Zippers

David Overton

1 August 2013

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

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’;

Motivation

But what is the equivalent for an immutable data structure? I.e. how do we keep trackof a local focus point within a data structure? E.g. XML document tree, text editorbuffer, 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.

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

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)

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)

*Main> let z = fromList [’a’, ’b’, ’c’]

*Main> z

("","abc")

a

ctx list

b

c

*Main> right z

("a","bc")a

ctx list

b

c

*Main> right $ right z

("ba","c")

a

ctx list

b c

*Main> right $ right $ right z

("cba","")

a

ctx list

b

c

List Zipper

Example (XMonad)

XMonad is a tiling window manager for X written in Haskell. The main data structurefor 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 ofworkspaces 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 allowwrapping when you reach the end of the window or workspace list.)

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

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)

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)

*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

*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

*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

Tree Zippers

Example (XML)

• XPath navigation around an XML document is basically a zipper – you have acurrent 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 | · · ·

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

Algebraic Data Types

Why are the types in languages such as Haskell sometimes referred to as AlgebraicData Types? One way to see the relationship to algebra is to convert the types toalgebraic expressions which represent the number of values that a type has.

data Void ⇔ 0 The empty typedata () = () ⇔ 1 The unit typedata Either a b = Left a | Right b ⇔ a + b A sum typetype 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 ⇔ 2data Maybe a = Nothing | Just a ⇔ 1 + atype Func a b = a -> b ⇔ ba An exponential typeInt ⇔ 232 or 264 (implementation dependent)

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)

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 alreadyknew!).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 + · · ·

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 squareroots of types!

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

One-Hole Contexts

DefinitionThe one-hole context of a parameterised type T (a) is the type of data structures youget if you remove one distinguished element of type a from a data structure of typeT (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 couldrepresent this is

data TripleOhc a = Left a a | Mid a a | Right a a

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

daa3

In fact, this this relationship holds for any parameterised type T (a).

ObservationThe type of the one-hole context of any parameterized type T (a) is the derivative type

d

daT (a) or ∂aT (a)

Zippers from One-Hole Contexts

ObservationFor 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 . . . ?

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]). Thedistinguished element is now pulled out of the second list and made explicit. This alsomeans that the zipper can’t be empty. However, this is still a list zipper and we derivedit using the differential calculus!

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])

Table of Contents

1 Motivation

2 List Zipper

3 Tree Zippers

4 Algebraic Data Types

5 Type Differentiation

6 Conclusion

Conclusion

• Zippers are a useful way of navigating and manipulating mutable data structuresin 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.