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Creating types

Products

You can make your own types like this:

data Square = Sq Int Int -- (1)!
  1. The choice of the names Square and Sq are both arbitrary. Both must be capitalized.
data Square where -- (1)!
    Sq :: Int -> Int -> Square
  1. This requires the GADT extension, which is included with GHC2021.

This creates a new type Square; values of type Square look like Sq i j, where i and j are Ints.

Sq is referred to as a data constructor, and is a Haskell function, with type shown explicitly in the "modern" version above.

repl example
> :t Sq 5 4
Sq 5 4 :: Square
> :t Sq 5 True
"Couldn't match expected type ‘Int’ with actual type ‘Bool’"
> :t Sq 
Sq :: Int -> (Int -> Square) -- (1)!
> :t Sq 3
Sq 3 :: Int -> Square -- (2)!

  1. Actually, the repl will drop the brackets, and show: Int -> Int -> Square.

  2. If this is unclear, see here for more info.

Tip

The type Square contains the same information as the product type (Int, Int). These types are different, in the sense that code which expects one will not work with the other, but it is easy to write loss-less functions between them:

fromSq :: Square -> (Int, Int)
fromSq (Sq i j) = (i, j)

toSq :: (Int, Int) -> Square
toSq (i, j) = Sq i j

Note

The number of types following Sq can be 0 or more. For example:

repl example
> data SquareAlt = SqAlt Int
> data SquareAlt2 = SqAlt2
> :t SqAlt

If the number of types is 1, you will see a suggestion to replace data with newtype. See more about newtype here

Records

You can also name entries:

data Entity = Sq {row :: Int, col :: Int}

data Entity where
  Sq :: {row :: Int, col :: Int} -> Entity

row and col are now accessing functions:

repl example
> entity = Sq 4 5
> :t entity
entity :: Entity
> row entity
4
> col entity
5
> :t row
row :: Entity -> Int
> :t col
col :: Entity -> Int

Sums

data Entity = Sq Int Int | Player Bool -- (1)!
  1. Entity is a type, but Sq and Player are values belonging to that type

The vertical bar | indicates that an Entity is either made with Sq or with Piece:

data Entity where
    Sq :: Int -> Int -> Entity -- (1)!
    Player :: Bool -> Entity
  1. This requires the GADT extension, which is included with GHC2021.

The newline indicates that an Entity is either made with Sq or with Player:

repl example
> Sq 4 6
Sq 4 6 :: Entity
> :t Player False
Player False :: Entity
> :t Player 
Player :: Bool -> Entity
Tip

The type Entity contains the same information as the type Either (Int, Int) Bool, and one can write loss-less functions between them:

fromEntity :: Entity -> Either (Int, Int) Bool
fromEntity (Sq i j) = Left (i, j)
fromEntity (Player bool) = Right bool

toEntity :: Either (Int, Int) Bool -> Entity
toEntity (Left (i ,j)) = Sq i j
toEntity (Right bool) = Player bool

Note

You can combine products and sums, using your own types:

data ChessPiece = Piece PieceType Color | SquareType Square -- (1)!
data Color = Black | White
data PieceType = Bishop | Rook
data Square = Sq Int Int
  1. Note that Haskell is unconcerned by the order of definitions: the definition of Color comes after its use in the definition of ChessPiece.

Parameterized types

One can also create types which take another type as a parameter:

data Piece c = Bishop c | Knight c | King c

This creates Piece Bool, Piece Int, and so on:

repl example
> data Piece c = Bishop c | Knight c | King c
> :t Knight True
Knight True :: Piece Bool
let four = 4 :: Int
> :t King four
King four :: Piece Int
> :t King (King True)
King (King True) :: Piece (Piece Bool)
Note

One can think of Piece as a function on types, which gives a type c, produces a type Piece c.

To make this idea explicit, one can say that Piece has kind Type -> Type, where kind is a name for the types that types themselves have.

Recursive types

data BinTree = Leaf Int | Branch BinTree BinTree

Here, BinTree is being used recursively in its own definition. Values of BinTree include:

repl example
> data BinTree = Leaf Int | Branch BinTree BinTree
> :t Leaf 4
Leaf 4 :: BinTree
> :t Leaf True
> :t Branch (Leaf 4) (Leaf 5)
Branch (Leaf 4) (Leaf 5) :: BinTree
> :t Branch (Leaf 3) ((Branch (Leaf 6) (Leaf 8)))
Branch (Leaf 3) ((Branch (Leaf 6) (Leaf 8))) :: BinTree

Recursive types can also be parametrized:

data BinTree a = Leaf a | Branch (BinTree a) (BinTree a)
repl example
data BinTree a = Leaf a | Branch (BinTree a) (BinTree a)
> :t Leaf True
Leaf True :: BinTree Bool
> :t Leaf ()
Leaf () :: BinTree ()
> :t Branch (Leaf True) (Leaf False)
Branch (Leaf True) (Leaf False) :: BinTree Bool
> :t Branch (Leaf True) (Leaf ())
"Couldn't match expected type ‘Bool’ with actual type ‘()’" -- (1)!
  1. The definition of Branch requires that the left and right branch be trees of the same type, which is why this doesn't work.

Here is a more complex recursive type and a program of that type:

data Machine a b = M (a -> (b, Machine a b))

machine :: Machine Int Int
machine = machine1 where 

    machine1 :: Machine Int Int
    machine1 = M (\i -> (i, if i > 10 then machine2 else machine1))

    machine2 :: Machine Int Int
    machine2 = M (\i -> (0, machine2))

Note

The list type can be defined recursively in this way:

data List a = EmptyList | HeadThenList a (List a)

In fact, the [a] type in Haskell is defined in this way, with the [1,2,3] being extra syntax for convenience:

data [] a = [] | a : [a] -- (1)!
  1. : is the data constructor, analogous to HeadThenList above, but written infix. [] is analogous to EmptyList.

Synonyms

One can also give new names to existing types:

type Number = Double

Note

Here, Number and Double are not distinguished as separate types by the compiler, so replacing one by the other in a type signature will always be fine. This would not be true for:

data Number = N Double

This can be useful for readability, particularly for quite complex types:

type Failure = Text
data Result = ...
type Program = Either Failure Result

Isomorphic types

Two types are isomorphic (or more colloquially, the same) if there are functions to convert between them in both directions that are loss-less:

repl example
data WrappedInt = MkW {getInt :: Int}

> :t MkW 
MkW :: Int -> WrappedInt -- one direction
> :t getInt 
getInt :: WrappedInt -> Int -- the other direction

> :t (getInt . MkW)
(getInt . MkW) :: Int -> Int
> (getInt . MkW) 4 -- (1)!
4

> :t (MkW . getInt)
(MkW . getInt) :: WrappedInt -> WrappedInt -- (2)!
  1. (getInt . MkW) is the identity function.
  2. (MkW . getInt) is also the identity function.

Types which are isomorphic can be regarded as "the same" for practical purposes, since you can always convert between them, although they may have different performance characteristics. They may also have different typeclass instances.

Reference table of isomorphic types

A type An isomorphic type
data WrappedInt = MkW Int Int
data Wrapped a = MkW a a
data Unit = U ()
data Pair = P Int Bool (Int, Bool)
data OneOf = I Int | B Bool Either Int Bool
a -> b -> c (a, b) -> c
Bool -> a (a, a)
Maybe a Either () a
Reader e a e -> a
State s a s -> (a, s)
Except err a Either err a
ReaderT e m a e -> m a
StateT s m a s -> m (a, s)
ExceptT err m a m (Either err a)

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