Applicative lenses

Posted on November 2, 2016

This is written in Literate Haskell.

module Applicative.Lenses where

import Control.Applicative
import Control.Monad
import Data.Functor
import Data.Monoid

import Bidirectional.Serialization.Two (Codec(..)) -- 2016-10-13
import Bidirectional.Serialization.Classes

I’m trying to figure out a relationship between the abstractions I previously brought up and lenses.

data Lens s t a b = Lens
  { get :: s -> a
  , put :: b -> s -> t

type Lens' s a = Lens s s a a

idL :: Lens' s s
idL = Lens id const

unitL :: Lens' s ()
unitL = ignorePut

ignorePut :: Lens s s () b
ignorePut = Lens (const ()) (const id)

There is a similarity to codecs. We import the definition of Codec from 2016-10-13.

Straightforward representation

Define the following wrapper types for get and put.

type MLensR = (->)
type MLensW s = Const (Endo s)
  -- Applicative but not Monad.

lensR :: (s -> a) -> MLensR s a
lensR = id

unMLensR :: MLensR s a -> s -> a
unMLensR = id

lensW :: (s -> s) -> MLensW s a
lensW = Const . Endo

unMLensW :: MLensW s a -> s -> s
unMLensW = appEndo . getConst

type MLens s = Codec (MLensR s) (MLensW s)

Then MLens s a a is isomorphic to Lens' s a. The isomorphism can in fact be generalized between MLens s b a and Lens s s a b.

fromLens' :: Lens' s a -> MLens s a a
fromLens' = fromLens

toLens' :: MLens s a a -> Lens' s a
toLens' = toLens

fromLens :: Lens s s a b -> MLens s b a
fromLens (Lens get put) = Codec
  (lensR get)
  (lensW . put)

toLens :: MLens s b a -> Lens s s a b
toLens (Codec parse produce) = Lens
  (unMLensR parse)
  (unMLensW . produce)

Based on that construction, we now have an Applicative interface for composing lenses “horizontally”, combining their views together.

  :: (Profunctor p, Applicative (p (b, b')))
  => p b a -> p b' a' -> p (b, b') (a, a')
pairP p p' = (,)
  <$> fst =. p
  <*> snd =. p'

pair :: MLens s b a -> MLens s b' a' -> MLens s (b, b') (a, a')
pair = pairP

pairLens :: Lens s s a b -> Lens s s a' b' -> Lens s s (a, a') (b, b')
pairLens l l' = toLens $ pair (fromLens l) (fromLens l')

However, their good behavior is not guaranteed.

oops :: Lens' s a -> Lens' s (a, a)
oops l = pairLens l l

-- PutGet fails.
oopsFails = get l (put l a s) /= a
    l = oops idL
    a = (True, False)
    s = False

The problem here is that the lenses overlap, so the views may not be modified independently.

Conversely, pairLens l l' is well-behaved if l :: Lens s a and l' :: Lens s a' are well-behaved and non-overlapping, i.e., if for all s, a and a', putting one view does not affect the other:

get l (put l' a' s) == get l s
get l' (put l a s) == get l' s

We actually only need one equality to hold to ensure the good behavior of pairLens l l', but determining which one (here, the second) relies on a careful examination of the order of updates in pair.

Detecting conflicts

Given two lenses l :: Lens s a and l' :: Lens s a', possibly overlapping, and some source s, we say that a pair (a, a') is a consistent update of s through l and l' if:

get l (put l' a' s) == get l s
get l' (put l a s) == get l' s

Two lenses are non-overlapping if and only if all updates through them are consistent.

For more flexibility, we shall allow ourselves to create partial lenses, which allow updates of overlapping views as long as they are consistent (actually, with a generalized definition of consistency).

We can “record observations” in order to forbid inconsistent modifications as in Applicative Bidirectional Programming with Lenses1, using a more elaborate MLensW2 type.

newtype MLensW2 s a = MLensW2
  { runMLensW2 :: s -> (s -> Bool) -> Maybe (s, s -> Bool, a)

instance Functor (MLensW2 s) where
  fmap = liftA

instance Applicative (MLensW2 s) where
  pure a = MLensW2 $ \s p -> Just (s, p, a)
  (<*>) = ap

instance Monad (MLensW2 s) where
  MLensW2 x_ >>= f = MLensW2 $ \s p -> do
    (s', p', x) <- x_ s p
    (s'', p'', y) <- runMLensW2 (f x) s' p'
    return (s'', p'', y)

MLensW2 is a composition of State (s, s -> Bool) and Maybe.

sourceW fetches the s component of the state.

sourceW :: MLensW2 s s
sourceW = MLensW2 $ \s p -> pure (s, p, s)

putW updates the source through a partial lens. putW is unsafe to use alone in general, with the risk of defining ill-behaved lenses. However, careful and fine grained use of putW can help optimize composite lenses by avoiding redundant checks.

-- PLens' defined below.
putW :: PLens' s a -> a -> MLensW2 s ()
putW l a = MLensW2 $ \s p -> do
  s' <- put l a s
  pure (s', p, ())

assertW checks that the updated source is consistent with previous observations and modifications.

assertW :: MLensW2 s ()
assertW = MLensW2 $ \s p ->
  guard (p s) $> (s, p, ())

matchW adds a another consistency constraint preventing a view of the source to be modified.

matchW :: Eq w => (s -> w) -> w -> MLensW2 s ()
matchW get w = MLensW2 $ \s p ->
  pure (s, \s' -> p s' && get s' == w, ())

We have an isomorphism between MLens2 and lenses with a partial put.

type MLens2 s = Codec (MLensR s) (MLensW2 s)
type PLens' s a = Lens s (Maybe s) a a

fromLens2 :: Eq a => PLens' s a -> MLens2 s a a
fromLens2 l = Codec
  (get l)
  (\a -> putW l a *> assertW *> matchW (get l) a $> a)

toLens2 :: MLens2 s b a -> Lens s (Maybe s) a b
toLens2 (Codec get produce) = Lens get put
    put b s = fmap fst3 (runMLensW2 (produce b) s (const True))

fst3 :: (a, b, c) -> a
fst3 (s, _, _) = s

We can also observe values in the source/state, without writing anything. Modifying a previously observed value is an error.

observe :: Eq w => (s -> w) -> MLens2 s () w
observe parse = Codec parse produce
    produce () = observeW parse

observeW :: Eq w => (s -> w) -> MLensW2 s w
observeW get = do
  s <- sourceW
  let w = get s
  matchW get w
  pure w

A functor from lenses to functions serves as “vertical” composition.

lift :: Lens' s a -> MLens2 z s s -> MLens2 z a a
lift l (Codec parse produce) = Codec
  (get l . parse)
  (\a -> do
    z <- sourceW
    let s = parse z
    produce (put l a s)
    pure a

-- A generalized version, though less efficient.
lift_ :: Lens s t a b -> MLens2 z t s -> MLens2 z b a
lift_ l (Codec parse produce) = Codec
  (get l . parse)
  (\b -> do
    z <- sourceW
    let s = parse z
    s <- produce (put l b s)
    pure (get l s)

  1. Applicative Bidirectional Programming with Lenses, K. Matsuda, M. Wang.