Why I Like Functional Programming

Shortly after Adelbert Chang entered college as a computer science major, he contemplated changing his major several times, usually in the direction of physics and mathematics. That all changed when he was introduced to functional programming — at once both mysterious and beautiful. In this talk, Adelbert will elaborate on his early perspectives on computer science, math, and physics, and discuss how he ended up so enamored with functional programming.

Transcript

1. Why I Like
   Functional Programming
   Adelbert Chang
   Machine Learning @ Box, Inc.
   

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2. Let’s go back..
   http://gauss.cs.ucsb.edu/home/images/UCSB-from-air.jpg
   

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4. 2011
   void insert(Node* &tree, int value) {
   if (!tree) return;
   Node *trav = tree, *parent;
   while (trav) {
   parent = trav;
   if (value < trav->data) trav = trav->left;
   else if (value > trav->data) trav = trav->right;
   else break;
   }
   if (value < parent->data)
   parent->left = new Node(value);
   else
   parent->right = new Node(value);
   }
   

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5. Physics and Math
   

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6. Physics and Math
   •Simplicity and consistency across problems
   

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7. Physics and Math
   •Simplicity and consistency across problems
   •Example: Deriving laws of motion from first principles
   

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8. Physics and Math
   •Simplicity and consistency across problems
   •Example: Deriving laws of motion from first principles
   •vf = v0 + a Δt
   

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9. Physics and Math
   •Simplicity and consistency across problems
   •Example: Deriving laws of motion from first principles
   •vf = v0 + a Δt
   •vavg = v0 + ½aΔt
   

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10. Physics and Math
    •Simplicity and consistency across problems
    •Example: Deriving laws of motion from first principles
    •vf = v0 + a Δt
    •vavg = v0 + ½aΔt
    •x = x0 + v0t + ½at2
    

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11. Deriving x = x0 + v0t + ½at2
    Physics and Math
    

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12. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Physics and Math
    

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13. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    Physics and Math
    

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14. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    Physics and Math
    

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15. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    Physics and Math
    

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16. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    vavg = ½(vf + v0)
    Physics and Math
    

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17. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    vavg = ½(vf + v0)
    a = Δv / Δt
    Physics and Math
    

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18. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    Physics and Math
    

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19. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    Physics and Math
    

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20. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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21. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    ½(v0 + aΔt + v0)
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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22. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    ½(v0 + aΔt + v0)
    ½(2v0 + a Δt)
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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23. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    ½(v0 + aΔt + v0)
    ½(2v0 + a Δt)
    vavg = v0 + ½aΔt
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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24. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    x = x0 + ((v0 + ½aΔt) * Δt)
    ½(v0 + aΔt + v0)
    ½(2v0 + a Δt)
    vavg = v0 + ½aΔt
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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25. Deriving x = x0 + v0t + ½at2
    vavg = Δx/Δt
    Δx = vavg * Δt
    (x - x0) = vavg * Δt
    x = x0 + (vavg * Δt)
    x = x0 + ((v0 + ½aΔt) * Δt)
    x0 + v0t + ½at2
    ½(v0 + aΔt + v0)
    ½(2v0 + a Δt)
    vavg = v0 + ½aΔt
    vavg = ½(vf + v0)
    a = Δv / Δt
    a = (vf - v0) / Δt
    vf - v0 = a Δt
    vf = v0 + a Δt
    Physics and Math
    

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27. 2012
    sealed abstract class Tree
    case class Branch(data: Int, left: Tree, right: Tree)
    extends Tree
    case class Leaf() extends Tree
    def insert(tree: Tree, value: Int): Tree =
    tree match {
    case Leaf() => Branch(value, Leaf(), Leaf())
    case b@Branch(d, l, r) =>
    if (value < d) Branch(d, insert(l, value), r)
    else if (value > d) Branch(d, l, insert(r, value)
    else b
    }
    

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28. Hmm.. functional programming seems pretty cool.
    

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29. But is it as cool as physics and math?
    Hmm.. functional programming seems pretty cool.
    

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30. Functional
    Programming
    

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31. Functional
    Programming
    programming with functions
    

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32. programming with pure functions
    Functional
    Programming
    

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33. programming with pure functions
    Functional
    Programming
    

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34. Functions
    

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35. def parseIntPartial(s: String): Int =
    s.toInt // can throw NumberFormatException
    Functions
    

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36. def parseIntPartial(s: String): Int =
    s.toInt // can throw NumberFormatException
    Functions
    def parseIntTotal(s: String): Option[Int] =
    try {
    Some(s.toInt)
    } catch {
    case nfe: NumberFormatException => None
    }
    

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37. Functions
    

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38. class Rng(var seed: Long) {
    def nextInt(): Int = {
    val int = getInt(seed)
    mutate(seed)
    int
    }
    }
    Functions
    

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39. class Rng(var seed: Long) {
    def nextInt(): Int = {
    val int = getInt(seed)
    mutate(seed)
    int
    }
    }
    case class Rng(seed: Long) {
    def nextInt: (Rng, Int) = {
    val int = getInt(seed)
    val newSeed = f(seed)
    (Rng(newSeed), int)
    }
    }
    Functions
    

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40. Referential Transparency
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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41. Referential Transparency
    An expression e is referentially transparent if
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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42. Referential Transparency
    An expression e is referentially transparent if
    for all programs p
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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43. Referential Transparency
    An expression e is referentially transparent if
    for all programs p
    every occurrence of e in p
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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44. Referential Transparency
    An expression e is referentially transparent if
    for all programs p
    every occurrence of e in p
    can be replaced with the result of evaluating e
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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45. Referential Transparency
    An expression e is referentially transparent if
    for all programs p
    every occurrence of e in p
    can be replaced with the result of evaluating e
    without changing the result of evaluating p.*
    * Taken from Higher Order blog post "What Purity Is and Isn't" May 29, 2015 http://blog.higher-order.com/blog/2012/09/13/what-purity-is-and-isnt/
    

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46. Referential Transparency
    

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47. x4 - 12x2 + 36 = 0
    Referential Transparency
    

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48. x4 - 12x2 + 36 = 0
    y = x2
    Referential Transparency
    

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49. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    Referential Transparency
    

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50. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    Referential Transparency
    

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51. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    Referential Transparency
    

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52. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    Referential Transparency
    

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53. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    (-(-12) ± √ (-12)2 - 4(1)(36)) / 2(1)
    Referential Transparency
    

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54. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    (-(-12) ± √ (-12)2 - 4(1)(36)) / 2(1)
    12 / 2
    Referential Transparency
    

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55. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    (-(-12) ± √ (-12)2 - 4(1)(36)) / 2(1)
    12 / 2
    y = 6
    Referential Transparency
    

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56. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    (-(-12) ± √ (-12)2 - 4(1)(36)) / 2(1)
    12 / 2
    y = 6
    x2 = 6
    Referential Transparency
    

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57. x4 - 12x2 + 36 = 0
    y = x2
    y2 - 12y + 36 = 0
    ax2 + bx + c = 0
    (-b ± √ b2 - 4ac) / 2a
    a = 1 b = -12 c = 36
    (-(-12) ± √ (-12)2 - 4(1)(36)) / 2(1)
    12 / 2
    y = 6
    x2 = 6
    x = ± √ 6
    Referential Transparency
    

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58. Referential Transparency
    

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59. def userInfo(id: UserId): Future[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => fetchData
    }
    Referential Transparency
    

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60. def userInfo(id: UserId): Future[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => fetchData
    }
    Referential Transparency
    

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61. def userInfo(id: UserId): Future[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => userInfo(userId)
    }
    Referential Transparency
    

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62. def userInfo(id: UserId): Future[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => userInfo(userId)
    }
    ???
    Referential Transparency
    

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63. def userInfo(id: UserId): Task[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => fetchData
    }
    Referential Transparency
    

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64. trait Task[A] {
    def unsafeRun(): A
    }
    def userInfo(id: UserId): Task[UserData]
    val userId = . . .
    val fetchData = userInfo(userId)
    fetchData.retry {
    case StatusCode(429) => fetchData
    }
    Referential Transparency
    

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65. Functions
    

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66. bar: B => C baz: C => D
    foo: A => B
    Functions
    

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67. baz.compose(bar).compose(foo): A => D
    bar: B => C baz: C => D
    foo: A => B
    Functions
    

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68. Tracking Effects
    

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69. Tracking Effects
    •Interesting programs will have effects
    •Optional values, error handling, asynchronous
    computation, input/output
    •Usual means aren’t quite nice
    

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70. Tracking Effects
    •Interesting programs will have effects
    •Optional values, error handling, asynchronous
    computation, input/output
    •Usual means aren’t quite nice
    Solution:
    

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71. Tracking Effects
    •Interesting programs will have effects
    •Optional values, error handling, asynchronous
    computation, input/output
    •Usual means aren’t quite nice
    Solution:
    Reify effects as values.
    

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72. Missing values
    

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73. // A value that might exist is either..
    sealed abstract class Option[A]
    // ..there
    final case class Some[A](a: A) extends Option[A]
    // .. or not there
    final case class None[A]() extends Option[A]
    Missing values
    

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74. def lookup(map: Map[Foo, Bar], foo: Foo): Option[Bar] =
    if (map.contains(foo)) Some(map(foo))
    else None
    Missing values
    

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75. Errors
    sealed abstract class Either[+E, +A]
    final case class Success[+A](a: A) extends Either[Nothing, A]
    final case class Failure[+E](e: E) extends Either[E, Nothing]
    

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76. sealed abstract class Error
    final case object UserDoesNotExist extends Error
    final case object InvalidToken extends Error
    def userInfo(uid: UserId,
    tk: Token): Either[Error, UserInfo] = . . .
    Errors
    

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77. Manipulating Effects
    

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78. Manipulating Effects
    •The type signature of our functions reflect the
    effects involved
    

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79. Manipulating Effects
    •The type signature of our functions reflect the
    effects involved
    def tokenFor(uid: UserId): Option[EncryptedToken]
    def decrypt(token: EncryptedToken): Token
    /** Client side */
    val encrypted: Option[EncryptedToken] =
    tokenFor(. . .)
    // Want EncryptedToken, have Option[EncryptedToken]
    val decrypted = decrypt(???)
    

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80. /** Apply a pure function to an effectful value */
    trait Functor[F[_]] {
    def map[A, B](fa: F[A])(f: A => B): F[B]
    }
    Manipulating Effects
    

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81. /** Apply a pure function to an effectful value */
    trait Functor[F[_]] {
    def map[A, B](fa: F[A])(f: A => B): F[B]
    }
    Manipulating Effects
    new Functor[Option] {
    def map[A, B](fa: Option[A])(f: A => B): Option[B] =
    fa match {
    case Some(a) => Some(f(a))
    case None() => None()
    }
    }
    

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82. def tokenFor(uid: UserId): Option[EncryptedToken]
    def decrypt(token: EncryptedToken): Token
    /** Client side */
    val encrypted: Option[EncryptedToken] =
    tokenFor(. . .)
    val decrypted = encrypted.map(tk => decrypt(tk))
    Manipulating Effects
    

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83. •Similar mechanisms for manipulating multiple
    effects
    def tokenFor(uid: UserId): Option[EncryptedToken]
    def decrypt(token: EncryptedToken): Token
    /** Client side */
    val encrypted: Option[EncryptedToken] =
    tokenFor(. . .)
    val decrypted = encrypted.map(tk => decrypt(tk))
    Manipulating Effects
    

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84. Lens
    

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85. Lens
    •Often want getters/setters when working with data
    

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86. Lens
    •Often want getters/setters when working with data
    •If getters/setters are per-object, cannot compose
    

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87. Lens
    •Often want getters/setters when working with data
    •If getters/setters are per-object, cannot compose
    •As with effects, reify as data
    •Getter: get an A field in an object S
    •Setter: change an A field in an object S
    

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88. Lens
    •Often want getters/setters when working with data
    •If getters/setters are per-object, cannot compose
    •As with effects, reify as data
    •Getter: get an A field in an object S
    •Setter: change an A field in an object S
    trait Lens[S, A] {
    def get(s: S): A
    def set(s: S, a: A): S
    }
    

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89. trait Lens[S, A] { outer =>
    def get(s: S): A
    def set(s: S, a: A): S
    def modify(s: S, f: A => A): S =
    set(s, f(get(s)))
    def compose[B](other: Lens[A, B]): Lens[S, B] =
    new Lens[S, B] {
    def get(s: S): B = other.get(get(s))
    def set(s: S, b: B): S =
    set(s, other.set(outer.get(s), b))
    }
    }
    Lens
    

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90. case class Employee(position: Position)
    object Employee {
    val position: Lens[Employee, Position] = . . .
    }
    case class Team(manager: Employee, . . .)
    object Team {
    val manager: Lens[Team, Employee] = . . .
    }
    Lens
    

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91. case class Employee(position: Position)
    object Employee {
    val position: Lens[Employee, Position] = . . .
    }
    case class Team(manager: Employee, . . .)
    object Team {
    val manager: Lens[Team, Employee] = . . .
    }
    /** Client side */
    val l: Lens[Team, Position] =
    Team.manager.compose(Employee.position)
    l.set(someTeam, somePosition)
    Lens
    

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92. Lens
    

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93. •Effect-ful modifications can be useful
    •Possibly failing: A => Option[A]
    •Many possible values: A => List[A]
    •Async: A => Task[A]
    Lens
    

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94. •Effect-ful modifications can be useful
    •Possibly failing: A => Option[A]
    •Many possible values: A => List[A]
    •Async: A => Task[A]
    trait Lens[S, A] {
    def modifyOption(s: S, f: A => Option[A]): Option[S]
    def modifyList(s: S, f: A => List[A]): List[S]
    def modifyTask(s: S, f: A => Task[A]): Task[S]
    }
    Lens
    

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95. trait Lens[S, A] {
    def modifyOption(s: S, f: A => Option[A]): Option[S] =
    f(get(s)).map(a => set(s, a))
    def modifyList(s: S, f: A => List[A]): List[S] =
    f(get(s)).map(a => set(s, a))
    def modifyTask(s: S, f: A => Task[A]): Task[S] =
    f(get(s)).map(a => set(s, a))
    }
    Lens
    

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96. trait Lens[S, A] {
    def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
    f(get(s)).map(a => set(s, a))
    }
    Lens
    

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97. trait Lens[S, A] {
    def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
    f(get(s)).map(a => set(s, a))
    }
    •Example functors:
    Lens
    

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98. trait Lens[S, A] {
    def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
    f(get(s)).map(a => set(s, a))
    }
    •Example functors:
    •Option, Either, Future, List, IO
    Lens
    

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99. trait Lens[S, A] {
    def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
    f(get(s)).map(a => set(s, a))
    }
    •Example functors:
    •Option, Either, Future, List, IO
    •What if we want to just modify without any effect?
    Lens
    

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100. trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     }
     •Example functors:
     •Option, Either, Future, List, IO
     •What if we want to just modify without any effect?
     •We want the F[A] to just be a plain A
     Lens
     

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101. trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     }
     •Example functors:
     •Option, Either, Future, List, IO
     •What if we want to just modify without any effect?
     •We want the F[A] to just be a plain A
     •Type-level identity function
     Lens
     

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102. trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     }
     •Example functors:
     •Option, Either, Future, List, IO
     •What if we want to just modify without any effect?
     •We want the F[A] to just be a plain A
     •Type-level identity function
     Lens
     def identity[A](a: A): A = a
     

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103. Id
     type Id[A] = A
     

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104. Id
     type Id[A] = A
     new Functor[Id] {
     def map[A, B](fa: Id[A])(f: A => B): Id[B] =
     }
     

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105. Id
     type Id[A] = A
     new Functor[Id] {
     def map[A, B](fa: Id[A])(f: A => B): Id[B] =
     }
     

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106. type Id[A] = A
     new Functor[Id] {
     def map[A, B](fa: A)(f: A => B): B =
     }
     Id
     

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107. type Id[A] = A
     new Functor[Id] {
     def map[A, B](fa: A)(f: A => B): B = f(fa)
     }
     Id
     

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108. trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     def modify(s: S, f: A => A): S = modifyF[Id](s, f)
     }
     Id
     

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109. Setting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     def modify(s: S, f: A => A): S = modifyF[Id](s, f)
     def set(s: S, a: A): S = modify(s, const(a))
     }
     

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110. Setting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     def modify(s: S, f: A => A): S = modifyF[Id](s, f)
     def set(s: S, a: A): S = modify(s, const(a))
     }
     def const[A, B](a: A)(b: B): A = a
     

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111. Setting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S] =
     f(get(s)).map(a => set(s, a))
     def modify(s: S, f: A => A): S = modifyF[Id](s, f)
     def set(s: S, a: A): S = modify(s, const(a))
     }
     def const[A, B](a: A)(b: B): A = a
     

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112. trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     def modify(s: S, f: A => A): S = modifyF[Id](s, f)
     def set(s: S, a: A): S = modify(s, const(a))
     }
     def const[A, B](a: A)(b: B): A = a
     Setting
     

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113. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     

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114. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     •Can we define `get` in terms of `modify`?
     

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115. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     •Can we define `get` in terms of `modify`?
     •`modifyF` gives us some F[S].. but we want an A
     

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116. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     •Can we define `get` in terms of `modify`?
     •`modifyF` gives us some F[S].. but we want an A
     •We need some way of "ignoring" the S
     parameter and still get an A back
     

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117. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     •Can we define `get` in terms of `modify`?
     •`modifyF` gives us some F[S].. but we want an A
     •We need some way of "ignoring" the S
     parameter and still get an A back
     •Type-level constant function
     

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118. Getting
     trait Lens[S, A] {
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     def get(s: S): A
     }
     •Can we define `get` in terms of `modify`?
     •`modifyF` gives us some F[S].. but we want an A
     •We need some way of "ignoring" the S
     parameter and still get an A back
     •Type-level constant function
     def const[A, B](a: A)(b: B): A = a
     

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119. Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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120. type Const[Z, A] = Z
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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121. type Const[Z, A] = Z
     new Functor[Const[Z, ?]] { // *
     def map[A, B](fa: Const[Z, A])(f: A => B): Const[Z, B]
     }
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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122. type Const[Z, A] = Z
     new Functor[Const[Z, ?]] { // *
     def map[A, B](fa: Const[Z, A])(f: A => B): Const[Z, B]
     }
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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123. new Functor[Const[Z, ?]] { // *
     def map[A, B](fa: Z)(f: A => B): Z =
     }
     type Const[Z, A] = Z
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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124. new Functor[Const[Z, ?]] { // *
     def map[A, B](fa: Z)(f: A => B): Z = fa
     }
     type Const[Z, A] = Z
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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125. trait Lens[S, Z] {
     def modifyF[F[_] : Functor](s: S, f: Z => F[Z]): F[S]
     def get(s: S): Z = {
     val const: Const[Z, S] =
     modifyF[Const[Z, ?]](s, z => z) // *
     const
     }
     }
     new Functor[Const[Z, ?]] { // *
     def map[A, B](fa: Z)(f: A => B): Z = fa
     }
     type Const[Z, A] = Z
     Getting
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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126. trait Lens[S, A] {
     /** Abstract */
     def modifyF[F[_] : Functor](s: S, f: A => F[A]): F[S]
     /** Implemented */
     def modify(s: S, f: A => A): S
     def get(s: S): A
     def set(s: S, a: A): S
     def compose[B](other: Lens[A, B]): Lens[S, B]
     }
     Lens
     

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127. Are Id and Const just party tricks?
     

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128. Traverse
     trait Traverse[F[_]] {
     def traverse[G[_] : Applicative, A, B]
     (fa: F[A])(f: A => G[B]): G[F[B]]
     }
     

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129. Traverse
     trait Traverse[F[_]] {
     def traverse[G[_] : Applicative, A, B]
     (fa: F[A])(f: A => G[B]): G[F[B]]
     }
     def validate[A]
     (data: List[A])(f: A => Option[B]): Option[List[B]]
     

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130. Traverse
     trait Traverse[F[_]] {
     def traverse[G[_] : Applicative, A, B]
     (fa: F[A])(f: A => G[B]): G[F[B]]
     }
     def scatterGather[A]
     (data: List[A])(f: A => Task[B]): Task[List[B]]
     def validate[A]
     (data: List[A])(f: A => Option[B]): Option[List[B]]
     

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131. Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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132. traverse: F[A] => (A => G[B]) => G[F[B]]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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133. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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134. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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135. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     F[A] => (A => B) => F[B]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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136. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     F[A] => (A => B) => F[B]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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137. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     F[A] => (A => B) => F[B]
     traverse[Const[Z, ?], A, B] // *, If Z can be "reduced"
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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138. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     F[A] => (A => B) => F[B]
     traverse[Const[Z, ?], A, B] // *, If Z can be "reduced"
     F[A] => (A => Const[Z, B]) => Const[Z, F[B]]
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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139. traverse: F[A] => (A => G[B]) => G[F[B]]
     traverse[Id, A, B]
     F[A] => (A => Id[B]) => Id[F[B]]
     F[A] => (A => B) => F[B]
     traverse[Const[Z, ?], A, B] // *, If Z can be "reduced"
     F[A] => (A => Const[Z, B]) => Const[Z, F[B]]
     F[A] => (A => Z) => Z
     Traverse
     * Const[Z, ?] syntax is valid due to kind-projector plugin https://github.com/non/kind-projector
     

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140. Interested?
     

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141. Interested?
     •Functional Programming
     •Cats, Scalaz
     •Argonaut, Atto, Monocle, Shapeless
     

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142. Interested?
     •Functional Programming
     •Cats, Scalaz
     •Argonaut, Atto, Monocle, Shapeless
     •Algebra
     •Algebird, Algebra, Breeze, Spire
     

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143. Interested?
     •Functional Programming
     •Cats, Scalaz
     •Argonaut, Atto, Monocle, Shapeless
     •Algebra
     •Algebird, Algebra, Breeze, Spire
     •Big Data
     •Scalding, Scoobi, Spark
     

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144. Interested?
     •Functional Programming
     •Cats, Scalaz
     •Argonaut, Atto, Monocle, Shapeless
     •Algebra
     •Algebird, Algebra, Breeze, Spire
     •Big Data
     •Scalding, Scoobi, Spark
     •Systems
     •Doobie, HTTP4S, Remotely, Scalaz-stream
     

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145. EOF
     

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