Data Types

To create a new Data Type we use the data keyword:

data Foo = Foo

x :: Foo
x = Foo

The type Foo has one inhabitant, Foo.

We can use the same name on the left, as the Data Type, and on the right as the Data Constructor because they are stored in different namespaces.

Product and Coproduct types

If some type can be either one or the other, but not both, it is a coproduct (sum type), also called union types. In Set Theory, an union operation is an “OR” operation.

The “hello world” of data types is defining our own version of true and false:

data Bool = T | F

f :: Bool
f = F

t :: Bool
t = T

Bool is the data type and T and F are the data constructors.

A given v :: Bool can be either F or T but not both at the same time. In Set Theory, a union is an OR operation, generally denoted by | or || in programming languages.

data ReasonToCancel
  = TooManyEmails
  | NotInterested
  | Other String

answer :: ReasonToCancel
answer = Other "I don't like email adds..."

The data constructor Other in ReasonToCancel data type is actually a function which implies:

Other :: String -> ReasonToCancel

That is, the “function” Other is a function from String to ReasonToCancel. It maps one type to another.

Type Variables

We can make the Other data constructor take a type we don’t know yet, instead of hard-coding it as String:

data WhyCancel a -- <1>
  = TooManyEmails
  | NotInterested
  | Other a -- <2>

becauseWithString :: WhyCancel String -- <3>
becauseWithString = Other "I don't like email ads."

type Reason = { code :: Int, text :: String }

becauseWithReason :: Reason -- <4>
becauseWithReason =
  { code: 7
  , text: "I don't like ads."
  }

Note the a type variable in 1 and 2. It means when we type something as WhyCancel, it takes a type variable, that is, some type, which is what we do in 3 and 4.

:kind

As explained in the book Haskell From First Principles

Kinds are types one level up.

In other words, kinds are types of types. Try this in the REPL:

> import WhyCancel

> :kind WhyCancel
Type -> Type

> :kind WhyCancel Int
Type

In the first case, we get Type -> Type, which means WhyCancel is not fully realised; it still requires some type to produce the final (concrete, realised) type. In the second case, we get Type, which means it has been fully realised and we are at a final, concrete type.

We can, of course, create a type alias for it:

> type T = WhyCancel String
> :kind T
Type

More examples

data Thing
  = Foo
  | Bar
  | Sth String

sth :: Thing
sth = Sth "Takes a String"

Thing is the data type, and Foo, Bar and Sth are data constructors. The data constructor Sth takes String. Now, consider this:

type Jedi = { id :: Int, name :: String }

data Thing a
  = Foo
  | Bar
  | Sth a

sthStr :: Thing String
sthStr = Sth "a string"

sthInt :: Thing Int
sthInt = Sth 1

sthJedi :: Thing Jedi
sthJedi = Sth { id: 1, name: "Ahsoka Tano" }

By making Thing take a polymorphic type variable, we can construct data of different types.