Sum Types

• Sum types enumerate all possible inhabitants
• Bool has 2 possibilities, Ordering has 3, …

data Bool = True | False

data Ordering = LT | EQ | GT

data Choice = Definitely | Possibly | NoWay

data Int = … | -1 | 0 | 1 | 2 | …

data Char = … | 'a' | 'b' | …

Product Types

• Product types are like structs, with fields that contain other types
• Choices has 3 * 3 == 9 possible inhabitants
• Can name these fields using record syntax (defines getters automatically)

data CoinFlip = CoinFlip Bool

data Choices = Choices Choice Choice

data Coord = Coord { x :: Double, y :: Double }

Sum of Products

• Possibly has (1 * 2) + 1 possibilities

data Possibly = Certainly Bool
              | Uncertain

data DrawCommand = Point Int Int
                 | Line Int Int Int Int
                 | Rect Int Int Int Int

data IntTree = Node Int IntTree IntTree
             | Leaf

Abstract Data Types

• Type variables are lowercase
• type creates aliases (can help readability)

data List a = Cons a (List a)
            | Nil

data Maybe a = Just a
             | Nothing

type IntList = List Int
type MaybeBool = Maybe Bool
type String = [Char]

Special Type Syntax

• Unit, Tuples, Lists, and Functions have special syntax
• Can also be written in prefix form

type Unit = ()

type ListOfInt = [Int]
type ListOfInt = [] Int

type AddFun = Int -> Int -> Int
type AddFun = Int -> (Int -> Int)
type AddFun = (->) Int ((->) Int Int)

type IntTuple = (Int, Int)
type IntTuple = (,) Int Int

Using Types

-- Terms can be annotated in-line
2 ^ (1 :: Int)

-- Bindings can be annotated
success :: a -> Maybe a
-- Constructors are terms
-- (and product constructors are functions)
success x = Just x

-- Constructors can be pattern matched
-- _ is a wildcard
case success True of
  Just True -> ()
  _         -> ()

:t shows type information

h> :t map
map :: (a -> b) -> [a] -> [b]
h> :t map (+1)
map (+1) :: Num b => [b] -> [b]
h> :t (>>=)
(>>=) :: Monad m => m a -> (a -> m b) -> m b

:i shows typeclass info

h> :i Num
class Num a where
  (+) :: a -> a -> a
  (*) :: a -> a -> a
  (-) :: a -> a -> a
  negate :: a -> a
  abs :: a -> a
  signum :: a -> a
  fromInteger :: Integer -> a
    -- Defined in `GHC.Num'
instance Num Integer -- Defined in `GHC.Num'
instance Num Int -- Defined in `GHC.Num'
instance Num Float -- Defined in `GHC.Float'
instance Num Double -- Defined in `GHC.Float'

:i shows term info

h> :info map
map :: (a -> b) -> [a] -> [b]   
-- Defined in `GHC.Base'
h> :info (>>=)
class Monad m where
  (>>=) :: m a -> (a -> m b) -> m b
  ...
    -- Defined in `GHC.Base'
infixl 1 >>=

:i shows type info

h> :info Int
data Int = ghc-prim:GHC.Types.I#
  ghc-prim:GHC.Prim.Int#
  -- Defined in `ghc-prim:GHC.Types'
instance Bounded Int -- Defined in `GHC.Enum'
instance Enum Int -- Defined in `GHC.Enum'
instance Eq Int -- Defined in `GHC.Classes'
instance Integral Int -- Defined in `GHC.Real'
instance Num Int -- Defined in `GHC.Num'
instance Ord Int -- Defined in `GHC.Classes'
instance Read Int -- Defined in `GHC.Read'
instance Real Int -- Defined in `GHC.Real'
instance Show Int -- Defined in `GHC.Show'

:l load a module

:r to reload

h> :! echo 'hello = print "hello"' > Hello.hs
h> :l Hello
[1 of 1] Compiling Main ( Hello.hs, interpreted )
Ok, modules loaded: Main.
h> hello
"hello"
h> :! echo 'hello = print "HELLO"' > Hello.hs
h> :r
[1 of 1] Compiling Main ( Hello.hs, interpreted )
Ok, modules loaded: Main.
h> hello
"HELLO"

map

map :: (a -> b) -> [a] -> [b]
map _ []     = []
map f (x:xs) = f x : map f xs

foldr

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr k z = go
   where
     go []     = z
     go (y:ys) = y `k` go ys

Pattern Matching

isJust :: Maybe a -> Bool
isJust (Just _) = True
isJust Nothing  = False

Pattern Matching

Haskell only implements linear patterns

-- DOES NOT WORK!
isEqual :: a -> a -> Bool
isEqual a a = True
isEqual _ _ = False

This isn't even possible! Only constructors can be pattern matched. Types have no built-in equality.

`Infix` and (Prefix)

-- Symbolic operators can be used
-- prefix when in (parentheses)
(+) a b

-- Named functions can be used
-- infix when in `backticks`
x `elem` xs

-- infixl, infixr define associativity
-- and precedence (0 lowest, 9 highest)
infixr 5 `append`
a `append` b = a ++ b

Functions & Lambdas

add :: Integer -> Integer -> Integer
add acc x = acc + x

sumFun :: [Integer] -> Integer
sumFun xs = foldl add 0 xs

sumLambda :: [Integer] -> Integer
sumLambda xs = foldl (\acc x -> acc + x) 0 xs

Functions & Lambdas

• Haskell only has functions of one argument
• a -> b -> c is really a -> (b -> c)
• f a b is really (f a) b
• Let's leverage that…

Guards

isNegative :: (Num a) => a -> Bool
isNegative x
  | x < 0     = True
  | otherwise = False

absoluteValue :: (Num a) => a -> Bool
absoluteValue x
  | isNegative x = -x
  | otherwise    = x

Typeclass Syntax

class Equals a where
  isEqual :: a -> a -> Bool

instance Equals Choice where
  isEqual Definitely Definitely = True
  isEqual Possibly   Possibly   = True
  isEqual NoWay      NoWay      = True
  isEqual _          _          = False

instance (Equals a) => Equals [a] where
  isEqual (a:as) (b:bs) = isEqual a b &&
                          isEqual as bs
  isEqual as     bs     = null as && null bs

Typeclass Syntax

{-
class Eq a where
  (==) :: a -> a -> Bool
-}

instance Eq Choice where
  Definitely == Definitely = True
  Possibly   == Possibly   = True
  NoWay      == NoWay      = True
  _          == _          = False

Typeclass Syntax

data Choice = Definitely
            | Possibly
            | NoWay
            deriving (Eq)

Typeclass Syntax

data Choice = Definitely
            | Possibly
            | NoWay
            deriving ( Eq, Ord, Enum, Bounded
                     , Show, Read )

QuickCheck

prop_intIdentity :: Int -> Bool
prop_intIdentity i = i == i

QuickCheck

$ ghci

λ> import Test.QuickCheck
λ> quickCheck (\i -> (i :: Int) == i)
+++ OK, passed 100 tests.

Do syntax (IO)

main :: IO ()
main = do
  secret <- readFile "/etc/passwd"
  writeFile "/tmp/passwd" secret
  return ()

Do syntax

do m
-- desugars to:
m

do a <- m
   return a
-- desugars to:
m >>= \a -> return a

do m
   return ()
-- desugars to:
m >> return ()

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = [ y | x <- xs, y <- f x ]

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = do
  x <- xs
  y <- f x
  return y

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = do
  x <- xs
  f x

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = xs >>= \x -> f x

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = xs >>= f

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap f xs = flip (>>=) f xs

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap = flip (>>=)

Do syntax ([a])

flatMap :: (a -> [b]) -> [a] -> [b]
flatMap = (=<<)

-- WordCount1.hs

main :: IO ()
main = do
  input <- getContents
  let wordCount = length (words input)
  print wordCount

-- WordCount2.hs

main :: IO ()
main =
  getContents >>= \input ->
    let wordCount = length (words input)
    in print wordCount

-- WordCount3.hs

main :: IO ()
main = getContents >>= print . length . words

Common combinators

-- Function composition
(.) :: (b -> c) -> (a -> b) -> a -> c
f . g = \x -> f (g x)

-- Function application (with a lower precedence)
($) :: (a -> b) -> a -> b
f $ x =  f x

{-# RULES
"ByteString specialise break (x==)" forall x.
    break ((==) x) = breakByte x
"ByteString specialise break (==x)" forall x.
    break (==x) = breakByte x
  #-}

GHC RULES

{-# RULES
"ByteString specialise break (x==)" forall x.
    break ((==) x) = breakByte x
"ByteString specialise break (==x)" forall x.
    break (==x) = breakByte x
  #-}

import Data.ByteString.Char8 (ByteString, break)

splitLine :: ByteString -> (ByteString, ByteString)
splitLine = break (=='\n')

GHC RULES

{-# RULES
"ByteString specialise break (x==)" forall x.
    break ((==) x) = breakByte x
"ByteString specialise break (==x)" forall x.
    break (==x) = breakByte x
  #-}

import Data.ByteString.Char8 (ByteString, break)

splitLine :: ByteString -> (ByteString, ByteString)
splitLine = breakByte '\n'

Non-Strict Evaluation

-- [1..] is an infinite list, [1, 2, 3, ...]
print (head (map (*2) [1..]))

if' :: Bool -> a -> a -> a
if' cond a b = case cond of
  True  -> a
  False -> b

(&&) :: Bool -> Bool -> Bool
a && b = case a of
  True  -> b
  False -> False

const :: a -> b -> a
const x = \_ -> x

fib :: [Integer]
fib = 0 : 1 : zipWith (+) fib (tail fib)

cycle :: [a] -> [a]
cycle xs = xs ++ cycle xs

iterate :: (a -> a) -> a -> [a]
iterate f x = x : iterate f (f x)

takeWhile :: (a -> Bool) -> [a] -> [a]
takeWhile _ [] = []
takeWhile p (x:xs)
  | p x       = x : takeWhile p xs
  | otherwise = []

module List where

data List a = Cons a (List a)
            | Nil

instance (Eq a) => Eq (List a) where
  (Cons a as) == (Cons b bs) = a == b && as == bs
  Nil         == Nil         = True
  _           == _           = False

instance Functor List where
  fmap _ Nil         = Nil
  fmap f (Cons x xs) = Cons (f x) (fmap f xs)

{-# LANGUAGE DeriveFunctor #-}

module List where

data List a = Cons a (List a)
            | Nil
            deriving (Eq, Functor)

import Data.List (sort)

newtype Down a = Down { unDown :: a }
                 deriving (Eq)

instance (Ord a) => Ord (Down a) where
  compare (Down a) (Down b) = case compare a b of
    LT -> GT
    EQ -> EQ
    GT -> LT

reverseSort :: Ord a => [a] -> [a]
reverseSort = map unDown . sort . map Down

Monoid

class Monoid a where
  mempty :: a
  mappend :: a -> a -> a

instance Monoid [a] where
  mempty = []
  mappend = (++)

infixr 6 <>
(<>) :: (Monoid a) => a -> a -> a
(<>) = mappend

Functor

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

instance Functor [] where
  fmap = map

instance Functor Maybe where
  fmap f (Just x) = Just (f x)
  fmap _ Nothing  = Nothing

infixl 4 <$>
(<$>) :: Functor f => (a -> b) -> f a -> f b
(<$>) = fmap

Applicative

class (Functor f) => Applicative f where
  pure :: a -> f a
  infixl 4 <*>
  (<*>) :: f (a -> b) -> f a -> f b

instance Applicative [] where
  pure x = [x]
  fs <*> xs = concatMap (\f -> map f xs) fs

instance Applicative Maybe where
  pure = Just
  Just f <*> Just x = Just (f x)
  _      <*> _      = Nothing

Monad

class Monad m where
  return :: a -> m a
  (>>=) :: m a -> (a -> m b) -> m b
  (>>)  :: m a -> m b -> m b
  ma >> mb = ma >>= \_ -> mb

instance Monad [] where
  return = pure
  m >>= f = concatMap f m

instance Monad Maybe where
  return = pure
  Just x  >>= f = f x
  Nothing >>= _ = Nothing

sum :: Num a => [a] -> a
sum []     = 0
sum (x:xs) = x + sum xs

sum :: Num [a] => [a] -> a
sum = go 0
  where
    go acc (x:xs) = go (acc + x) (go xs)
    go acc []     = acc

sum :: Num [a] => [a] -> a
sum = go 0
  where
    go acc _
      | acc `seq` False = undefined
    go acc (x:xs)       = go (acc + x) (go xs)
    go acc []           = acc

{-# LANGUAGE BangPatterns #-}

sum :: Num [a] => [a] -> a
sum = go 0
  where
    go !acc (x:xs) = go (acc + x) (go xs)
    go  acc []     = acc