It was all well and good to see the patterns in tiny “toy” functions. Needless to say, these patterns appear regularly in “real” code, if only you know to look for them. Next, we will develop a small library that “swizzles” text files.

We will start with a beginner’s version that is riddled with explicit recursion.

Next, we will try to spot the patterns and eliminate recursion using higher-order functions.

Finally, we will parameterize the code so that we can both “swizzle” and “unswizzle” without having to duplicate code.

Needless to say, the code can be cleaned up even more, and I encourage you to do so. For example, rewrite the code that swizzles `Char`

so that instead of using association lists, it uses the more efficient `Map k v`

(maps from keys `k`

to values `v`

) type in the standard library module `Data.Map`

.

Recall that Haskell allows you to create brand new data types like this one from lecture 1.

`> data Shape = Rectangle Double Double `

> | Polygon [(Double, Double)]

Values of this type are either two doubles *tagged* with `Rectangle`

`ghci> :type (Rectangle 4.5 1.2)`

(Rectangle 4.5 1.2) :: Shape

or a list of pairs of `Double`

values *tagged* with `Polygon`

`ghci> :type (Polygon [(1, 1), (2, 2), (3, 3)])`

(Polygon [(1, 1), (2, 2), (3, 3)]) :: Shape

One can think of these values as being inside special “boxes” with the appropriate tags.

However, Haskell (and other functional languages), allow you to define datatypes *recursively* much like functions can be defined recursively. For example, consider the type

`> data IntList = IEmpty `

> | IOneAndMore Int IntList

> deriving (Show)

(Ignore the bit about `deriving`

for now.) What does a value of type `IntList`

look like? As before, you can *only* obtain them through the constructors. Here, we have the constructor `IEmpty`

takes no arguments and returns an `IntList`

, that is

`ghci> :type IEmpty `

IEmpty :: IntList

Now, that we have at least one value of type `IntList`

we can make more by using the other constructor

`> is1 = IOneAndMore 1 IEmpty`

> is2 = IOneAndMore 2 is1

> is3 = IOneAndMore 3 is2

and so on. Suppose we ask Haskell to *show* us `is3`

we get

`ghci> is3`

IOneAndMore 3 (IOneAndMore 2 (IOneAndMore 1 IEmpty))

ghci> :type is3

is3 :: IntList

However, the presence of recursion does not change what the values really are, now you simply have boxes within boxes.

Of course, there is no reason why the type definition must have only one *recursive* occurrence of the type. You can encode general *trees* like

`> data IntTree = ILeaf Int`

> | INode IntTree IntTree

> deriving (Show)

Here, each value is either a simple *leaf* which is a box containing an `Int`

labeled `ILeaf`

, such as each of

`> it1 = ILeaf 1 `

> it2 = ILeaf 2

or an *internal node* which is a box containing two trees, a left and right tree, and marked with a tag `INode`

, such as each of

`> itt = INode (ILeaf 1) (ILeaf 2)`

> itt' = INode itt itt

> itt'' = INode itt' itt'

Now, if we ask Haskell

`ghci> itt'`

INode (INode (ILeaf 1) (ILeaf 2)) (INode (ILeaf 1) (ILeaf 2))

ghci> :type itt''

itt' :: IntTree

Needless to say, you can have multiple branching factors, for example you can define 2–3 trees over integer values as

`> data Int23T = ILeaf0 `

> | INode2 Int Int23T Int23T

> | INode3 Int Int23T Int23T Int23T

> deriving (Show)

An example value of type Int23T would be

`> i23t = INode3 0 t t t`

> where t = INode2 1 ILeaf0 ILeaf0

which looks like

and has the type

`ghci> :type i23t `

i23t :: Int23T

We could go on and define versions of `IntList`

that stored, say, `Char`

and `Double`

values instead.

`> data CharList = CEmpty `

> | COneAndMore Char CharList

> deriving (Show)

>

> data DoubleList = DEmpty

> | DOneAndMore Char CharList

> deriving (Show)

and we could go on and on and do the same for trees and 2–3 trees and so on. But that would be truly lame, because we would (mostly) repeating ourselves. This looks like a job for abstraction!

The song remains the same: as when finding abstract *computation* patterns, we can find abstract *data* patterns by identifying the bits that are different and turning them into parameters. Here, the bit that is different is the underlying *base* data stored in each box of the structure. So, we will turn that base datatype into a *type parameter* that is passed as input to the type constructor.

`> data List a = Empty `

> | OneAndMore a (List a)

> deriving (Show)

Now, as before, we may define each of the types as simply *instances* of the above parameterized type

`type IntList = List Int`

type CharList = List Char

type DoubleList = List Double

Similarly, it is instructive to look at the types of the constructors

`ghci> :type Empty `

List a

That is, the `Empty`

tag is a value of *any* kind of list, and

`ghci> :type OneAndMore `

a -> List a -> List a

That is, the constructor two arguments: of type `a`

and `List a`

and returns a value of type `List a`

. Here’s how you might construct values of type `List Int`

(note that you can use the binary constructor function in infix form)

`> l1 = OneAndMore 'a' `OneAndMore` Empty`

> l2 = OneAndMore 'b' `OneAndMore` l1

> l3 = OneAndMore 'c' `OneAndMore` l2

Of course, this is pretty much how the “built-in” lists are defined in Haskell, except that `Empty`

is called `[]`

and `OneAndMore`

is called `:`

.

One can use parameterized types to generalize the definition of the other data structures that we saw. For example, trees

`> data Tree a = Leaf a `

> | Node (Tree a) (Tree a)

> deriving (Show)

and *2–3* trees

`> data Tree23 a = Leaf0 `

> | Node2 (Tree23 a) (Tree23 a)

> | Node3 (Tree23 a) (Tree23 a) (Tree23 a)

> deriving (Show)

Of course, there is no reason to limit ourselves to a single type parameter! We might define a type

`data Map k v = Emp `

| Bnd k v (Map k v) (Map k v)

deriving (Show)

What do you think such a type would be useful for?

The `Tree a`

corresponds to trees of values of type `a`

. If `a`

is the *type parameter*, then what is `Tree`

? A *function* that takes a type `a`

as input and returns a type `Tree a`

as output! But wait, if `List`

is a “function” then what is its type? A *kind* is the “type” of a type.

`ghci> :kind Int`

Int :: *

ghci> :kind Char

Char :: *

ghci> :kind Bool

Bool :: *

Thus, `List`

is a function from any “type” to any other "type, and so

`ghci> :kind List`

List :: * -> *

We will not dwell too much on this now. As you might imagine, they allow for all sorts of abstractions over how to construct data. See this for more information about kinds.

We have (and will continue) to see that trees are crucial pillar upon which many data structures are built in Haskell. The workhorse, lists, are infact a kind of tree with exactly one child. Lets write some functions over trees.

First, here is a function that computes the *height* of a tree.

`height :: Tree a -> Int`

height (Leaf _) = 0

height (Node l r) = 1 + max (height l) (height l)

The readability of the above, makes an English description redundant! Good old recursion. You have a base case for the `Leaf`

pattern (namely `0`

) and a recursive or inductive case for the `Node`

pattern (namely the larger of the recursively computed heights of the left and right subtrees.) Lets give it a whirl

`> st1 = Node (Leaf "cat") (Leaf "doggerel") `

> st2 = Node (Leaf "piglet") (Leaf "hippopotamus")

> st3 = Node st1 st2

`ghci> height st1`

1

ghci> height st3

2

How do we compute the *number* of leaf elements in the tree?

`size :: Tree a -> Int`

size (Leaf _) = 1

size (Node l r) = (size l) + (size l)

How about a function that gathers all the elements that occur as leaves of the tree:

`toList :: Tree a -> [a] `

toList (Leaf x) = [x]

toList (Node l r) = (toList l) ++ (toList r)

Did you spot the pattern? What are the different bits? The base value and the operation? So we can write a tree folding routine as

`treeFold op b (Leaf x) = b`

treeFold op b (Node l r) = (treeFold op b l)

`op`

(treeFold op b r)

Does that work? Well, we can easily check that

`size = treeFold (+) 1`

height = treeFold max 0

But what about `toList`

? Urgh. Does. Not. Work. We painted ourselves into a corner. For `size`

and `height`

the base value is a constant, but for `toList`

the base value *depends on* the actual datum at the leaf! Thus, we need *two* operations: one to combine the results of the left and right subtrees, and another to give us the value at a leaf. In other words, the base `b`

needs to be a *function* that takes as input the leaf value and returns as output the result of folding over the leaf.

`> treeFold op b (Leaf x) = b x`

> treeFold op b (Node l r) = (treeFold op b l)

> `op`

> (treeFold op b r)

Now we can write the first `height`

and `size`

as

`> height = treeFold max (const 0)`

> size = treeFold (+) (const 1)

where `const 0`

and `const 1`

are the respective base functions that ignore the leaf value and just always return `0`

and `1`

respectively. Can you guess the definition of `const`

?

What about the problematic `toList`

? Easy enough

`> toList = treeFold (++) (\x -> [x]) `

What about the equivalent of `map`

for `Tree`

s ? We could write the recursive definition

`treeMap :: (a -> b) -> Tree a -> Tree b`

treeMap f (Leaf x) = Leaf (f x)

treeMap f (Node l r) = Node (treeMap f l) (treeMap f r)

but recursion is **HARD TO READ** do we really have to use it ?

**DO IN CLASS**

Sweet! see how we just used the constructors as functions! Lets take it out for a spin.

`ghci> st1`

Node (Leaf "cat") (Leaf "doggerel")

ghci> treeMap length st1

Node (Leaf 3) (Leaf 8)

ghci> st2

Node (Leaf "piglet") (Leaf "hippopotamus")

ghci> treeMap reverse st2

Node (Leaf "telgip") (Leaf "sumatopoppih")