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LGtk/ADT lenses

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The flollowing partial lenses are defined for lists:
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The following partial lenses are defined for lists:
   
 
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Revision as of 23:26, 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 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.

Question:

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 following 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.

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 a 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))
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
.

3.1.1 List lens 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. See the interpretation subsection below.
  • 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:

ADT.png

On the figure, edges are lenses and nodes are references. One possible definition of references is described in LGtk/Semantics#References. How
s'
can be created and how
s
and
s'
can be kept in sync is an important but separate question. LGtk/Semantics#Dependent_reference_creation describes a possible solution.

3.2 Example: ADT with repeated record fields

Consider the following ADT:

data X a
  = X1 { y :: Int, z :: a }
  | X2 { y :: Int, v :: Char }
Note that the
y
field is defined twice.

For the ADT lens, first define an auxiliary enum type for the constructor tags:

data XTag = X1Tag | X2Tag
The definition of
XTag
is kind of inevitable if we would like to edit the constructor tags. We could use
Bool
, but that would not scale to more constructors. We could use
Int
too, but with less static checks from the type system. The definition of the lens toolbox for
X
:
xLens :: Lens (XTag, (Int, (a, Char))) (X a)
xLens = lens get set where
 
    get (X1Tag, (y, (z, _))) = X1 y z
    get (X2Tag, (y, (_, v))) = X2 y v
 
    set (X1 y z) (_, (_, (_, v))) = (X1Tag, (y, (z, v)))
    set (X2 y v) (_, (_, (z, _))) = (X2Tag, (y, (z, v)))

Remarks:

  • Instead of
    (XTag, (Int, (a, Char)))
    , we could use
    (XTag, (Int, a, Char))
    or
    (XTag, Int, a, Char)
    too. This is an implementation detail.
  • xLens
    remembers the value of
    y
    if we change between the constructor tags. This is the intended behaviour.
  • xLens
    remembers the values of
    v
    and
    z
    fields if we change between the constructor tags. This is the intended behaviour.

3.2.1 Interpreation

The intended behaviour can be justified if we interpret lenses as abstract editors. If we would like to define an editor of an
X
value, the state of the editor would be
(XTag, (Int, (a, Char)))
, and one could retrive the actual
X
value by
xLens
from the state. The editor would be the composition of the following simpler editors:
  • An elementary editor for
    XTag
    (maybe a combo box or a checkbox) which would be connected to the editor state by
    fstLens
    .
  • An elementary editor for an
    Int
    (maybe a text box or a slider) which would be connected to the editor state by
    fstLens . sndLens
    .
  • An editor for an
    a
    typed value which would be connected to the editor state by
    fstLens . sndLens . sndLens
    .
  • An editor for a
    Char
    (maybe a combo box or a text box or a virtual keyboard) which would be connected to the editor state by
    fstLens . sndLens . sndLens . sndLens
    .

Now, the intended behaviour is the following:

  • If the user fills in an
    Int
    value for
    y
    , this value should remain the same after changing the
    XTag
    value.
  • The
    a
    value editor should be active only if the
    XTag
    value is
    X1Tag
    .
  • The
    Char
    editor should be active only if the
    XTag
    value is
    X2Tag
    .
  • If the user fills in an
    a
    value for
    z
    when the
    XTag
    value is
    X1Tag
    , and the user changes
    X1Tag
    to
    X2Tag
    and then back to
    X1Tag
    , the
    a
    value should be the same as before (consider a complex value which is hard to re-create). Similar holds for the
    Char
    value.

3.3 Generic ADT lens

In the generic case, consider the following ADT:

data X a1 ... aN
  = X1 { x11 :: t11, x12 :: t12, ... }
  | X2 { x21 :: t21, x22 :: t22, ... }
  ...
  | XM { xM1 :: tM1, xM2 :: tM2, ... }
Suppose that the set of different field names are
f1 :: s1
,
f2 :: s2
, ...,
fK :: sK
. Let
cij
equal to
k
iff
xij
is equal to
fk
.

Define an auxiliary enum type for the constructor tags:

data XTag = X1Tag | X2Tag | ... | XMTag
The definition of the lens toolbox for
X
:
xLens :: Lens (XTag, (s1, (s2, ...))) (X a1 ... aN)
xLens = lens get set where
 
    get (X1Tag, (s1, (s2, ...))) = X1 s{c11} s{c12} ...
    ...
    get (XMTag, (s1, (s2, ...))) = X2 s{cM1} s{cM2} ...
 
    -- replace second occurrence of the same variable name by _ in pattern
    set (X1 s{c11} s{c12} ...) (_, (s1, (s2, ...))) = (X1Tag, (s1, (s2, ...)))
    ...
    set (XM s{cM1} s{cM2} ...) (_, (s1, (s2, ...))) = (XMTag, (s1, (s2, ...)))

4 Summary

With the proposed ADT lenses, one can do arbitrary edit actions on ADT values in a user-friendly way (both distinct and common ADT parts are remembered when switching between constructors).

5 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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