The document discusses zippers, which are data structures that help navigate and update immutable data structures efficiently. It introduces list and tree zippers, demonstrating their utility with examples such as Haskell implementations for window managers and XML navigation. The conclusion highlights the usefulness of zippers in functional programming, the algebraic nature of data types, and the potential for deriving zippers through type differentiation.

1. Zippers
David Overton
1 August 2013

2. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

3. Motivation
Navigating around in and updating mutable data structures is easy:
• Keep a pointer to your current location in the data structure;
• Update the pointer to move around the data structure;
• Modify the data structure using destructive update via the pointer.
Example (C Array)
char my_array[1024];
char *p = my_array;
// Move right
p++;
// Move left
p--;
// Update focused element
*p = ’A’;

4. Motivation
But what is the equivalent for an immutable data structure? I.e. how do we keep track
of a local focus point within a data structure? E.g. XML document tree, text editor
buffer, window manager stack (e.g. XMonad), AVL tree.
For a list we could use an integer to represent the element we are interested in:
type ListFocus a = (Int, [a])
but this requires traversing from the start of the list every time – an O(n) operation.
For more complex types, this is even less practical.
Instead, we can use a data structure called a zipper.

5. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

6. List Zipper
-- A list zipper is a pair consisting of the front
-- of the list (reversed) and the rear of the list.
type ListZipper a = ([a], [a])
-- Convert a list to a zipper.
-- Start at the front of the list.
fromList :: [a] → ListZipper a
fromList xs = ([], xs)

7. List Zipper
-- Move about in the zipper.
right, left :: ListZipper a → ListZipper a
right (xs, y:ys) = (y:xs, ys)
left (x:xs, ys) = (xs, x:ys)
-- Modify the focused element.
modify :: (a → a) → ListZipper a → ListZipper a
modify f (xs, y:ys) = (xs, (f y):ys)
modify _ zs = zs
-- When we are finished, convert back to a list.
toList :: ListZipper a → [a]
toList ([], ys) = ys
toList zs = toList (left zs)

8. *Main> let z = fromList [’a’, ’b’, ’c’]
*Main> z
("","abc")
a
ctx list
b
c

9. *Main> right z
("a","bc")
a
ctx list
b
c

10. *Main> right $ right z
("ba","c")
a
ctx list
b c

11. *Main> right $ right $ right z
("cba","")
a
ctx list
b
c

12. List Zipper
Example (XMonad)
XMonad is a tiling window manager for X written in Haskell. The main data structure
for managing workspace and window focus uses nested list zippers.
type StackSet a = ListZipper (Workspace a)
type Workspace a = ListZipper (Window a)
This allows XMonad to keep track of the focused workspace in an ordered list of
workspaces and also keep track of the focused window on each workspace.
(This is a simplification of the actual types used. The zippers that XMonad uses allow
wrapping when you reach the end of the window or workspace list.)

13. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

14. Binary Tree
data Tree a
= Empty
| Node (Tree a) a (Tree a)
deriving Show
type Context a = Either (a, Tree a) (a, Tree a)
type TreeZipper a = ([Context a], Tree a)
fromTree :: Tree a → TreeZipper a
fromTree t = ([], t)
right, left, up :: TreeZipper a → TreeZipper a
right (ctx, Node l a r) = (Right (a, l) : ctx, r)
left (ctx, Node l a r) = (Left (a, r) : ctx, l)

15. Binary Tree
up (Left (a, r) : ctx, l) = (ctx, Node l a r)
up (Right (a, l) : ctx, r) = (ctx, Node l a r)
modify :: (a → a) → TreeZipper a → TreeZipper a
modify f (ctx, Node l a r) = (ctx, Node l (f a) r)
modify f z = z
toTree :: TreeZipper a → Tree a
toTree ([], t) = t
toTree z = toTree (up z)

16. *Main> let t = Node
(Node
(Node Empty ’d’ Empty)
’b’
(Node Empty ’e’ Empty))
’a’
(Node
(Node Empty ’f’ Empty)
’c’
(Node Empty ’g’ Empty))
*Main> fromTree t
a
cb
ed gf
ctx tree

17. *Main> right $ fromTree t
( [Right (’a’,
Node
(Node Empty ’d’ Empty)
’b’
(Node Empty ’e’ Empty))],
Node
(Node Empty ’f’ Empty)
’c’
(Node Empty ’g’ Empty) )
Right
a c
b
ed
gf
ctx tree

18. *Main> left $ right $ fromTree t
( [ Left (’c’, Node Empty ’g’ Empty),
Right (’a’,
Node
(Node Empty ’d’ Empty)
’b’
(Node Empty ’e’ Empty))],
Node Empty ’f’ Empty )
Right
Left
a
c
b
ed
g
f
ctx tree

19. Tree Zippers
Example (XML)
• XPath navigation around an XML document is basically a zipper – you have a
current context and methods to navigate to neighbouring nodes.
• One implementation of this in Haskell is the rose tree zipper from HXT.
data NTree a = NTree a [NTree a] -- A rose tree
data NTCrumb = NTC [NTree a] a [NTree a] -- "Breadcrumb" for context
-- The zipper itself
data NTZipper a = NTZ { ntree :: NTree a, context :: [NTCrumb a] }
-- And using it to represent XML
type XmlTree = NTree XNode
type XmlNavTree = NTZipper XNode
data XNode = XText String | XTag QName [XmlTree] | XAttr QName | · · ·

20. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

21. Algebraic Data Types
Why are the types in languages such as Haskell sometimes referred to as Algebraic
Data Types? One way to see the relationship to algebra is to convert the types to
algebraic expressions which represent the number of values that a type has.
data Void ⇔ 0 The empty type
data () = () ⇔ 1 The unit type
data Either a b = Left a | Right b ⇔ a + b A sum type
type Pair a b = (a, b) ⇔ a.b A product type
Technically, Haskell types form an algebraic structure called a semiring.
Some more examples:
data Bool = False | True ⇔ 2
data Maybe a = Nothing | Just a ⇔ 1 + a
type Func a b = a -> b ⇔ ba An exponential type
Int ⇔ 232 or 264 (implementation dependent)

22. Algebraic Laws
The usual algebraic laws (for semirings) hold, up to type “equivalence”:
0 + a = a ⇔ Either Void a ∼= a
a + b = b + a ⇔ Either a b ∼= Either b a
0.a = 0 ⇔ (Void, a) ∼= Void
1.a = a ⇔ ((), a) ∼= a
a.b = b.a ⇔ (a, b) ∼= (b, a)
a.(b + c) = a.b + a.c ⇔ (a, Either b c) ∼= Either (a,b) (a,c)
(cb)a = cb.a ⇔ a -> b -> c ∼= (a, b) -> c
a2 = a.a ⇔ Bool -> a ∼= (a, a)

23. Algebra of the List Type
data List a = Nil | Cons a (List a)
L(a) = 1 + a.L(a)
= 1 + a.(1 + a.L(a))
= 1 + a + a2(1 + a.L(a))
= 1 + a + a2 + a3 + a4 + · · ·
So a list has either 0 elements or 1 element or 2 elements or . . . (which we already
knew!).
Alternatively, solving for L(a):
L(a) − a.L(a) = 1
(1 − a).L(a) = 1
L(a) =
1
1 − a
It doesn’t seem to make much sense to do subtraction or division on types. But look
at the Taylor series expansion:
1
1 − a
= 1 + a + a2 + a3 + a4 + · · ·

24. Algebra of Tree Types
data Tree a = Empty | Node (Tree a) a (Tree a)
T(a) = 1 + a.T(a)2
a.T(a)2 − T(a) + 1 = 0
T(a) =
1 −
√
1 − 4.a
2.a
= 1 + a + 2a2 + 5a3 + 14a4 + · · ·
via the quadratic formula and Taylor series expansion. But now we are taking square
roots of types!

25. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

26. One-Hole Contexts
Definition
The one-hole context of a parameterised type T(a) is the type of data structures you
get if you remove one distinguished element of type a from a data structure of type
T(a) and somehow mark the hole where the element came from.
Example
type Triple a = (a, a, a)
The one-hole contexts are ( , a, a), (a, , a), and (a, a, ). A type that could
represent this is
data TripleOhc a = Left a a | Mid a a | Right a a

27. One-Hole Contexts
Look at the algebraic types:
type Triple a = (a, a, a) ⇔ a3
type TripleOhc a = Left a a | Mid a a | Right a a ⇔ 3a2
Notice that
3a2
=
d
da
a3
In fact, this this relationship holds for any parameterised type T(a).
Observation
The type of the one-hole context of any parameterized type T(a) is the derivative type
d
da
T(a) or ∂aT(a)

28. Zippers from One-Hole Contexts
Observation
For any parameterized type T(a), the type a.∂aT(a) is a zipper for type T(a).
Example
The type
type TripleZipper a = (a, TripleOhc a)
is a zipper for the type Triple a.
TripleZipper(a) = a.3a2
= 3a3
= 3.Triple(a)
But what about some more interesting types . . . ?

29. List Zipper
L(a) =
1
1 − a
∂aL(a) =
1
(1 − a)2
= L(a)2
ZL(a) = a.∂aL(a) = a.L(a)2
type ListZipper a = (a, [a], [a])
This is slightly different from our original list zipper type ([a], [a]). The
distinguished element is now pulled out of the second list and made explicit. This also
means that the zipper can’t be empty. However, this is still a list zipper and we derived
it using the differential calculus!

30. Tree Zipper
T(a) = 1 + a.T(a)2
∂aT(a) = T(a)2 + 2a.T(a).∂aT(a)
∂aT(a) =
T(a)2
1 − 2a.T(a)
= T(a)2.L(2a.T(a))
ZT (a) = a.∂aT(a)
= a.T(a)2.L(2a.T(a))
type Context a = Either (a, Tree a) (a, Tree a)
type TreeZipper a = (a, Tree a, Tree a, [Context a])

31. Table of Contents
1 Motivation
2 List Zipper
3 Tree Zippers
4 Algebraic Data Types
5 Type Differentiation
6 Conclusion

32. Conclusion
• Zippers are a useful way of navigating and manipulating mutable data structures
in real-world functional programs.
• Algebraic data types really are algebraic.
• Differentiation of types can lead to automatic derivation of zippers.
• Nobody really knows why this works yet — active research.