lens: from the ground up

A look at functional lenses in general, and the haskell "lens" library specifically. This talk walks through why lenses are encoded the way they are, and builds up to demonstrations of the lens library.

The demonstration code is available at https://github.com/markhibberd/lens-talk (see script.org and src/*.hs)

1. lens @markhibberd from the ground up
2. motivations
3. the ground up
4. data Lens a b = Lens { get :: a -> b , set :: b -> a -> a } intuitions
5. data Lens a b = Lens { get :: a -> b , set :: b -> a -> a } intuitions set-get ==> get l (set l b a) == b ! get-set ==> set l (get l a) a == a ! set-set ==> set l c (set l b a) == set l c a Pierce’s laws!
6. but… modify :: Lens a b -> (b -> b) -> a -> a modify l f a = set l (f (get l a)) a ! compose :: Lens a b -> Lens b c -> Lens a c compose l j = Lens (\a -> get j (get l a)) (\c a -> set l (set j c (get l a)) a)
7. but… modify :: Lens a b -> (b -> b) -> a -> a modify l f a = set l (f (get l a)) a ! compose :: Lens a b -> Lens b c -> Lens a c compose l j = Lens (\a -> get j (get l a)) (\c a -> set l (set j c (get l a)) a) efficiency matters!
8. but… data Wedge a = Wedge { _name :: String, _val :: a } ! name :: Lens (Wedge a) String name = Lens _name (\n w -> w { _name = n }) ! value :: Lens (Wedge a) a value = Lens _val (\v w -> w { _val = v })
9. but… data Wedge a = Wedge { _name :: String, _val :: a } ! name :: Lens (Wedge a) String name = Lens _name (\n w -> w { _name = n }) ! value :: Lens (Wedge a) a value = Lens _val (\v w -> w { _val = v }) polymorphic update matters
10. but… data Safety = Safety { _readOnly :: String } ! readOnly :: Lens Safety String readOnly = Lens _readOnly (error "don't do this")
11. but… read only / write only matters data Safety = Safety { _readOnly :: String } ! readOnly :: Lens Safety String readOnly = Lens _readOnly (error "don't do this")
12. but… (&&&) :: Lens a b -> Lens a c -> Lens a (b, c) (&&&) l j = Lens (\a -> (get l a, get j a)) (\(b, c) a -> set j c (set l b a))
13. but… (&&&) :: Lens a b -> Lens a c -> Lens a (b, c) (&&&) l j = Lens (\a -> (get l a, get j a)) (\(b, c) a -> set j c (set l b a)) not a lens
14. but… (&&&) :: Lens a b -> Lens a c -> Lens a (b, c) (&&&) l j = Lens (\a -> (get l a, get j a)) (\(b, c) a -> set j c (set l b a)) composition matters not a lens
15. but… data OneOf = First String | Second Int ! first :: Lens OneOf (Maybe String) first = undefined ! first :: Lens OneOf (Maybe String) first = undefined
16. but… data OneOf = First String | Second Int ! first :: Lens OneOf (Maybe String) first = undefined ! first :: Lens OneOf (Maybe String) first = undefined not a lens
17. but… data OneOf = First String | Second Int ! first :: Lens OneOf (Maybe String) first = undefined ! first :: Lens OneOf (Maybe String) first = undefined not a lens partiality matters
18. failed experiments data Store s a = Store (s -> a) s ! data Lens a b = Lens (a -> Store b a) ! ! ! ! ! ! ! !
19. data Store s a = Store (s -> a) s ! data Lens a b = Lens (a -> Store b a) ! get :: Lens a b -> a -> b get (Lens l) a = case l a of Store _ s -> s ! set :: Lens a b -> b -> a -> a set (Lens l) b a = case l a of Store f _ -> f b failed experiments
20. data Store s a = Store (s -> a) s ! data Lens a b = Lens (a -> Store b a) ! get :: Lens a b -> a -> b get (Lens l) a = case l a of Store _ s -> s ! set :: Lens a b -> b -> a -> a set (Lens l) b a = case l a of Store f _ -> f b polymorphic update matters composition matters failed experiments partiality matters read only / write only matters
21. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board
22. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board ! set :: Lens’ a b -> b -> a -> a set = error “can we?” ! get :: Lens’ a b -> a -> b get = error “can we?”
23. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = let x = const $ Identity b -- :: b -> Identity b in undefined ! !
24. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = let x = const $ Identity b -- :: b -> Identity b y = l x -- :: a -> Identity a in runIdentity z !
25. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = let x = const $ Identity b -- :: b -> Identity b y = l x -- :: a -> Identity a z = y a -- :: Identity a in undefined
26. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = let x = const $ Identity b -- :: b -> Identity b y = l x -- :: a -> Identity a z = y a -- :: Identity a in runIdentity z
27. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = runIdentity (l (const $ Identity b) a) ! ! !
28. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b a = runIdentity . l (const $ Identity b) $ a ! ! !
29. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Identity a = Identity { runIdentity :: a } ! set :: Lens’ a b -> b -> a -> a set l b = runIdentity . l (const $ Identity b) ! ! !
30. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype ??? a = ??? ! get :: Lens’ a b -> a -> b get l a = undefined ! ! !
31. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = undefined ! ! !
32. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = undefined ! ! ! ! instance Functor (Const x) where fmap _ = Const . runConst
33. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = let x = Const -- :: b -> Const b b in undefined ! !
34. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = let x = Const -- :: b -> Const b b y = l x -- :: a -> Const b a in undefined !
35. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = let x = Const -- :: b -> Const b b y = l x -- :: a -> Const b a z = y a -- :: Const b a in undefined
36. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = let x = Const -- :: b -> Const b b y = l x -- :: a -> Const b a z = y a -- :: Const b a in runConst z
37. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = runConst (l Const a) ! ! !
38. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l a = runConst . l Const $ a ! ! !
39. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board newtype Const x a = Const x ! get :: Lens’ a b -> a -> b get l = runConst . l Const ! ! !
40. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board set :: Lens’ a b -> b -> a -> a set l b = runIdentity . l (const $ Identity b) ! get :: Lens’ a b -> a -> b get l = runConst . l Const ! ! ! !
41. ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a back to the drawing board set :: Lens’ a b -> b -> a -> a set l b = runIdentity . l (const $ Identity b) ! get :: Lens’ a b -> a -> b get l = runConst . l Const ! modify :: Lens’ a b -> (b -> b) -> a modify l f = runIdentity . l (Identity . f) !
42. functional elegance
43. composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a
44. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b y :: Lens b c x . y :: Lens a c ! ! ! ! !
45. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c x . y :: Lens a c ! ! ! ! !
46. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c ! ! ! ! !
47. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! ! ! ! !
48. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! ! !
49. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j !
50. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i !
51. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b
52. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b, j :: a -> f a
53. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b, j :: a -> f a, h :: c -> f c
54. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b, j :: a -> f a, h :: c -> f c x . y :: h -> j
55. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b, j :: a -> f a, h :: c -> f c x . y :: h -> j :: (c -> f c) -> (a -> f a)
56. function composition ! type Lens’ a b = forall f. Functor f => (b -> f b) -> a -> f a x :: Lens a b :: (b -> f b) -> (a -> f a) y :: Lens b c :: (c -> f c) -> (b -> f b) x . y :: Lens a c :: (c -> f c) -> (a -> f a) ! (.) :: (i -> j) -> (h -> i) -> h -> j ! x :: i -> j, y :: h -> i i :: b -> f b, j :: a -> f a, h :: c -> f c x . y :: h -> j :: (c -> f c) -> (a -> f a)
57. ! type Lens a a’ b b’ = forall f. Functor f => (b -> f b’) -> a -> f a’ ! type Lens’ a b = Lens a a b b polymorphic update
58. ! type Lens a a’ b b’ = forall f. Functor f => (b -> f b’) -> a -> f a’ ! type Simple f a b = f a a b b ! type Lens’ = Simple Lens polymorphic update
59. ! type Lens a a’ b b’ = forall f. Functor f => (b -> f b’) -> a -> f a’ polymorphic update set :: Lens’ a b -> b -> a -> a set l b = runIdentity . l (const $ Identity b) ! get :: Lens’ a b -> a -> b get l = runConst . l Const ! modify :: Lens’ a b -> (b -> b) -> a modify l f = runIdentity . l (Identity . f) !
60. ! type Lens a a’ b b’ = forall f. Functor f => (b -> f b’) -> a -> f a’ polymorphic update set :: Lens a a’ b b’ -> b’ -> a -> a’ set l b = runIdentity . l (const $ Identity b) ! get :: Lens a a’ b b’ -> a -> b’ get l = runConst . l Const ! modify :: Lens a a’ b b’ -> (b -> b’) -> a’ modify l f = runIdentity . l (Identity . f) !
61. ! type Lens a a’ b b’ = forall f. Functor f => (b -> f b’) -> a -> f a’ polymorphic update set :: Lens a a’ b b’ -> b’ -> a -> a’ set l b = runIdentity . l (const $ Identity b) ! get :: Lens a a’ b b’ -> a -> b’ get l = runConst . l Const ! modify :: Lens a a’ b b’ -> (b -> b’) -> a’ modify l f = runIdentity . l (Identity . f) !
62. type Getter s a = forall r. (a -> Const r a) -> s -> Const r s ! type Setter s t a b = (a -> Identity b) -> s -> Identity t mirrored lenses
63. multiplate type Traversal a a’ b b’ = forall f. Applicative f => (b -> f b’) -> a -> f a’
64. type Traversal a a’ b b’ = forall f. Applicative f => (b -> f b’) -> a -> f a’ multiplate set :: Traversal a a’ b b’ -> b’ -> a -> a’ set l b = runIdentity . l (const $ Identity b) ! get :: Traversal a a’ b b’ -> a -> b’ get l = runConst . l Const ! modify :: Traversal a a’ b b’ -> (b -> b’) -> a’ modify l f = runIdentity . l (Identity . f) !
65. type Prism s t a b = forall p f. (Choice p, Applicative f) => p a (f b) -> p s (f t) prisms / co-lens
66. type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t) iso
67. code
68. gotchas
69. references • Pierce’s original work on lenses / bidirectional programming: • http://www.cis.upenn.edu/~bcpierce/papers/index.shtml#Lenses • van Laarhoven’s original write-up: • http://www.twanvl.nl/blog/haskell/cps-functional-references • O’Connor’s coining of the term “van Laarhoven lens” and the insight into polymorphic update: • http://r6.ca/blog/20120623T104901Z.html • O’Connor’s multiplate paper (a.k.a. traversals): • http://arxiv.org/pdf/1103.2841v2.pdf • Kmett’s expanding on mirrored lens insights (a.k.a. read-only / write only): • http://comonad.com/reader/2012/mirrored-lenses/ • lens website, lots of links and video: • http://lens.github.io/ • git repo: • https://github.com/ekmett/lens • hackage: • http://hackage.haskell.org/package/lens
70. fin @markhibberd