Simplicity in composition

Transcript

1. Simplicity in composition
   Adelbert Chang
   @adelbertchang
   Scala World 2017
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2. Composition
   A: type (set) of values
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3. Composition
   A: type (set) of values
   ⊕: A × A → A
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4. Composition
   A: type (set) of values
   ⊕: A × A → A
   1A
   : identity for ⊕
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5. Composition
   A: type (set) of values
   ⊕: A × A → A
   1A
   : identity for ⊕
   (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)
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6. Composition
   A: type (set) of values
   ⊕: A × A → A
   1A
   : identity for ⊕
   (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)
   x = x ⊕ 1A
   = 1A
   ⊕ x
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7. (Z, +, 0)
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8. ([a], +
   +, [])
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9. (A → A, ◦, a → a)
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10. trait Monoid[A] {
    def combine(x: A, y: A): A
    def empty: A
    }
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11. 1 2 3 4 5 6 7 8
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15. 36
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16. 1 2 3 4 5 6 7 8
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17. 3 7 11 15
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18. 10 26
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19. 36
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20. 1 2 3 4 5 6 7 8
    3 7 11 15
    10 26
    36
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21. Associative composition allows for modular
    decomposition and reasoning.
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22. Composing programs
    A
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23. Composing programs
    F[A]
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24. Composing programs
    (F[A], F[A]) => F[A]
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25. Composing programs
    (F[A], F[B]) => F[?]
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26. Composing programs
    (F[A], F[B]) => F[(A, B)]
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27. Composing programs
    F( ): type of program
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28. Composing programs
    F( ): type of program
    ⊗ : F(A) × F(B) → F((A, B))
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29. Composing programs
    F( ): type of program
    ⊗ : F(A) × F(B) → F((A, B))
    η : A → F(A)
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30. Composing programs
    F( ): type of program
    ⊗ : F(A) × F(B) → F((A, B))
    η : A → F(A)
    (fa ⊗ fb) ⊗ fc ∼
    = fa ⊗ (fb ⊗ fc)
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31. Composing programs
    F( ): type of program
    ⊗ : F(A) × F(B) → F((A, B))
    η : A → F(A)
    (fa ⊗ fb) ⊗ fc ∼
    = fa ⊗ (fb ⊗ fc)
    fa ∼
    = fa ⊗ ηUnit
    ∼
    = ηUnit
    ⊗ fa
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32. Composing programs
    def zipOption[A, B]
    (oa: Option[A], ob: Option[B]): Option[(A, B)] =
    (oa, ob) match {
    case (Some(a), Some(b)) => Some((a, b))
    case _ => None
    }
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33. Composing programs
    def zipOption[A, B]
    (oa: Option[A], ob: Option[B]): Option[(A, B)] =
    (oa, ob) match {
    case (Some(a), Some(b)) => Some((a, b))
    case _ => None
    }
    def pureOption[A](a: A): Option[A] = Some(a)
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34. Composing programs
    def zipList[A, B]
    (la: List[A], lb: List[B]): List[(A, B)] =
    la match {
    case Nil => Nil
    case h :: t => lb.map((h, _)) ++ zipList(t, lb)
    }
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35. Composing programs
    def zipList[A, B]
    (la: List[A], lb: List[B]): List[(A, B)] =
    la match {
    case Nil => Nil
    case h :: t => lb.map((h, _)) ++ zipList(t, lb)
    }
    def pureList[A](a: A): List[A] = List(a)
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36. Composing programs
    def zipFunction[A, B, X]
    (f: X => A, g: X => B): X => (A, B) =
    (x: X) => (f(x), g(x))
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37. Composing programs
    def zipFunction[A, B, X]
    (f: X => A, g: X => B): X => (A, B) =
    (x: X) => (f(x), g(x))
    def pureFunction[A, X](a: A): X => A =
    (_: X) => a
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38. /** Same as Applicative */
    trait Monoidal[F[_]] {
    def zip[A, B](fa: F[A], fb: F[B]): F[(A, B)]
    def pure[A](a: A): F[A]
    /*
    def map[A, B](fa: F[A])(f: A => B): F[B]
    */
    }
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40. Composing programs
    (F[A], F[B]) => F[(A, B)]
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41. Composing independent programs
    (F[A], F[B]) => F[(A, B)]
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42. Composing programs
    F[A]
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43. Composing dependent programs
    (F[A], A => F[B]) => F[B]
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44. Composing dependent programs
    F( ): type of program
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
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45. Composing dependent programs
    F( ): type of program
    >
    >=: (F(A) × A → F(B)) → F(B)
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
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46. Composing dependent programs
    F( ): type of program
    >
    >=: (F(A) × A → F(B)) → F(B)
    η : A → F(A)
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 33 / 67
    

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47. Composing dependent programs
    F( ): type of program
    >
    >=: (F(A) × A → F(B)) → F(B)
    η : A → F(A)
    (fa >
    >= f ) >
    >= g = fa >
    >= (f >=> g)1
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
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48. Composing dependent programs
    F( ): type of program
    >
    >=: (F(A) × A → F(B)) → F(B)
    η : A → F(A)
    (fa >
    >= f ) >
    >= g = fa >
    >= (f >=> g)1
    f (x) = η(x) >
    >= f
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 33 / 67
    

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49. Composing dependent programs
    F( ): type of program
    >
    >=: (F(A) × A → F(B)) → F(B)
    η : A → F(A)
    (fa >
    >= f ) >
    >= g = fa >
    >= (f >=> g)1
    f (x) = η(x) >
    >= f
    fa = fa >
    >= η
    1>
    =
    >: (A → F(B) × B → F(C)) → A → F(C)
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 33 / 67
    

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50. Composing dependent programs
    def flatMapOption[A, B]
    (oa: Option[A], f: A => Option[B]): Option[B] =
    oa match {
    case Some(a) => f(a)
    case None => None
    }
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51. Composing dependent programs
    def flatMapList[A, B]
    (la: List[A], f: A => List[B]): List[B] =
    la match {
    case Nil => Nil
    case h :: t => f(h) ++ flatMapList(t, f)
    }
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52. Composing dependent programs
    def flatMapFunction[A, B, X]
    (fa: X => A, f: A => (X => B)): X => B =
    (x: X) => f(fa(x))(x)
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53. trait Monad[F[_]] {
    def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
    def pure[A](a: A): F[A]
    }
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54. trait Monad[F[_]] extends Monoidal[F] {
    def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
    def pure[A](a: A): F[A]
    }
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55. A nicer monad
    (fa >
    >= f ) >
    >= g = fa >
    >= (f >=> g)
    f (x) = η(x) >
    >= f
    fa = fa >
    >= η
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56. A nicer monad
    f : A → F(B)
    g : B → F(C)
    h : C → F(D)
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57. A nicer monad
    f : A → F(B)
    g : B → F(C)
    h : C → F(D)
    (f >=> g) >=> h = f >=> (g >=> h)
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58. A nicer monad
    f : A → F(B)
    g : B → F(C)
    h : C → F(D)
    (f >=> g) >=> h = f >=> (g >=> h)
    f = f >=> η = η >=> f
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59. User info Friends Cookies Logs
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60. Category theory studies the algebra of composition.
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61. Category theory
    objects: A, B, C ...
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62. Category theory
    objects: A, B, C ...
    arrows: f : A → B, g : B → C ...
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63. Category theory
    objects: A, B, C ...
    arrows: f : A → B, g : B → C ...
    1A
    : A → A
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64. Category theory
    objects: A, B, C ...
    arrows: f : A → B, g : B → C ...
    1A
    : A → A
    (f ◦ g) ◦ h = f ◦ (g ◦ h)
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65. Category theory
    objects: A, B, C ...
    arrows: f : A → B, g : B → C ...
    1A
    : A → A
    (f ◦ g) ◦ h = f ◦ (g ◦ h)
    f = f ◦ 1A
    = 1A
    ◦ f
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66. Category theory
    B
    A C
    g
    1B
    f
    g◦f
    1A
    1C
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67. Category theory
    B
    A C
    g
    1B
    f
    g◦f
    1A
    1C
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68. Category theory
    B
    A C
    g
    1B
    f
    g◦f
    1A
    1C
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69. Category theory: monoids
    •
    x
    y
    x⊕y
    1A
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70. Category theory: monoids
    •
    x
    y
    x⊕y
    1A
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71. Category theory: monoids
    •
    x
    y
    x⊕y
    1A
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72. Category theory: monoids
    •
    x
    y
    x⊕y
    1A
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73. Category theory: monads
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74. Category theory: monads
    F(B)
    A F(C)
    f
    ?◦f
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75. Category theory: monads
    F(B)
    A F(C)
    f
    ?◦f
    >
    >=: (F(B) × B → F(C)) → F(C)
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76. Category theory: monads
    F(B)
    A F(C)
    f
    ?◦f
    >
    >=: (B → F(C)) → F(B) → F(C)
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77. Category theory: monads
    F(B)
    A F(C)
    g∗
    f
    g∗◦f
    >
    >=: (B → F(C)) → F(B) → F(C)
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78. Category theory: monads
    B
    A C
    g
    1B
    B→F(B)
    f
    g ◦f
    1A
    A→F(A) 1C
    C→F(C)
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79. Category theory: monads
    B
    A C
    g
    1B
    B→F(B)
    f
    g ◦f
    1A
    A→F(A) 1C
    C→F(C)
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80. Category theory: monads
    B
    A C
    g
    1B
    B→F(B)
    f
    g ◦f
    1A
    A→F(A) 1C
    C→F(C)
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81. Category theory: monads
    B
    A C
    g
    1B
    B→F(B)
    f
    g ◦f
    1A
    A→F(A) 1C
    C→F(C)
    η : X → F(X)
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82. User info Friends Cookies Logs
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83. What if we have different monoids, monads, etc?
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84. (Monoid[A], Monoid[B]) => Monoid[?]
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85. Product composition: monoids
    (Monoid[A], Monoid[B]) => Monoid[(A, B)]
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86. Product composition: monoids
    def combine(x: (A, B), y: (A, B)): (A, B) = {
    val a = x._1 combine y._1
    val b = x._2 combine y._2
    (a, b)
    }
    0Assuming Monoid[A] and Monoid[B]
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87. Product composition: monoids
    def combine(x: (A, B), y: (A, B)): (A, B) = {
    val a = x._1 combine y._1
    val b = x._2 combine y._2
    (a, b)
    }
    def empty: (A, B) = (empty[A], empty[B])
    0Assuming Monoid[A] and Monoid[B]
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88. (Monoidal[F], Monoidal[G]) => Monoidal[?]
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89. Product composition: lax monoidal functors
    type L[X] = (F[X], G[X])
    (Monoidal[F], Monoidal[G]) => Monoidal[L]
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90. Product composition: lax monoidal functors
    def zip[A, B](x: (F[A], G[A]),
    y: (F[B], G[B])): (F[(A, B)], G[(A, B)]) = {
    val f = x._1 zip y._1
    val g = x._2 zip y._2
    (f, g)
    }
    0Assuming Monoidal[F] and Monoidal[G]
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91. Product composition: lax monoidal functors
    def zip[A, B](x: (F[A], G[A]),
    y: (F[B], G[B])): (F[(A, B)], G[(A, B)]) = {
    val f = x._1 zip y._1
    val g = x._2 zip y._2
    (f, g)
    }
    def pure[A](a: A): (F[A], G[A]) = (a.pure[F], a.pure[G])
    0Assuming Monoidal[F] and Monoidal[G]
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92. Product composition: monads
    type L[X] = (F[X], G[X])
    (Monad[F], Monad[G]) => Monad[L]
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93. Product composition: monads
    def flatMap[A, B]
    (xa: (F[A], G[A]))(ff: A => (F[B], G[B])):
    (F[(A, B)], G[(A, B)]) = {
    val f = xa._1.flatMap(a => ff(a)._1)
    val g = xa._2.flatMap(a => ff(a)._2)
    (f, g)
    }
    0Assuming Monad[F] and Monad[G]
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94. Nested composition: lax monoidal functors
    type L[X] = F[G[X]]
    (Monoidal[F], Monoidal[G]) => Monoidal[L]
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95. User input Password reqs Features Latest posts
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96. Nested composition: lax monoidal functors
    def zip[A, B](fga: F[G[A]], fgb: F[G[B]]): F[G[(A, B)]] = {
    val fp: F[(G[A], G[B])] = fga zip fgb
    fp.map { case (ga, gb) => ga zip gb }
    }
    0Assuming Monoidal[F] and Monoidal[G]
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97. Nested composition: lax monoidal functors
    def zip[A, B](fga: F[G[A]], fgb: F[G[B]]): F[G[(A, B)]] = {
    val fp: F[(G[A], G[B])] = fga zip fgb
    fp.map { case (ga, gb) => ga zip gb }
    }
    def pure[A](a: A): F[G[A]] =
    a.pure[G].pure[F]
    0Assuming Monoidal[F] and Monoidal[G]
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 62 / 67
    

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98. Nested composition: monads
    type L[X] = F[G[X]]
    (Monad[F], Monad[G]) => Monad[L]
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 63 / 67
    

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99. Nested composition: monads
    type L[X] = F[G[X]]
    (Monad[F], Monad[G]) => Monad[L]
    Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 64 / 67
    

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100. Nested composition: monads
     def flatMap[A, B](fga: F[G[A]](f: A => F[G[B]]): F[G[B]] =
     0Assuming Monad[F] and Monad[G]
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 65 / 67
     

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101. Nested composition: monads
     def flatMap[A, B](fga: F[G[A]](f: A => F[G[B]]): F[G[B]] =
     fga.flatMap { (ga: G[A]) =>
     ???
     }
     0Assuming Monad[F] and Monad[G]
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 65 / 67
     

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102. Review
     Associative composition allows for modular decomposition and
     reasoning
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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103. Review
     Associative composition allows for modular decomposition and
     reasoning
     Monoids compose A’s, lax monoidal/applicative functors compose
     independent programs, monads compose dependent programs
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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104. Review
     Associative composition allows for modular decomposition and
     reasoning
     Monoids compose A’s, lax monoidal/applicative functors compose
     independent programs, monads compose dependent programs
     Category theory studies composition, and each of the above can
     be framed as a category
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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105. Review
     Associative composition allows for modular decomposition and
     reasoning
     Monoids compose A’s, lax monoidal/applicative functors compose
     independent programs, monads compose dependent programs
     Category theory studies composition, and each of the above can
     be framed as a category
     Monoids compose
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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106. Review
     Associative composition allows for modular decomposition and
     reasoning
     Monoids compose A’s, lax monoidal/applicative functors compose
     independent programs, monads compose dependent programs
     Category theory studies composition, and each of the above can
     be framed as a category
     Monoids compose
     Lax monoidal/applicative functors compose
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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107. Review
     Associative composition allows for modular decomposition and
     reasoning
     Monoids compose A’s, lax monoidal/applicative functors compose
     independent programs, monads compose dependent programs
     Category theory studies composition, and each of the above can
     be framed as a category
     Monoids compose
     Lax monoidal/applicative functors compose
     Monads in general do not compose
     See: monad transformers, extensible (algebraic) effects
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 66 / 67
     

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108. References
     Equational reasoning at scale - Gabriel Gonzalez
     http://www.haskellforall.com/2014/07/
     equational-reasoning-at-scale.html
     Invariant Shadows - Michael Pilquist
     http://mpilquist.github.io/blog/2015/06/22/
     invariant-shadows-part-2/
     All About Applicative - Me!
     https://www.youtube.com/watch?v=Mn7BtPALFys
     A categorial view of computational effects - Emily Riehl
     https://www.youtube.com/watch?v=6t6bsWVOIzs
     Conceptual Mathematics - F. William Lawvere and Stephen H.
     Schanuel
     Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 67 / 67
     

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