Making our Own Typeclasses

Algebraic Data types intro

Let’s see how Bool is implemented.

data Bool = False | True

a bool is either False or True.

Here’s how Int is defined:

data Int = -214783648 | ... | -1 | 0 | 1 | ... | 214783647

Let’s make our own type to represent a shape:

data Shape = Circle Float Float Float | Rectangle Float Float Float Float

A Shape can either be a Circle that takes three Float values, or a Rectangle, which takes four Float values.

ghci> :t Circle
Circle :: Float -> Float -> Float -> Shape
ghci> :t Rectangle
Rectangle :: Float -> Float -> Float -> Float -> Shape

Let’s make a function that takes a Shape and returns its area.

surface :: Shape -> Float
surface (Circle _ _ r) = pi * r ^ 2
surface (Rectangle x1 y1 x2 y2) = (abs $ x2 - x2) * (abs $ y2 - y1)

Here’s how that works:

ghci> surface $ Circle 10 20 10
314.15927
ghci> surface $ Rectangle 0 0 100 100
10000.0

We can’t print out the circle, until we have it derive show:

data Shape = Circle Float Float Float | Rectangle Float Float Float Float deriving (Show)

Now we can print out the Circle:

ghci> Circle 10 20 5
Circle 10.0 20.0 5.0
ghci> Rectangle 50 230 60 90
Rectangle 50.0 230.0 60.0 90.0

Value constructors are functions, so we can map them and partially apply them:

ghci> map (Circle 10 20) [4,5,6,6]
[Circle 10.0 20.0 4.0,Circle 10.0 20.0 5.0,Circle 10.0 20.0 6.0,Circle 10.0 20.0 6.0]

Let’s make an intermediate Point to simplify our definition:

data Point = Point Float Float deriving (Show)
data Shape = Circle Point Float | Rectangle Point Point deriving (Show)

We can simplify our surface definition now:

surface :: Shape -> Float
surface (Circle _ r) = pi * r ^ 2
surface (Rectangle (Point x1 y1) (Point x2 y2)) = (abs $ x2 - x1) * (abs $ y2 - y1)

Now this works fine.

ghci> surface (Rectangle (Point 0 0) (Point 100 100))
10000.0
ghci> surface (Circle (Point 0 0) 24)
1809.5574

We can add a nudge function to move it around as well:

nudge :: Shape -> Float -> Float -> Shape
nudge (Circle (Point x y) r) a b = Circle (Point (x+a) (y+b)) r
nudge (Rectangle (Point x1 y1) (Point x2 y2)) a b = Rectangle (Point (x1+a) (y1+b)) (Point (x2+a) (y2+b))

ghci> nudge (Circle (Point 34 34) 10) 5 10
Circle (Point 39.0 44.0) 10.0

How about some base constructors?

baseCircle :: Float -> Shape
baseCircle r = Circle (Point 0 0) r

baseRect :: Float -> Float -> Shape
baseRect width height = Rectangle (Point 0 0) (Point width height)

And their use:

ghci> nudge (baseRect 40 100) 60 23
Rectangle (Point 60.0 23.0) (Point 100.0 123.0)

How would we export this?

module Shapes
( Point(..)
, Shape(..)
, surface
, nudge
, baseCircle
, baseRect
) where

Record Syntax

Here’s a data type that describes a person: Their: first name last name age height phone number favorite ice cream flavor

data Person = Person String String Int Float String String deriving (Show)

It’s very cumbersome to use this without labeling them.

ghci> let guy = Person "Buddy" "Finklestein" 43 184.2 "526-2928" "Chocolate"
ghci> guy
Person "Buddy" "Finklestein" 43 184.2 "526-2928" "Chocolate"

Here’s how we create value constructors for everything:

firstName :: Person -> String
firstName (Person firstname _ _ _ _ _) = firstname

lastName :: Person -> String
lastName (Person _ lastname _ _ _ _) = lastname

age :: Person -> Int
age (Person _ _ age _ _ _) = age

height :: Person -> Float
height (Person _ _ _ height _ _) = height

phoneNumber :: Person -> String
phoneNumber (Person _ _ _ _ number _) = number

flavor :: Person -> String
flavor (Person _ _ _ _ _ flavor) = flavor

This is a bit painful and difficult to remember, so Haskell provided us with record syntax:

data Person = Person {
  firstName :: String
, lastName :: String
, age :: Int
, height :: Float
, phoneNumber :: String
, flavor :: String
} deriving (Show)

This also provides us with getters for that person.

So we can do this just like the previous example:

ghci> let guy = Person "Buddy" "Finklestein" 43 184.2 "526-2928" "Chocolate"ghci> firstName guy
"Buddy"

Record syntax is generally preferred when the order of values is vague, and it’s not obvious which field is which.

Type parameters

A value constructor can take a value (or set of values) of some types and return a value of a different type.

The Maybe type is defined like this:

data Maybe a = Nothing | Just a

If this type is Null, it returns Left, whereas if it holds something, it returns Right which is Just a.

Since Maybe is generic, it works with any type.

Playing around with that:

ghci> Just "Haha"
Just "Haha"
ghci> Just 84
Just 84
ghci> :t Just "Haha"
Just "Haha" :: Maybe [Char]
ghci> :t Just 84
Just 84 :: (Num t) => Maybe t
ghci> :t Nothing
Nothing :: Maybe a
ghci> Just 10 :: Maybe Double
Just 10.0

This is useful for some type definitions, like Map.

data (Ord k) => Map k v = ...

We can require that the Ord typeclass has to be satisfied for a key, so the Map can be ordered.

Let’s create a Vector 3D vector type and add operations for it. In the type, don’t add the constraint, because we have to for the functions.

data Vector a = Vector a a a deriving (Show)

vplus :: (Num t) => Vector t -> Vector t -> Vector t
(Vector i j k) `vplus` (Vector l m n) = Vector (i+l) (j+m) (k+n)

vectMult :: (Num t) => Vector t -> t -> Vector t
(Vector i j k) `vectMult` m = Vector (i*m) (j*m) (k*m)

scalarMult :: (Num t) => Vector t -> Vector t -> t
(Vector i j k) `scalarMult` (Vector l m n) = i*l + j*m + k*n

Derived Instances

Typeclasses are more like interfaces with constraints that can be automatically derived.

data Person = Person {
firstName :: String
, lastName :: String
, age :: Int
} deriving (Show, Read, Eq)

We’ll create a person type and then derive Show, Read and Eq for it.

ghci> let mikeD = Person {firstName = "Michael", lastName = "Diamond", age = 43}
ghci> let adRock = Person {firstName = "Adam", lastName = "Horovitz", age = 41}
ghci> let mca = Person {firstName = "Adam", lastName = "Yauch", age = 44}
ghci> mca == adRock
False
ghci> mikeD == adRock
False
ghci> mikeD == mikeD
True
ghci> mikeD == Person {firstName = "Michael", lastName = "Diamond", age = 43}
True

Show is like toString, as it creates a string representation of our type, whereas read turns a string representation of our type into the type.

ghci> read "Person {firstName ="Michael", lastName ="Diamond", age = 43}" :: Person
Person {firstName = "Michael", lastName = "Diamond", age = 43}

Ord is another typeclass that allows for creating an order.

If we make our Bool typeclass like this:

data Bool = False | True deriving (Ord)

We can compare them:

ghci> True `compare` False
GT
ghci> True > False
True
ghci> True < False
False

Likewise, Maybe derives Ord as well.

data Maybe a = Nothing | Just a deriving (Ord)

ghci> Nothing < Just 100
True
ghci> Nothing > Just (-49999)
False
ghci> Just 3 `compare` Just 2
GT
ghci> Just 100 > Just 50
True

We can use Algebraic Data Types to make enumerations of the Enum (a type with no value constructor), and Bounded, which has a lowest possible value and a highest possible value.

data Day = Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday deriving (Eq, Ord, Show, Read, Bounded, Enum)

We can use Show and Read:

ghci> Wednesday
Wednesday
ghci> show Wednesday
"Wednesday"
ghci> read "Saturday" :: Day
Saturday

We can use Eq and Ord:

ghci> Saturday == Sunday
False
ghci> Saturday == Saturday
True
ghci> Saturday > Friday
True
ghci> Monday `compare` Wednesday
LT

We can use Bounded:

ghci> minBound :: Day
Monday
ghci> maxBound :: Day
Sunday

We can use succ and Pred and make ranges:

ghci> succ Monday
Tuesday
ghci> pred Saturday
Friday
ghci> [Thursday .. Sunday]
[Thursday,Friday,Saturday,Sunday]
ghci> [minBound .. maxBound] :: [Day]
[Monday,Tuesday,Wednesday,Thursday,Friday,Saturday,Sunday]

Type Synonyms

Remember that the String type means [Char].

In Haskell that could be represented like this:

type String = [Char]

Likewise, we could simplify our Data.Map module’s types:

phoneBook :: [(String,String)]
phoneBook = [
   (
  ,(
  ,(
  ,(
  ,(
  ,(
  ]

We know that this is a mapping of String -> String, but we could make a type synonym and make that easier to understand.

type PhoneNumber = String
type Name = String
type PhoneBook = [(Name,PhoneNumber)]

Type synonyms can also be parameterized:

type AssocList k v = [(k,v)]

Let’s talk about the Either type:

data Either a b = Left a | Right b deriving (Eq, Ord, Read, Show)

We can pattern match Left and Right based on different stuff and see which one of them it was:

ghci> Right 20
Right 20
ghci> Left "w00t"
Left "w00t"
ghci> :t Right 'a'
Right 'a' :: Either a Char
ghci> :t Left True
Left True :: Either Bool b

Recursive Data Structures

Let’s make a List data type with Algebraic Data Types

data List a = Empty | Cons a (List a) deriving (Show, Read, Eq, Ord)

Cons is the : operator. Therefore, these are the same:

ghci> 4 `Cons` (5 `Cons` Empty)
[4,5]
ghci> 4: (5 : [])
[4,5]

We can also define how high the precedence is of our operator :-:.

infixr 5 :-:
data List a = Empty | a :-: (List a) deriving (Show, Read, Eq, Ord)

Let’s define our own ++ operator:

infixr 5  .++
(.++) :: List a -> List a -> List a
Empty .++ ys = ys
(x :-: xs) .++ ys = x :-: (xs .++ ys)

Let’s make a binary search tree:

data Tree a = EmptyTree | Node a (Tree a) (Tree a) deriving (Show, Read, Eq)

Here are two functions: One makes a singleton tree, and another one makes a function to insert an element into a tree.

singleton :: a -> Tree a
singleton x = Node x EmptyTree EmptyTree

treeInsert :: (Ord a) => a -> Tree a -> Tree a
treeInsert x EmptyTree = singleton x
treeInsert x (Node a left right)
  | x == a = Node x left right
  | x < a  = Node a (treeInsert x left) right
  | x > a  = Node a left (treeInsert x right)

Here’s a function that checks if an Element is in a tree:

treeElem :: (Ord a) => a -> Tree a -> Bool
treeElem x EmptyTree = False
treeElem x (Node a left right)
  | x == a = True
  | x < a = treeElem x left
  | x > a = TreeElem x right

And let’s start inserting values into the tree:

ghci> let nums = [8,6,4,1,7,3,5]
ghci> let numsTree = foldr treeInsert EmptyTree nums
ghci> numsTree
Node 5 (Node 3 (Node 1 EmptyTree EmptyTree) (Node 4 EmptyTree EmptyTree)) (Node 7 (Node 6 EmptyTree EmptyTree) (Node 8 EmptyTree EmptyTree))

Typeclasses 102

Let’s see how the Eq typeclass is implemented:

class Eq a where
  (==) :: a -> a -> Bool
  (/=) :: a -> a -> Bool
  x == y = not (x /= y)
  x /= y = not (x == y)

A type conforms to the Eq typeclass if it has a definition for == and /= and doesn’t have the same implementation for == and /=.

Let’s write a Traffic Light class:

data TrafficLight = Red | Yellow | Green

Normally we could derive it automatically, but let’s write it out:

instance Eq TrafficLight where
  Red == Red = True
  Green == Green = True
  Yellow == Yellow = True
  _ == _ = False

Because Eq is defined in terms of each other, we can stop right here. Eq is fulfilled, because it can figure out the definition for /=.

Here’s how show would be written by hand:

instance Show TrafficLight where
    show Red = "Red light"
    show Yellow = "Yellow light"
    show Green = "Green light"

Let’s override Eq for Maybe.

instance (Eq m) => Eq (Maybe m) where
  Just x == Just y = x == y
  Nothing == Nothing = True
  _ == _ = False

Functors

The functor typeclass allows you to use a function on every instance of something:

fmap is the defined like this:

class Functor f where
  fmap :: (a -> b) -> f a -> f b

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