Simplicity in composition

1.

Simplicity in composition Adelbert Chang @adelbertchang Scala World 2017 Adelbert
Chang (@adelbertchang) Simplicity in composition Scala World 2017 1 / 67

2.

Composition A: type (set) of values Adelbert Chang (@adelbertchang) Simplicity
in composition Scala World 2017 2 / 67

3.

Composition A: type (set) of values ⊕: A × A
→ A Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 2 / 67

4.

→ A 1A : identity for ⊕ Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 2 / 67

5.

→ A 1A : identity for ⊕ (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 2 / 67

6.

→ A 1A : identity for ⊕ (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) x = x ⊕ 1A = 1A ⊕ x Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 2 / 67

7.

(Z, +, 0) Adelbert Chang (@adelbertchang) Simplicity in composition Scala
World 2017 3 / 67

8.

([a], + +, []) Adelbert Chang (@adelbertchang) Simplicity in composition
Scala World 2017 4 / 67

9.

(A → A, ◦, a → a) Adelbert Chang (@adelbertchang)
Simplicity in composition Scala World 2017 5 / 67

10.

trait Monoid[A] { def combine(x: A, y: A): A def
empty: A } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 6 / 67

11.

1 2 3 4 5 6 7 8 Adelbert Chang
(@adelbertchang) Simplicity in composition Scala World 2017 7 / 67

12.

3 3 4 5 6 7 8 Adelbert Chang (@adelbertchang)

13.

6 4 5 6 7 8 Adelbert Chang (@adelbertchang) Simplicity

14.

15 6 7 8 Adelbert Chang (@adelbertchang) Simplicity in composition

15.

36 Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017
11 / 67

16.

1 2 3 4 5 6 7 8 Adelbert Chang

17.

3 7 11 15 Adelbert Chang (@adelbertchang) Simplicity in composition

18.

10 26 Adelbert Chang (@adelbertchang) Simplicity in composition Scala World
2017 14 / 67

19.

36 Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017
15 / 67

20.

1 2 3 4 5 6 7 8 3 7
11 15 10 26 36 Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 16 / 67

21.

Associative composition allows for modular decomposition and reasoning. Adelbert Chang

22.

Composing programs A Adelbert Chang (@adelbertchang) Simplicity in composition Scala
World 2017 18 / 67

23.

Composing programs F[A] Adelbert Chang (@adelbertchang) Simplicity in composition Scala
World 2017 19 / 67

24.

Composing programs (F[A], F[A]) => F[A] Adelbert Chang (@adelbertchang) Simplicity

25.

Composing programs (F[A], F[B]) => F[?] Adelbert Chang (@adelbertchang) Simplicity

26.

Composing programs (F[A], F[B]) => F[(A, B)] Adelbert Chang (@adelbertchang)

27.

Composing programs F( ): type of program Adelbert Chang (@adelbertchang)

28.

Composing programs F( ): type of program ⊗ : F(A)
× F(B) → F((A, B)) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 23 / 67

29.

× F(B) → F((A, B)) η : A → F(A) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 23 / 67

30.

× F(B) → F((A, B)) η : A → F(A) (fa ⊗ fb) ⊗ fc ∼ = fa ⊗ (fb ⊗ fc) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 23 / 67

31.

× F(B) → F((A, B)) η : A → F(A) (fa ⊗ fb) ⊗ fc ∼ = fa ⊗ (fb ⊗ fc) fa ∼ = fa ⊗ ηUnit ∼ = ηUnit ⊗ fa Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 23 / 67

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 } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 24 / 67

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) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 24 / 67

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) } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 25 / 67

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) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 25 / 67

36.

Composing programs def zipFunction[A, B, X] (f: X => A,
g: X => B): X => (A, B) = (x: X) => (f(x), g(x)) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 26 / 67

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 Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 26 / 67

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] */ } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 27 / 67

39.

User input Password reqs Features Latest posts Sign-up form Featured
Welcome page Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 28 / 67

40.

Composing programs (F[A], F[B]) => F[(A, B)] Adelbert Chang (@adelbertchang)

41.

Composing independent programs (F[A], F[B]) => F[(A, B)] Adelbert Chang

42.

Composing programs F[A] Adelbert Chang (@adelbertchang) Simplicity in composition Scala
World 2017 31 / 67

43.

Composing dependent programs (F[A], A => F[B]) => F[B] Adelbert

44.

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

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) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 33 / 67

46.

(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

47.

(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) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 33 / 67

48.

(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

49.

(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

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 } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 34 / 67

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) } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 35 / 67

52.

Composing dependent programs def flatMapFunction[A, B, X] (fa: X =>
A, f: A => (X => B)): X => B = (x: X) => f(fa(x))(x) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 36 / 67

53.

trait Monad[F[_]] { def flatMap[A, B](fa: F[A])(f: A => F[B]):
F[B] def pure[A](a: A): F[A] } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 37 / 67

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] } Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 38 / 67

55.

A nicer monad (fa > >= f ) > >=
g = fa > >= (f >=> g) f (x) = η(x) > >= f fa = fa > >= η Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 39 / 67

56.

A nicer monad f : A → F(B) g :
B → F(C) h : C → F(D) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 40 / 67

57.

B → F(C) h : C → F(D) (f >=> g) >=> h = f >=> (g >=> h) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 40 / 67

58.

B → F(C) h : C → F(D) (f >=> g) >=> h = f >=> (g >=> h) f = f >=> η = η >=> f Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 40 / 67

59.

User info Friends Cookies Logs Proﬁle Recommendations News feed Adelbert

60.

Category theory studies the algebra of composition. Adelbert Chang (@adelbertchang)

61.

Category theory objects: A, B, C ... Adelbert Chang (@adelbertchang)

62.

Category theory objects: A, B, C ... arrows: f :
A → B, g : B → C ... Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 43 / 67

63.

A → B, g : B → C ... 1A : A → A Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 43 / 67

64.

A → B, g : B → C ... 1A : A → A (f ◦ g) ◦ h = f ◦ (g ◦ h) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 43 / 67

65.

A → B, g : B → C ... 1A : A → A (f ◦ g) ◦ h = f ◦ (g ◦ h) f = f ◦ 1A = 1A ◦ f Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 43 / 67

66.

Category theory B A C g 1B f g◦f 1A
1C Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 44 / 67

67.


68.


69.

Category theory: monoids • x y x⊕y 1A Adelbert Chang

70.


71.


72.


73.

Category theory: monads Adelbert Chang (@adelbertchang) Simplicity in composition Scala
World 2017 46 / 67

74.

Category theory: monads F(B) A F(C) f ?◦f Adelbert Chang

75.

Category theory: monads F(B) A F(C) f ?◦f > >=:
(F(B) × B → F(C)) → F(C) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 46 / 67

76.

Category theory: monads F(B) A F(C) f ?◦f > >=:
(B → F(C)) → F(B) → F(C) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 47 / 67

77.

Category theory: monads F(B) A F(C) g∗ f g∗◦f >
>=: (B → F(C)) → F(B) → F(C) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 48 / 67

78.

Category theory: monads B A C g 1B B→F(B) f
g ◦f 1A A→F(A) 1C C→F(C) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 49 / 67

79.


80.


81.

g ◦f 1A A→F(A) 1C C→F(C) η : X → F(X) Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 49 / 67

82.

User info Friends Cookies Logs Proﬁle Recommendations News feed Adelbert

83.

What if we have diﬀerent monoids, monads, etc? Adelbert Chang

84.

(Monoid[A], Monoid[B]) => Monoid[?] Adelbert Chang (@adelbertchang) Simplicity in composition

85.

Product composition: monoids (Monoid[A], Monoid[B]) => Monoid[(A, B)] Adelbert Chang

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 54 / 67

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 54 / 67

88.

(Monoidal[F], Monoidal[G]) => Monoidal[?] Adelbert Chang (@adelbertchang) Simplicity in composition

89.

Product composition: lax monoidal functors type L[X] = (F[X], G[X])
(Monoidal[F], Monoidal[G]) => Monoidal[L] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 56 / 67

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 57 / 67

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 57 / 67

92.

Product composition: monads type L[X] = (F[X], G[X]) (Monad[F], Monad[G])
=> Monad[L] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 58 / 67

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 59 / 67

94.

Nested composition: lax monoidal functors type L[X] = F[G[X]] (Monoidal[F],
Monoidal[G]) => Monoidal[L] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 60 / 67

95.

User input Password reqs Features Latest posts Sign-up form Featured
Welcome page Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 61 / 67

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] Adelbert Chang (@adelbertchang) Simplicity in composition Scala World 2017 62 / 67

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

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

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

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

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

102.

Review Associative composition allows for modular decomposition and reasoning Adelbert

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

104.

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

105.

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

106.

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

107.

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

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

Simplicity in composition

Simplicity in composition

Want more?

Transcript