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)

Transcript

1. lens
   @markhibberd
   from the ground up
   

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2. motivations
   

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3. the ground up
   

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4. data Lens a b = Lens {
   get :: a -> b
   , set :: b -> a -> a
   }
   intuitions
   

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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!
   

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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)
   

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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!
   

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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 })
   

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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
   

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10. but…
    data Safety =
    Safety { _readOnly :: String }
    !
    readOnly :: Lens Safety String
    readOnly =
    Lens _readOnly (error "don't do this")
    

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11. but… read only / write only matters
    data Safety =
    Safety { _readOnly :: String }
    !
    readOnly :: Lens Safety String
    readOnly =
    Lens _readOnly (error "don't do this")
    

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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))
    

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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
    

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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
    

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15. but…
    data OneOf = First String | Second Int
    !
    first :: Lens OneOf (Maybe String)
    first = undefined
    !
    first :: Lens OneOf (Maybe String)
    first = undefined
    

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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
    

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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
    

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18. failed experiments
    data Store s a =
    Store (s -> a) s
    !
    data Lens a b =
    Lens (a -> Store b a)
    !
    !
    !
    !
    !
    !
    !
    !
    

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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
    

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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
    

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21. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    

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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?”
    

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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
    !
    !
    

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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
    !
    

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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
    

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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
    

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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)
    !
    !
    !
    

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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
    !
    !
    !
    

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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)
    !
    !
    !
    

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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
    !
    !
    !
    

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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
    !
    !
    !
    

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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
    

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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
    !
    !
    

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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
    !
    

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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
    

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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
    

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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)
    !
    !
    !
    

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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
    !
    !
    !
    

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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
    !
    !
    !
    

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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
    !
    !
    !
    !
    

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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)
    !
    

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42. functional elegance
    

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43. composition
    !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    

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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
    !
    !
    !
    !
    !
    

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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
    !
    !
    !
    !
    !
    

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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
    !
    !
    !
    !
    !
    

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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)
    !
    !
    !
    !
    !
    

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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
    !
    !
    !
    

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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
    !
    

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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
    !
    

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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
    

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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
    

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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
    

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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
    

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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)
    

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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)
    

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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
    

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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
    

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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)
    !
    

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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)
    !
    

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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)
    !
    

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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
    

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63. multiplate
    type Traversal a a’ b b’ =
    forall f. Applicative f =>
    (b -> f b’) -> a -> f a’
    

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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)
    !
    

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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
    

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66. type Iso s t a b =
    forall p f. (Profunctor p, Functor f) =>
    p a (f b) -> p s (f t)
    iso
    

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67. code
    

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68. gotchas
    

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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
    

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70. fin
    @markhibberd
    

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