LGtk/ADT lenses

Problem description

Lenses provide uniform and compositional way to view and edit data structures.

For example, one can view and edit pairs with fstLens and sndLens. These two lenses form a toolbox for editing pairs in the sense that given pairs p :: (a, b) and q :: (a, b), by succesive get and set operations p can be changed to be equivalent to q:

q == setL fstLens (getL fstLens q) (setL sndLens (getL sndLens q) p)

Similarly, there is a toolbox of lenses for records which toolbox contains one lens for each record field.

Are there a toolbox of lenses for algebraic data types with multiple constructors?

Existing solutions

Partial lenses

The data-lens library provides partial lenses which are isomorphic to

type PartialLens a b = (a -> Maybe b, a -> Maybe (b -> a))

The flollowing partial lenses are defined for lists:

headLens :: PartialLens [a] a
headLens = (get, set)
 where
  get [] = Nothing
  get (h:t) = Just h

  set [] = Nothing
  set (h:t) = Just (:t)

tailLens :: PartialLens [a] [a]
tailLens = (get, set)
 where
  get [] = Nothing
  get (h:t) = Just t

  set [] = Nothing
  set (h:t) = Just (h:)

Unfortunately headLens and tailLens does not provide a complete toolbox, one cannot change an empty list to a non-empty list with them, for example.

ADT lenses

The proposed solution, summarized:

As a lens toolbox for an ADT, use a lens whose codomain is the ADT and whose domain is tuple of the constructor tag and the ADT components.

Let's see specific examples before the generic descripton of the proposed lens.

Example: List lens

The lens for lists which forms a complete toolbox:

import Data.Lens.Common

listLens :: Lens (Bool, (a, [a])) [a]
listLens = lens get set where

    get (False, _) = []
    get (True, (l, r)) = l: r

    set [] (_, x) = (False, x)
    set (l: r) _ = (True, (l, r))

Here Bool is used as the constructor tag, and a and [a] are the components of the list (head and tail). Intead of a triple we use two pairs, so we reach the parts with fstLens and sndLens.

Usage

Suppose that we have a state s of type

type S = (Bool, (Int, [Int]))

We can view and edit the list through the following lenses:

listLens :: Lens S [Int] edits the complete list.
fstLens :: Lens S Bool edits the top level constructor of the list: False corresponds to [] and True corresponds to (:).
headLens = fstLens . sndLens :: Lens S Int edits the head of the list.
tailLens = sndLens . sndLens :: Lens S [Int] edits the tail of the list.

Remarks:

If the top level constructor of the list is [], the head and the tail of the list can still be edited; the change will only be visible through listLens when the constructor is changed back to (:). This may seem to be odd, but for many applications this is the right behaviour.
For editing the tail of the tail of the list, we need an s' :: S such that s viewed through tailLens is the same as s' viewed through listLens. Explained on a figure:

How s' can be created and how s and s' can be kept in sync is a related but different question. See LGtk/Semantics#Dependent_reference_creation.

Links and references

I have not seen this technique described before. Please help to extend the list of papers / blog entries, where this or similar technique is used.

[Reddit comments]

LGtk/ADT lenses

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