =<< f x —- (<=<) = flip (>=>) (=<<) :: Monad m => (a -> m b) -> m a -> m b f =<< x = x >>= f —- (=<<) = flip (>>=) )BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒
     ࣮ࡍͷίʔυ flip(flatMap)
144. (=<<) :: (a -> m b) -> m a -> m b ͜ͷܕͷҙຯ͸Կʁ )BTLFMMʹ͓͚Δ
     flip(flatMap) == (=<<)
145. (=<<) :: (a -> m b) -> m a -> m b ͜ͷܕͷҙຯ͸Կʁ ʮa -> m bʯͷࣹ͔Β
     ʮm a -> m bʯͷࣹʹࣸؔ͢਺ )BTLFMMʹ͓͚Δ
     flip(flatMap) == (=<<)
146. A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB F G G(fT ) f
147. A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB F G = < < ֦ுԋࢉ
148. A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB = < < ֦ுԋࢉ F G ֦ுؔख
149. F G f A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT G(fT ) T2(B) T(f ) μB G(fT ) = μB ∘ T(f) ӈล͸
     
     
     
     
     Λ
     Ҿ਺ʹऔΔؔ਺ f
150. Ϟφυɾ֦ுԋࢉ = < < A T(B) T(A) B A fT T2(B) T(f ) μB μB ∘ T(f) f = < < = def f
151. Ϟφυɾ֦ுԋࢉ f A T(B) T(A) B A fT T2(B) T(f ) μB f = < < Λ࢖Θͣ Λ࢖ͬͯϞφυΛදݱ͢Δ μ = < < = < < ֦ுԋࢉ = < <
152. ΫϥΠεϦɾτϦϓϧ wࣗݾؔख
     wࣗવม׵
     w֦ுԋࢉ (T, η, = < < ) T : C → C η : I → T = < <: (A → T(B)) → T(A) → T(B) (g = < < ) ∘ (f = < < ) = (((g = < < ) ∘ f) = < < ) (η = < < ) = 1T ݁߹཯ ୯Ґ཯ (f = < < ) ∘ η = f Ϟφυଇ ,MFJTMJ
     4UBS
     ͱ΋ݴ͏ f* def = (f = < < )
153. ΫϥΠεϦɾτϦϓϧ
     ʹϞφυͷ
     join
     Λ࢖Θͣ
     
     flip(flatMap)
     Λ࢖ͬͨผͷܗ (T, η, = < < )
154. )BTLFMMʹ͓͚Δ֦ுԋࢉ (=<<) :: Monad m => (a -> m b) -> m a -> m b (=<<) f = join . fmap f μB ∘ T(f) f = < < = def = < <
155. ࢀߟɿjoin, <=<, =<< ͷؔ܎ -- from `join` join :: (Monad m) => m (m a) -> m a join x = id =<< x join x = (id <=< id) x -- from `<=<` (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \a -> g =<< f a g <=< f = join . fmap g . f -- from `=<<` (=<<) :: Monad m => (a -> m b) -> m a -> m b f =<< m = (f <=< id) m f =<< m = (join . fmap f) m
156. wΫϥΠεϦ߹੒ wϞφυflatMap ࠨ୯Ґ཯ return >=> g ≡ g ӈ୯Ґ཯ f >=> return ≡ f ݁߹཯ (f >=> g) >=> h ≡ f >=> (g >=> h) ࠨ୯Ґ཯ return a >>= f ≡ f a ӈ୯Ґ཯ m
     
     >>= return ≡ m ݁߹཯ (m >>= f) >>= g ≡ m >>= (\x -> f x >>= g) Ϟφυଇ
     (flatMap / >>=) (>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c) (>>=) :: m a -> (a -> m b) -> m b (<=<)
     ͷٯ޲͖
     ࠨ͔Βӈ΁ͷΫϥΠεϦ߹੒ f >=> g = \a -> f a >>= g
157. ·ͱΊ
158. ·ͱΊ wϞφυ
     wؔखݍɺࣗݾؔखͷݍ
     wϞϊΠυɺϞϊΠυର৅ɺϞϊΠμϧݍ
     wࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅
     wΫϥΠεϦɾτϦϓϧ
     wjoin
     ͷ୅ΘΓʹ
     flip(flatMap) /
     =<<
     Λ࢖͏
159. (T, μ, η) (T, η, = < < )
161. ͞ΒʹֶͿͨΊʹ w ετϦϯάਤ
     4USJOH
     %JBHSBN .POBET
     .BEF
     %J⒏DVMU
 4USJOH
     EJBHSBNT
     BEKVODUJPOT
     BOE
     NPOBET
     
     :PV5VCF
162. ͞ΒʹֶͿͨΊʹ wϞφυม׵ࢠ
     .POBE
     5SBOTGPSNFS
     wࣗ༝Ϟφυ
     'SFF
     .POBE
     w'SFFS
     
     0QFSBUJPOBMϞφυ
     ࣗ༝ؔख
     
     ࣗ༝Ϟφυ
     wࢦඪ෇͖Ϟφυ
     *OEFYFE
     .POBE
     w&YUFOTJCMF
     
     'SFFS
     &GGFDUT
     'SFFS
     
     0QFO
     6OJPO
     w(SBEFEϞφυ
     ϞφυͷҰൠԽ
163. ࢀߟจݙ wϞφυ
     
     ΢ΥʔΫεϧʔ
     )BTLFMM
     wϞφυʹ͍ͭͯ
     ϞϊΠυɾϞϊΠυݍɾϞϊΠυର৅ɾϞφυ
     
     ٙ೦͸୳ڀͷಈػͰ͋Γɺ୳ڀͷ།Ұͷ໨త͸৴ ೦ͷ֬ఆͰ͋Δɻ
     wϓϩάϥϛϯάʹ͓͚ΔϞφυͷॳظͷྺ࢙ʹ͍ͭͯ
     
     ࠶ؼͷ൓෮
     wʮϞφυ͸୯ͳΔࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅ͩΑɻԿ͔໰୊Ͱ΋ʁʯ
     
     2JJUB
     w)BTLFMMͷ.POBEΛݍ࿦ͷ.POBE,MFJTMJ
     5SJQMFͱରൺ͢Δ
     
     Ί΋Ί΋
     wϞφυͷఆٛͱ͔
     
     ᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه
     wϞϊΠυ͔ΒϞφυΛ࡞Δ
     
     ᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه
     w.POBET
     .BEF
     %JGpDVMU
     w/PUJPOT
     PG
     $PNQVUBUJPO
     BT
     .POPJET
164. Thanks! Yasuhiro Inami @inamiy