# Spotting Patterns In The “Real” World

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.

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

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

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

## Exercise

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`.

# Recursive Types

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)]``````

## Quiz

What is the type of `Rectangle` ?

1. `Shape`
2. `Double`
3. `Double -> Double -> Shape`
4. `(Double, Double) -> Shape`
5. `[(Double, Double)] -> Shape`

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'``````

``````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

# Parameterized Types

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)``````

Empty :: List a OneAndMore :: a -> List a -> List a

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)``````

## Kinds

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 :: * -> *``````

## Quiz

What is the kind of `->`? That, is what does GHCi say if we type

``ghci> :kind (->) ``
1. `*`
2. `* -> *`
3. `* -> * -> *`

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.

# Computing Over Trees

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?

``````ht (Leaf _)     = 0
ht (Node l r)   = 1 + max (ht l) (ht r)

size            :: Tree a -> Int
size (Leaf _)   = 1
size (Node l r) = (size l) + (size l)

foo :: (a -> b) -> (b -> b-> b) -> Tree a -> b
foo b op (Leaf x)   = b x
foo b op (Node l r) = (foo b op l) `op` (foo b op r)

bar               :: (b -> b-> b) -> Tree b -> b
bar op (Leaf x)   = x
bar op (Node l r) = (bar op l) `op` (bar op r)

ht   = foo (\_ -> 0)    (\xl xr -> 1 + max xl xr)
sz   = foo (\_ -> 1)    (+)
elts = foo (\x -> [x])  (++)

elts :: Tree a -> [a]
elts = foo [] (:)
elts = foo [] (++)``````

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)``````

## Computation Pattern: Tree Fold

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

## Quiz

What does `treeFold (+) 1 t` return?

1. `0`
2. the largest element in the tree
3. the height of the tree
4. the number-of-leaves of the tree
5. type error

## Quiz

What does `treeFold max 0 t` return?

1. `0`
2. the largest element in the tree
3. the height of the tree
4. the number-of-leaves of the tree
5. type error

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 `size` as

``> size   = treeFold (+) (const 1)``

How would you write `height` ?

``> height = treeFold undefined undefined``

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` ?

## Quiz

Which of the following implements `toList :: Tree a -> [a]`

1. `toList = treeFold (:) []`
2. `toList = treeFold (++) []`
3. `toList = treeFold (:) (\x -> [x])`
4. `toList = treeFold (++) (\x -> [x])`
5. none of the above.

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")``````