This document discusses the concept of lenses in functional programming, specifically in Scala, focusing on their implementation through the Monocle library. It explains how lenses facilitate the manipulation of nested data structures while reducing verbosity and complexity. The document also covers advanced topics like polymorphic lenses, isomorphisms, and the creation of lenses using van Laarhoven's theory, highlighting various functional programming principles and techniques.

1. Optics with Monocle
Modeling the part and the whole
By Ilan Godik (@Nightra on IRC)

2. About me
▪ A CS Undergraduate at Haifa University
▪ A contributor to the Monocle library
▪ Functional programming lover.

3. What are lenses?
trait Lens[S,A]{
def get(s: S): A
def set(a: A)(s: S): S
}
In the simplest model, it's just a pair of a getter and a setter:
get
S
A1
S
A2
set
Lens[S, A]
A2

4. Classic lens example
person.copy(
address = person.address.copy(
street = person.address.street.copy(
name = person.address.street.name.capitalize
)
)
)
Updating nested structures is verbose and painful
case class Person(fullName: String, address: Address)
case class Address(city: String, street: Street)
case class Street(name: String, number: Int)

5. Monocle Lenses
(address composeLens street composeLens name).modify(_.capitalize)(person)
case class Person(fullName: String, address: Address)
case class Address(city: String, street: Street)
case class Street(name: String, number: Int)
val address = Lenser[Person](_.address)
val street = Lenser[Address](_.street)
val name = Lenser[Street](_.name)
Define lenses once, and compose as you wish

6. Lenses Macro annotation
(address composeLens street composeLens name).modify(_.capitalize)(person)
@Lenses case class Person(fullName: String, address: Address)
@Lenses case class Address(city: String, street: Street)
@Lenses case class Street(name: String, number: Int)
import Person._, Address._, Street._
Awesome, but not IDE friendly.

7. The object graph
Person
String Address
String Street
String Int

8. Code size growth
▪ Vanilla Scala: O(h2)
▪ Lenses: O(h)
Where h is the height of the object graph

9. Composition: Follow the arrows
Person
String Address
String Street
String Int

10. Lens composition
We compose lenses exactly how we would do it with copy.
S
set
Lens[S,A] compose Lens[A,B]
A
B2
get
S
A
B1
get B2

11. Lens composition
We compose lenses exactly how we would do it with copy.
S
set
Lens[S,A] compose Lens[A,B]
A
B2
get
S
A
B1
get B2

12. Lens composition
We compose lenses exactly how we would do it with copy.
S
set
Lens[S,A] compose Lens[A,B]
A
B2
get
S
A
B1
get B2
A
B2

13. Lens composition
We compose lenses exactly how we would do it with copy.
S
set
Lens[S,A] compose Lens[A,B]
A
B2
get
S
A
B1
get B2
A
B2

14. Polymorphic Lenses
Can we change the type of part of the structure while setting?
first.set("Hello")((1,2)) == ("Hello",2)

15. Polymorphic Lenses
Can we change the type of part of the structure while setting?
trait PLens[S, T, A, B] {
def get(s: S): A
def set(b: B)(s: S): T
} get
S
A
T
B
B
set
PLens[S, T, A, B]

16. Example: Unit conversion
Meters CM MM
?
Lens[Meters,CM]
Lens[Meters,MM]
Optics as different points of view of data

17. Isomorphisms
Do we get more power if our lens is bidirectional?
Meters CM MM
Iso[Meters,CM]
Iso[Meters,MM]

18. Isomorphisms
A B
Isomorphisms as a bijection: A function with an inverse

19. Prisms
What if we don’t have such a nice correspondence?
?
?
A B

20. Prisms
What if we don’t have such a nice correspondence?
?
?
IntString
A B

21. Prisms
What if we don’t have such a nice correspondence?
?
?
IntString
IntString
Option[Int]String
A B

22. Prisms
What if we don’t have such a nice correspondence?
?
?
IntString
IntString
Option[Int]String
Laws:
1. If there is an answer, going
back must give the source.
2. If we go back, there must
be an answer, which is
the source.
A B

23. Property Testing
Laws => Automated property testing
Int =String
Option[Int] =String
_.toString
Try(_.toInt).toOption

24. Property Testing
” .toInt = 9“

25. Property Testing
” .toInt = 9“
WAT

26. Prism Laws
?
IntString
A B
“9”
9

27. Example: Double binding
°C30 °F86
Iso[Celcius, Fahrenheit]
Prism[String, °F]Prism[String, °C]

28. Example: Double binding
°C30 °F86
Iso[Celcius, Fahrenheit]
Prism[String, °F]Prism[String, °C]

29. Example: Double binding
°C30 °F86
Iso[Celcius, Fahrenheit]
Prism[String, °F]Prism[String, °C]

30. A bit of theory:
Van Laarhoven Lenses
Is it possible to unify all the lens functions?
▪ get: S => A
▪ set: A => S => S
▪ modify: (A => A) => (S => S)

31. Is it possible to unify all the lens functions?
▪ get: S => A
▪ set: A => S => S
▪ modify: (A => A) => (S => S)
▪ modifyMaybe: (A => Option[A]) => (S => Option[S])
▪ modifyList: (A => List[A]) => (S => List[S])
A bit of theory:
Van Laarhoven Lenses

32. Functors
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}

33. List Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
Functor[List]{
def map[A,B](f: A => B)(list: List[A]): List[B] =
list.map(f)
}

34. Option Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
Functor[Option]{
def map[A,B](f: A => B)(opt: Option[A]): Option[B] = opt match {
case None => None
case Some(a) => Some(f(a))
}
}

35. Van Laarhoven Lenses
The answer:
Functor[F] => (A => F[A]) => (S => F[S])

36. Van Laarhoven Lenses
The answer:
Functor[F] => (A => F[A]) => (S => F[S])
▪
▪
▪ :
▪ modifyMaybe: (A => Option[A]) => (S => Option[S])
▪ modifyList: (A => List[A]) => (S => List[S])

37. Van Laarhoven Lenses
The answer:
Functor[F] => (A => F[A]) => (S => F[S])
▪ get: S => A
▪ set: A => S => S
▪ modify: (A => A) => (S => S)
▪
▪

38. Van Laarhoven Lenses
The answer:
lens: Functor[F] => (A => F[A]) => (S => F[S])
type Id[A] = A
lens: (A => Id[A]) => (S => Id[S])
lens: (A => A) => (S => S)
modify = lens[Id]

39. Identity Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
type Id[A] = A
Functor[Id]{
def map[A,B] (f: A => B)(a: Id[A]): Id[B]
}

40. Identity Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
type Id[A] = A
Functor[Id]{
def map[A,B] (f: A => B)(a: Id[A]): Id[B]
def map[A,B] (f: A => B)(a: A): B = f(a)
}

41. Van Laarhoven Lenses
The answer:
lens: Functor[F] => (A => F[A]) => (S => F[S])
set(b) = modify(_ => b)
get: S => A = ???

42. Van Laarhoven Lenses
The answer:
lens: Functor[F] => (A => F[A]) => (S => F[S])
set(b) = modify(_ => b)
get: S => A = ???

43. Van Laarhoven Lenses
The answer:
lens: Functor[F] => (A => F[A]) => (S => F[S])
type Const[X][T] = X
F = Const[A]
lens: (A => Const[A][A]) => (S => Const[A][S])
lens: (A => A) => (S => A)
get = lens[Const[A]](a => a)

44. Const Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
type Const[X][T] = X
Functor[Const[X]]{
def map[A,B] (f: A => B)(fa: Const[X][A]): Const[X][B]
}

45. Const Functor
trait Functor[F[_]]{
def map[A,B](f: A => B)(fa: F[A]): F[B]
}
type Const[X][T] = X
Functor[Const[X]]{
def map[A,B] (f: A => B)(fa: Const[X][A]): Const[X][B]
def map[A,B] (f: A => B)(x: X): X = x
}

46. Van Laarhoven Lenses
The answer:
Functor[F] => (A => F[A]) => (S => F[S])
▪ get: S => A
▪ set: A => S => S
▪ modify: (A => A) => (S => S)
▪ modifyMaybe: (A => Option[A]) => (S => Option[S])
▪ modifyList: (A => List[A]) => (S => List[S])

47. Creating Van Laarhoven Lenses
lens: Functor[F] => (A => F[A]) => (S => F[S])
def get: S => A
def set: A => S => S
def lens[F[_]:Functor](f: A => F[A])(s: S): F[S] =
f(get(s)). map (a => set(a)(s))
A
F[A]
(A => S) =>
F[A] => F[S]
(A => S)
F[S]
Shortly:
lens f s = f(get(s)).map(a => set(a)(s))

48. Creating Van Laarhoven Lenses
lens: Functor[F] => (A => F[A]) => (S => F[S])
type Id[A] = A
F = Id
lens: (A => A) => (S => S)
lens f s = f(get(s)).map(a => set(a)(s))
[ A ] [ A => S ]
modify f s = set(f(get(s)))(s)
set x s = modify(_ => x) s
= set(x)(s)

49. Creating Van Laarhoven Lenses
lens: Functor[F] => (A => F[A]) => (S => F[S])
type Const[A][X] = A
F = Const[A]
lens: (A => A) => (S => A)
lens f s = f(get(s)).map(a => set(a)(s))
[ A ] [ A => S ]
get’ f s = f(get(s))
get s = get’(id)(s) = get(s)

50. Monocle
▪ Provides lots of built-in optics and functions
▪ Macros for creating lenses
▪ Monocle used different models of lenses over time
▪ Current lens representation: PLens – Van Laarhoven hybrid.
▪ Performance: limited by scala function values – pretty good.

51. Resources
▪ Monocle on github
▪ Simon Peyton Jones’s lens talk at Scala Exchange 2013
▪ Edward Kmett on Lenses with the State Monad

52. Thank you!

53. Extra Slides

54. ASTs - Lenses for APIs
We can simplify our mental model with lenses
case class Person(fullName: String, address: Address)
Case class
decleration
Class
name
List[Field name -> Type]

55. ASTs - Lenses for APIs
We can simplify our mental model with lenses
case class Person(fullName: String, address: Address)
Case class
decleration
Class
name
List[Field name -> Type]
Public
constructor
Lens[Complex model, Simple model]
val

56. Polymorphic Optic Instances
How can we use the same lens for many types?
first.set("Hello")((1,2)) == ("Hello",2)
first.set("Hello")((1,2,3)) == ("Hello",2,3)
first.set("Hello")((1,2,3,4)) == ("Hello",2,3,4)

57. Example: Json
The structure of a specific Json object isn’t defined at the type system;
Each element can be a String or a Number or an Array or an Object.
We want to process Json assuming our specific structure, and let the system handle failures for
us.
We can define Prism[Json, String], Prism[Json, Double], Prism[Json, List[Json]]
and Prism[Json, Map[String, Json]].
Now we can compose these prisms to go deep inside a Json object and manipulate it.

58. * Polymorphic Lens Composition
get
S
A
T
B
B
set
PLens[S, T, A, B]
get
A
C
B
D
D
set
PLens[A, B, C, D]
get
S
C
T
D
D
set
PLens[S, T, C, D]