lens: from the ground up (in clojure)

A look at functional lenses in general, and how they would work in an uni-typed language like clojure. This talk walks through why lenses are useful, how to encode lenses, and builds up a small lens library in clojure.

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

This talk is derived from a more in-depth presentation into the haskell "lens" library: https://speakerdeck.com/markhibberd/lens-from-the-ground-up

Transcript

1. lens
   @markhibberd
   from the ground up
   

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2. motivations
   

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3. (defrecord Address [street city postcode])
   !
   (defrecord Person [name age address])
   !
   (defrecord User [uid username identity password])
   records
   

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4. (def home (->Address "road" "place" 4000))
   !
   (def mark (->Person "mark" 99 home))
   !
   (def user (->User 3365 "mth" mark
   "6aefd2842be62cd470709b27aedc7db7"))
   records
   (defrecord Address [street city postcode])
   !
   (defrecord Person [name age address])
   !
   (defrecord User [uid username identity password])
   

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5. !
   (def updated (update-in user [:identity]
   (fn [person] (update-in person [:address]
   (fn [address] (update-in address [:postcode]
   #(+ 4 %)))))))
   records
   (defrecord Address [street city postcode])
   !
   (defrecord Person [name age address])
   !
   (defrecord User [uid username identity password])
   

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6. !
   (update-in user
   [:identity :address :postcode] #(+ 2 %))
   records
   (defrecord Address [street city postcode])
   !
   (defrecord Person [name age address])
   !
   (defrecord User [uid username identity password])
   Can we do better?!
   

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7. the ground up
   

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8. ; get :: a -> b
   ; set :: a -> b -> a
   !
   (defrecord Lens
   [getter setter])
   !
   (defn lget [lens a]
   ((:getter lens) a))
   !
   (defn lset [lens a b]
   ((:setter lens) a b))
   intuitions
   

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9. ; get :: a -> b
   ; set :: a -> b -> a
   !
   (defrecord Lens
   [getter setter])
   !
   (defn lget [lens a]
   ((:getter lens) a))
   !
   (defn lset [lens a b]
   ((:setter lens) a b))
   set-get ==>
   (get l (set l a b)) == b
   !
   get-set ==>
   (set l a (get l a)) == a
   !
   set-set ==>
   (set l (set l b a) c) ==
   (set l a c)
   intuitions
   Pierce’s laws!
   

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10. but…
    !
    (defn lmodify [lens a func]
    ((:setter lens) a (func ((:getter lens) a))))
    !
    (defn lcompose [lens-a-b lens-b-c]
    (->Lens
    (fn [a] (lget lens-b-c (lget lens-a-b a)))
    (fn [a c] (lset lens-a-b a
    (lset lens-b-c
    (lget lens-a-b a) c)))))
    

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11. but…
    efficiency matters
    composition matters
    partiality matters
    elegance matters
    

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12. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    haskell
    

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13. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    haskell
    !
    !
    clojure
    

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14. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    !
    (deffn set [lens a b] (“can we?”))
    !
    (deffn get [lens a] (“can we?”))
    

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15. !
    (defprotocol Functor
    (fmap [functor f]
    "fmap :: f a -> (a -> b) -> f b"))
    aside: what is a functor
    

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16. !
    (defprotocol Functor
    (fmap [functor f]
    "fmap :: f a -> (a -> b) -> f b"))
    !
    (defrecord List [wrapped]
    Functor (fmap [functor f]
    (List.
    (map f (:wrapped functor)))))
    aside: what is a functor
    

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17. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -set [lens a] (???))
    !
    !
    !
    

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18. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -set [lens a b]
    (let [b-ib (fn [bb] (->Id b)) -- b -> Id b
    ]
    (“???”)))
    

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19. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -set [lens a b]
    (let [b-ib (fn [bb] (->Id b)) -- b -> Id b
    a-ia (lens b-ib) -- a -> Id a
    ]
    (“???”)))
    

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20. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -set [lens a b]
    (let [b-ib (fn [bb] (->Id b)) -- b -> Id b
    a-ia (lens b-ib) -- a -> Id a
    ia (a-ia a)] -- Id a
    (“???”)))
    

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21. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -set [lens a b]
    (let [b-ib (fn [bb] (->Id b)) -- b -> Id b
    a-ia (lens b-ib) -- a -> Id a
    ia (a-ia a)] -- Id a
    (:runId ia))) -- a
    

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22. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord ??? [???]
    Functor
    (fmap [functor f] ???))
    !
    (deffn -get [lens a] (???))
    !
    !
    !
    !
    

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23. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Const [runConst]
    Functor
    (fmap [functor f]
    (Const. (:runConst functor))))
    !
    (deffn -get [lens a] (???)
    !
    !
    !
    

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24. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Const [runConst]
    Functor
    (fmap [functor f]
    (Const. (:runConst functor))))
    !
    (deffn -get [lens a]
    (let [b-cb (fn [b] (->Const b)) -- b -> Const b
    ]
    (???)))
    

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25. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Const [runConst]
    Functor
    (fmap [functor f]
    (Const. (:runConst functor))))
    !
    (deffn -get [lens a]
    (let [b-cb (fn [b] (->Const b)) -- b -> Const b
    a-ca (lens b-cb) -- a -> Const a
    ]
    (???)))
    

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26. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Const [runConst]
    Functor
    (fmap [functor f]
    (Const. (:runConst functor))))
    !
    (deffn -get [lens a]
    (let [b-cb (fn [b] (->Const b)) -- b -> Const b
    a-ca (lens b-cb) -- a -> Const a
    ca (a-ca a)] -- Const a
    (???)))
    

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27. !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    back to the drawing board
    (defrecord Const [runConst]
    Functor
    (fmap [functor f]
    (Const. (:runConst functor))))
    !
    (deffn -get [lens a]
    (let [b-cb (fn [b] (->Const b)) -- b -> Const b
    a-ca (lens b-cb) -- a -> Const a
    ca (a-ca a)] -- Const a
    (:runConst ca))) -- 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
    (defrecord Id [runId]
    Functor
    (fmap [functor f]
    (Id. (f (:runId functor)))))
    !
    (deffn -modify [lens a f]
    (let [b-ib (fn [bb] (->Id (f bb))) -- b -> Id b
    a-ia (lens b-ib) -- a -> Id a
    ia (a-ia a)] -- Id a
    (:runId ia))) -- a
    

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29. functional elegance
    

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

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31. function composition
    !
    type Lens’ a b =
    forall f. Functor f => (b -> f b) -> a -> f a
    x :: Lens a b
    y :: Lens b c
    (comp x y) :: Lens a c
    !
    !
    !
    !
    !
    

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32. 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
    (comp x y) :: Lens a c
    !
    !
    !
    !
    !
    

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33. 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)
    (comp x y) :: Lens a c
    !
    !
    !
    !
    !
    

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34. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    !
    !
    !
    !
    

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35. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    !
    !
    

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36. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    (defn comp [f g]
    (fn [a] (f (g a))))
    

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37. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j
    !
    

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38. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    !
    

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39. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    i :: b -> f b
    

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40. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    i :: b -> f b, j :: a -> f a
    

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41. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (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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42. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    i :: b -> f b, j :: a -> f a, h :: c -> f c
    (comp x y) :: h -> j
    

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43. 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)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    i :: b -> f b, j :: a -> f a, h :: c -> f c
    (comp x y) :: h -> j :: (c -> f c) -> (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 :: (b -> f b) -> (a -> f a)
    y :: Lens b c :: (c -> f c) -> (b -> f b)
    (comp x y) :: Lens a c :: (c -> f c) -> (a -> f a)
    !
    (comp) :: (i -> j) -> (h -> i) -> h -> j
    !
    x :: i -> j, y :: h -> i
    i :: b -> f b, j :: a -> f a, h :: c -> f c
    (comp x y) :: h -> j :: (c -> f c) -> (a -> f a)
    

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

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47. 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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48. code
    

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49. 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
    • this talk in haskell:
    • https://speakerdeck.com/markhibberd/lens-from-the-ground-up
    

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50. fin
    @markhibberd
    

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