# LGtk/ADT lenses

### From HaskellWiki

m (→Example: List lens) |
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The proposed solution, summarized: | The proposed solution, summarized: | ||

− | ''' | + | '''As a lens toolbox for an ADT, use a lens whose ''co''domain 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 === | === Example: List lens === | ||

Line 77: | Line 79: | ||

set (l: r) _ = (True, (l, r)) | set (l: r) _ = (True, (l, r)) | ||

</haskell> | </haskell> | ||

+ | |||

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

=== Usage === | === Usage === | ||

Line 86: | Line 90: | ||

</haskell> | </haskell> | ||

− | |||

− | We can edit the list | + | We can view and edit the list through the following lenses: |

+ | |||

+ | * <hask>listLens :: Lens S [Int]</hask> edits the '''complete list'''. | ||

+ | * <hask>fstLens :: Lens S Bool</hask> edits the '''top level constructor''' of the list: <hask>False</hask> corresponds to <hask>[]</hask> and <hask>True</hask> corresponds to <hask>(:)</hask>. | ||

+ | * <hask>headLens = fstLens . sndLens :: Lens S Int</hask> edits the '''head''' of the list. | ||

+ | * <hask>tailLens = sndLens . sndLens :: Lens S [Int]</hask> edits the '''tail''' of the list. | ||

− | + | Remarks: | |

− | + | ||

− | + | ||

− | + | * If the top level constructor of the list is <hask>[]</hask>, the head and the tail of the list can still be edited; the change will only be visible through <hask>listLens</hask> when the constructor is changed back to <hask>(:)</hask>. 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 <hask>s' :: S</hask> such that <hask>s</hask> viewed through <hask>tailLens</hask> is the same as <hask>s'</hask> viewed through <hask>listLens</hask>. Explained on a figure: | ||

[[Image:ADT.png]] | [[Image:ADT.png]] | ||

− | How <hask>s'</hask> | + | How <hask>s'</hask> can be created and how <hask>s</hask> and <hask>s'</hask> can be kept in sync is a related but different question. See [[LGtk/Semantics#Dependent_reference_creation]]. |

== Links and references == | == Links and references == |

## Revision as of 21:48, 7 June 2013

## Contents |

## 1 Problem description

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

For example, one can view and edit pairs withq == 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?

## 2 Existing solutions

### 2.1 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:)

### 2.2 Other solutions

*Please help to extend the list of known solutions.*

## 3 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.

### 3.1 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))

### 3.2 Usage

Suppose that we have a statetype S = (Bool, (Int, [Int]))

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

- edits thelistLens :: Lens S [Int]
**complete list**. - edits thefstLens :: Lens S Bool
**top level constructor**of the list:corresponds toFalseand[]corresponds toTrue.(:) - edits theheadLens = fstLens . sndLens :: Lens S Int
**head**of the list. - edits thetailLens = sndLens . sndLens :: Lens S [Int]
**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[]when the constructor is changed back tolistLens. 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 such thats' :: Sviewed throughsis the same astailLensviewed throughs'. Explained on a figure:listLens

## 4 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.

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