CSE 230 W11 - Higher-Order Programming II



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 having to 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)]

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.

Datatypes are Boxed-and-Tagged Values

Datatypes are Boxed-and-Tagged Values

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.

Recursively Nested Boxes

Recursively Nested 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

Integer 2–3 Tree

Integer 2–3 Tree

and has the type

ghci> :type i23t 
i23t :: Int23T

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)

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?

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 :: * -> *

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?

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)

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

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 Trees ? 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")