\[\newcommand{\andalso}{\quad\quad} \newcommand{\infabbrev}[2]{\infax{#1 \quad\eqdef\quad #2}} \newcommand{\infrule}[2]{\displaystyle \dfrac{#1}{#2}} \newcommand{\ar}{\rightarrow} \newcommand{\Int}{\mathtt{Int}} \newcommand{\Bool}{\mathtt{Bool}} \newcommand{\becomes}{\Downarrow} \newcommand{\trule}[1]{(\textbf{#1})} \newcommand{\FV}[1]{\mathtt{fv}(#1)} \newcommand{\FTV}[1]{\mathtt{ftv}(#1)} \newcommand{\BV}[1]{\mathtt{bv}(#1)} \newcommand{\compiles}[1]{\text{C}\llbracket{#1}\rrbracket} \newcommand{\exec}[1]{\text{E}\llbracket{#1}\rrbracket} \renewcommand{\t}[1]{\mathtt{#1}} \newcommand{\ite}[3]{\text{if }#1\text{ then }#2\text{ else }#3} \]

Haskell Basics

Let us now survey a few of the core concepts that will be used throughout the text. This will be a very fast and informal discussion. If you are familiar with all of these concepts then it is very likely you will be able to read the entirety of this tutorial and focus on the subject domain and not the supporting code. The domain material itself should largely be accessible to an ambitious high school student or undergraduate; and requires nothing more than a general knowledge of functional programming.

Functions

Functions are the primary building block of all of Haskell logic.

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

In Haskell all functions are pure. The only thing a function may do is return a value.

All functions in Haskell are curried. For example, when a function of three arguments receives less than three arguments, it yields a partially applied function, which, when given additional arguments, yields yet another function or the resulting value if all the arguments were supplied.

g :: Int -> Int -> Int -> Int
g x y z = x + y + z

h :: Int -> Int
h = g 2 3

Haskell supports higher-order functions, i.e., functions which take functions as arguments and yield other functions. For example the compose function takes two functions as arguments f and g and returns the composite function of applying f then g.

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

Datatypes

Constructors for datatypes come in two flavors: sum types and product types.

A sum type consists of multiple options of type constructors under the same type. The two cases can be used at all locations the type is specified, and are discriminated using pattern matching.

data Sum = A Int | B Bool

A product type combines multiple fields into the same type.

data Prod = Prod Int Bool

Records are a special product type that, in addition to generating code for the constructors, generates a special set of functions known as selectors which extract the values of a specific field from the record.

data Prod = Prod { a :: Int , b :: Bool }

-- a :: Prod -> Int
-- b :: Prod -> Bool

Sums and products can be combined.

data T1
  = A Int Int
  | B Bool Bool

The fields of a datatype may be parameterized, in which case the type depends on the specific types the fields are instantiated with.

data Maybe a = Nothing | Just a

Values

A list is a homogeneous, inductively defined sum type of linked cells parameterized over the type of its values.

data List a = Nil | Cons a (List a)
a = [1,2,3]
a = Cons 1 (Cons 2 (Cons 3 Nil))

List have special value-level syntax:

(:) = Cons
[]  = Nil
(1 : (2 : (3 : []))) = [1,2,3]

A tuple is a heterogeneous product type parameterized over the types of its two values.

Tuples also have special value-level syntax.

data Pair a b = Pair a b
a = (1,2)
a = Pair 1 2
(,) = Pair

Pattern matching

Pattern matching allows us to discriminate on the constructors of a datatype, mapping separate cases to separate code paths and binding variables for each of the fields of the datatype.

data Maybe a = Nothing | Just a

maybe :: b -> (a -> b) -> Maybe a -> b
maybe n f Nothing  = n
maybe n f (Just a) = f a

Top-level pattern matches can always be written identically as case statements.

maybe :: b -> (a -> b) -> Maybe a -> b
maybe n f x = case x of
  Nothing -> n
  Just a  -> f a

Wildcards can be placed for patterns where the resulting value is not used.

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

Subexpression in the pattern can be explicitly bound to variables scoped on the right hand side of the pattern match.

f :: Maybe (Maybe a) -> Maybe a
f (Just x @ (Just _)) = x

List and tuples have special pattern syntax.

length :: [a] -> Int
length []     = 0
length (x:xs) = 1 + (length xs)
fst :: (a, b) -> a
fst (a,b) = a

Patterns may be guarded by predicates (functions which yield a boolean). Guards only allow the execution of a branch if the corresponding predicate yields True.

filter :: (a -> Bool) -> [a] -> [a]
filter pred []     = []
filter pred (x:xs)
  | pred x         = x : filter pred xs
  | otherwise      =     filter pred xs

Recursion

In Haskell all iteration over data structures is performed by recursion. Entering a function in Haskell does not create a new stack frame, the logic of the function is simply entered with the arguments on the stack and yields result to the register. In the case where a function returns an invocation of itself invoked in the tail position the resulting logic is compiled identically to while loops in other languages, via a jmp instruction instead of a call.

sum :: [Int] -> Int
sum ys = go ys 0
  where
    go (x:xs) i = go xs (i+x)
    go [] i = i

Functions can be defined to recurse mutually on each other.

even 0 = True
even n = odd (n-1)

odd 0 = False
odd n = even (n-1)

Laziness

A Haskell program can be thought of as being equivalent to a large directed graph. Each edge represents the use of a value, and each node is the source of a value. A node can be:

  • A thunk, i.e., the application of a function to values that have not been evaluated yet
  • A thunk that is currently being evaluated, which may induce the evaluation of other thunks in the process
  • An expression in weak head normal form, which is only evaluated to the outermost constructor or lambda abstraction

The runtime has the task of determining which thunks are to be evaluated by the order in which they are connected to the main function node. This is the essence of all evaluation in Haskell and is called graph reduction.

Self-referential functions are allowed in Haskell. For example, the following functions generate infinite lists of values. However, they are only evaluated up to the depth that is necessary.

-- Infinite stream of 1's
ones = 1 : ones

-- Infinite count from n
numsFrom n = n : numsFrom (n+1)

-- Infinite stream of integer squares
squares = map (^2) (numsfrom 0)

The function take consumes an infinite stream and only evaluates the values that are needed for the computation.

take :: Int -> [a] -> [a]
take n _  | n <= 0 =  []
take n []          =  []
take n (x:xs)      =  x : take (n-1) xs
take 5 squares
-- [0,1,4,9,16]

This also admits diverging terms (called bottoms), which have no normal form. Under lazy evaluation, these values can be threaded around and will never diverge unless actually forced.

bot = bot

So, for instance, the following expression does not diverge since the second argument is not used in the body of const.

const 42 bot

The two bottom terms we will use frequently are used to write the scaffolding for incomplete programs.

error :: String -> a
undefined :: a

Higher-Kinded Types

The "type of types" in Haskell is the language of kinds. Kinds are either an arrow (k -> k') or a star (*).

The kind of Int is *, while the kind of Maybe is * -> *. Haskell supports higher-kinded types, which are types that take other types and construct a new type. A type constructor in Haskell always has a kind which terminates in a *.

-- T1 :: (* -> *) -> * -> *
data T1 f a = T1 (f a)

The three special types (,), (->), [] have special type-level syntactic sugar:

(,) Int Int   =  (Int, Int)
(->) Int Int  =  Int -> Int
[] Int        =  [Int]

Typeclasses

A typeclass is a collection of functions which conform to a given interface. An implementation of an interface is called an instance. Typeclasses are effectively syntactic sugar for records of functions and nested records (called dictionaries) of functions parameterized over the instance type. These dictionaries are implicitly threaded throughout the program whenever an overloaded identifier is used. When a typeclass is used over a concrete type, the implementation is simply spliced in at the call site. When a typeclass is used over a polymorphic type, an implicit dictionary parameter is added to the function so that the implementation of the necessary functionality is passed with the polymorphic value.

Typeclasses are "open" and additional instances can always be added, but the defining feature of a typeclass is that the instance search always converges to a single type to make the process of resolving overloaded identifiers globally unambiguous.

For instance, the Functor typeclass allows us to "map" a function generically over any type of kind (* -> *) and apply it to its internal structure.

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

instance Functor [] where
  fmap f []     = []
  fmap f (x:xs) = f x : fmap f xs

instance Functor ((,) a) where
  fmap f (a,b) = (a, f b)

Operators

In Haskell, infix operators are simply functions, and quite often they are used in place of alphanumerical names when the functions involved combine in common ways and are subject to algebraic laws.

infixl 6 +
infixl 6 -
infixl 7 /
infixl 7 *

infixr 5 ++
infixr 9 .

Operators can be written in section form:

(x+) =  \y -> x+y
(+y) =  \x -> x+y
(+)  =  \x y -> x+y

Any binary function can be written in infix form by surrounding the name in backticks.

(+1) `fmap` [1,2,3] -- [2,3,4]

Monads

A monad is a typeclass with two functions: bind and return.

class Monad m where
  bind   :: m a -> (a -> m b) -> m b
  return :: a -> m a

The bind function is usually written as an infix operator.

infixl 1 >>=

class Monad m where
  (>>=)  :: m a -> (a -> m b) -> m b
  return :: a -> m a

This defines the structure, but the monad itself also requires three laws that all monad instances must satisfy.

Law 1

return a >>= f = f a

Law 2

m >>= return = m

Law 3

(m >>= f) >>= g = m >>= (\x -> f x >>= g)

Haskell has a level of syntactic sugar for monads known as do-notation. In this form, binds are written sequentially in block form which extract the variable from the binder.

do { a <- f ; m } = f >>= \a -> do { m }
do { f ; m } = f >> do { m }
do { m } = m

So, for example, the following are equivalent:

do
  a <- f
  b <- g
  c <- h
  return (a, b, c)
f >>= \a ->
  g >>= \b ->
    h >>= \c ->
      return (a, b, c)

Applicatives

Applicatives allow sequencing parts of some contextual computation, but do not bind variables therein. Strictly speaking, applicatives are less expressive than monads.

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

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

Applicatives satisfy the following laws:

pure id <*> v = v                             -- Identity
pure f <*> pure x = pure (f x)                -- Homomorphism
u <*> pure y = pure ($ y) <*> u               -- Interchange
u <*> (v <*> w) = pure (.) <*> u <*> v <*> w  -- Composition

For example:

example1 :: Maybe Integer
example1 = (+) <$> m1 <*> m2
  where
    m1 = Just 3
    m2 = Nothing

Instances of the Applicative typeclass also have available the functions *> and <*. These functions sequence applicative actions while discarding the value of one of the arguments. The operator *> discards the left argument, while <* discards the right. For example, in a monadic parser combinator library, the *> would discard the value of the first argument but return the value of the second.

Monoids

Monoids provide an interface for structures which have an associative operation (mappend, there is also the synonym <>) and a neutral (also: unit or zero) element (mempty) for that operation.

class Monoid a where
  mempty :: a
  mappend :: a -> a -> a
  mconcat :: [a] -> a

The canonical example is the list type with concatenation as the operation and the empty list as zero.

import Data.Monoid

a :: [Integer]
a = [1,2,3] <> [4,5,6]

b :: [Integer]
b = ([1,2,3] <> mempty) <> (mempty <> [4,5,6])

Deriving

Instances for typeclasses like Read, Show, Eq and Ord can be derived automatically by the Haskell compiler.

data PlatonicSolid
  = Tetrahedron
  | Cube
  | Octahedron
  | Dodecahedron
  | Icosahedron
  deriving (Show, Eq, Ord, Read)
example = show Icosahedron
example = read "Tetrahedron"
example = Cube == Octahedron
example = sort [Cube, Dodecahedron]

IO

A value of type IO a is a computation which, when performed, does some I/O before returning a value of type a. The notable feature of Haskell is that IO is still functionally pure; a value of type IO a is simply a value which stands for a computation which, when invoked, will perform IO. There is no way to peek into its contents without running it.

For instance, the following function does not print the numbers 1 to 5 to the screen. Instead, it builds a list of IO computations:

fmap print [1..5] :: [IO ()]

We can then manipulate them as an ordinary list of values:

reverse (fmap print [1..5]) :: [IO ()]

We can then build a composite computation of each of the IO actions in the list using sequence_, which will evaluate the actions from left to right. The resulting IO computation can be evaluated in main (or the GHCi repl, which effectively is embedded inside of IO).

>> sequence_ (fmap print [1..5]) :: IO ()
1
2
3
4
5

>> sequence_ (reverse (fmap print [1..5])) :: IO ()
5
4
3
2
1

The IO monad is wired into the runtime with compiler support. It is a special case and most monads in Haskell have nothing to do with effects in this sense.

putStrLn :: String -> IO ()
print    :: Show a => a -> IO ()

The type of main is always IO ().

main :: IO ()
main = do
  putStrLn "Enter a number greater than 3: "
  x <- readLn
  print (x > 3)

The essence of monadic IO in Haskell is that effects are reified as first class values in the language and reflected in the type system. This is one of foundational ideas of Haskell, although it is not unique to Haskell.

Monad Transformers

Monads can be combined together to form composite monads. Each of the composite monads consists of layers of different monad functionality. For example, we can combine an error-reporting monad with a state monad to encapsulate a certain set of computations that need both functionalities. The use of monad transformers, while not always necessary, is often one of the primary ways to structure modern Haskell programs.

class MonadTrans t where
  lift :: Monad m => m a -> t m a

The implementation of monad transformers is comprised of two different complementary libraries, transformers and mtl. The transformers library provides the monad transformer layers and mtl extends this functionality to allow implicit lifting between several layers.

To use transformers, we simply import the Trans variants of each of the layers we want to compose and then wrap them in a newtype.

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

import Control.Monad.Trans
import Control.Monad.Trans.State
import Control.Monad.Trans.Writer

newtype Stack a = Stack { unStack :: StateT Int (WriterT [Int] IO) a }
  deriving (Monad)

foo :: Stack ()
foo = Stack $ do
  put 1                  -- State layer
  lift $ tell [2]        -- Writer layer
  lift $ lift $ print 3  -- IO Layer
  return ()

evalStack :: Stack a -> IO [Int]
evalStack m = execWriterT (evalStateT (unStack m) 0)

As illustrated by the following stack diagram:

Using mtl and GeneralizedNewtypeDeriving, we can produce the same stack but with a simpler forward-facing interface to the transformer stack. Under the hood, mtl is using an extension called FunctionalDependencies to automatically infer which layer of a transformer stack a function belongs to and can then lift into it.

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

import Control.Monad.Trans
import Control.Monad.State
import Control.Monad.Writer

newtype Stack a = Stack { unStack :: StateT Int (WriterT [Int] IO) a }
  deriving (Monad, MonadState Int, MonadWriter [Int], MonadIO)

foo :: Stack ()
foo = do
  put 1             -- State layer
  tell [2]          -- Writer layer
  liftIO $ print 3  -- IO Layer
  return ()

evalStack :: Stack a -> IO [Int]
evalStack m = execWriterT (evalStateT (unStack m) 0)

StateT

The state monad allows functions within a stateful monadic context to access and modify shared state.

put    :: s -> State s ()          -- set the state value
get    :: State s s                -- get the state
gets   :: (s -> a) -> State s a    -- apply a function over the state, and return the result
modify :: (s -> s) -> State s ()   -- set the state, using a modifier function

Evaluation functions often follow the naming convention of using the prefixes run, eval, and exec:

execState :: State s a -> s -> s         -- yield the state
evalState :: State s a -> s -> a         -- yield the return value
runState  :: State s a -> s -> (a, s)    -- yield the state and return value

For example:

import Control.Monad.State

test :: State Int Int
test = do
  put 3
  modify (+1)
  get

main :: IO ()
main = print $ execState test 0

ReaderT

The Reader monad allows a fixed value to be passed around inside the monadic context.

ask   :: Reader r r                            -- get the value
asks  :: (r -> a) -> Reader r a                -- apply a function to the value, and return the result
local :: (r -> r) -> Reader r a -> Reader r a  -- run a monadic action, with the value modified by a function

For example:

import Control.Monad.Reader

data MyContext = MyContext
  { foo :: String
  , bar :: Int
  } deriving (Show)

computation :: Reader MyContext (Maybe String)
computation = do
  n <- asks bar
  x <- asks foo
  if n > 0
    then return (Just x)
    else return Nothing

ex1 :: Maybe String
ex1 = runReader computation $ MyContext "hello" 1

ex2 :: Maybe String
ex2 = runReader computation $ MyContext "haskell" 0

WriterT

The writer monad lets us emit a lazy stream of values from within a monadic context. The primary function tell adds a value to the writer context.

tell :: (Monoid w) => w -> Writer w ()

The monad can be evaluated returning the collected writer context and optionally the returned value.

execWriter :: (Monoid w) => Writer w a -> w
runWriter  :: (Monoid w) => Writer w a -> (a, w)
import Control.Monad.Writer

type MyWriter = Writer [Int] String

example :: MyWriter
example = do
  tell [1..5]
  tell [5..10]
  return "foo"

output :: (String, [Int])
output = runWriter example

ExceptT

The Exception monad allows logic to fail at any point during computation with a user-defined exception. The exception type is the first parameter of the monad type.

throwError :: e -> Except e a
runExcept  :: Except e a -> Either e a

For example:

import Control.Monad.Except

type Err = String

safeDiv :: Int -> Int -> Except Err Int
safeDiv a 0 = throwError "Divide by zero"
safeDiv a b = return (a `div` b)

example :: Either Err Int
example = runExcept $ do
  x <- safeDiv 2 3
  y <- safeDiv 2 0
  return (x + y)

Kleisli Arrows

The additional combinators for monads ((>=>), (<=<)) compose two different monadic actions in sequence. (<=<) is the monadic equivalent of the regular function composition operator (.) and (>=>) is just flip (<=<).

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c

The monad laws can be expressed equivalently in terms of Kleisli composition.

(f >=> g) >=> h   =   f >=> (g >=> h)
return >=> f      =   f
f >=> return      =   f

Text

The usual String type is a singly-linked list of characters, which, although simple, is not efficient in storage or locality. The letters of the string are not stored contiguously in memory and are instead allocated across the heap.

The Text and ByteString libraries provide alternative efficient structures for working with contiguous blocks of text data. ByteString is useful when working with the ASCII character set, while Text provides a text type for use with Unicode.

The OverloadedStrings extension allows us to overload the string type in the frontend language to use any one of the available string representations.

class IsString a where
  fromString :: String -> a

pack :: String -> Text
unpack :: Text -> String

So, for example:

{-# LANGUAGE OverloadedStrings #-}
import qualified Data.Text as T

str :: T.Text
str = "bar"

Cabal & Stack

To set up an existing project with a sandbox, run:

$ cabal sandbox init

This will create the .cabal-sandbox directory, which is the local path GHC will use to look for dependencies when building the project.

To install dependencies from Hackage, run:

$ cabal install --only-dependencies

Finally, configure the library for building:

$ cabal configure

Now we can launch a GHCi shell scoped with the modules from the project in scope:

$ cabal repl

Resources

If any of these concepts are unfamiliar, there are some external resources that will try to explain them. The most thorough is the Stanford course lecture notes.

There are some books as well, but your mileage may vary with these. Much of the material is dated and only covers basic programming and not "programming in the large".