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Typeclasses

Typeclasses allow the user to add constraints to universally quantified types:

notEqual :: Eq a => (a, a) -> Bool 
notEqual (x,y) = not (x == y)

notEqual :: forall a. Eq a => (a, a) -> Bool -- (1)!
notEqual (x,y) = not (x == y)
  1. Requires the ExplicitForall extension.

The type of notEqual states: for any type a such that a is an instance of the Eq typeclass, given a pair of as, this will return a Bool.

Note

Eq a is not a type, but rather a different kind of entity, called a constraint.

Eq is referred to as a typeclass.

Basics

Typeclasses, such as Eq, are defined as follows:

class  Eq a  where
    (==) :: a -> a -> Bool --(1)!
  1. The actual definition has a second method, (/=), omitted here for clarity.

Here, (==) is a method of the Eq typeclass.

To make a type be an instance of a type class, one writes a definition of the method(s) for the type in question:

instance Eq Bool where
    True == True = True
    False == False = True
    _ == _ = False

Automatically deriving instances

repl example
> data Piece = Bishop | Knight deriving (Eq, Ord, Show)

> Bishop == Knight
False
> show Bishop
"Bishop"
> import Data.List
> sort [Knight, Bishop]
[Bishop,Knight]

Different instances for the same type

A single type can have at most one instance for a typeclass.

However, two types which are isomorphic can have different instances:

import qualified Data.Text as T

data InterspersedText = MkI {getText :: T.Text} deriving Show -- (1)!

instance Semigroup InterspersedText where  --(2)!
    (MkI t1) <> (MkI t2) = MkI 
        $ T.concat 
        $ fmap (\(c1, c2) -> T.pack [c1,c2]) 
        $ T.zip t1 t2

  1. InterspersedText and Text contain the same data in the sense that MkI :: Text -> InterspersedText and getText :: InterspersedText -> Text map back and forth losslessly.

  2. This isn't a sensible instance in practice and is only exemplary, not least because it breaks the associativity law for Semigroup

InterspersedText and Text contain the same information, but have different Semigroup instances.

repl example
> (MkI "foo") <> (MkI "bar")
MkI "fboaor"

> "foo" <> "bar"
"foobar"

For a practical example, see the Sum and Product types. The type Sum Int is isomorphic to Int, but they have different Monoid instances.

repl example
-- the (Sum Int) Monoid
> Sum 4 -- (1)!
Sum {getSum = 4}
> :t Sum 4
Sum 4 :: Num a => Sum a
> Sum 4 <> Sum 5
Sum {getSum = 9}

-- the (Product Int) Monoid
> Product 4
Product {getProduct = 4}
> :t Product 4
Product 4 :: Num a => Product a
> Product 4 <> Product 5
Product {getProduct = 20}
  1. Sum is both the function which maps a number to a value of the Sum type, and the name of the type. See this page on punning.
Note

One other useful example is the logarithm type Log, from the log-domain package.

> import Numeric.Log
> :i Log -- (1)!
newtype Log a = Exp {ln :: a} -- (2)!
...
instance RealFloat a => Num (Log a) --(3)!

> Exp 0 
1.0 -- (4)!
> Exp 0 * Exp 0 -- (5)!
2.0
  1. :i prompts the repl to give info about an expression.
  2. The definition of the Log type.
  3. Read this as saying: "if the type a is an instance of RealFloat, then the type Log a is an instance of Num. See this section for more.
  4. Exp 0 represents the real number 1, but under the hood, stores it in log-space as 0.
  5. The definition of * (which is part of the instance for Num) adds the log-space numbers, rather than multiplying the real-space numbers.

Constraint implication (classes)

One typeclass may depend on another:

class Semigroup a where
    (<>) :: a -> a -> a

class Semigroup a => Monoid a where
    mempty :: a

What this means is that any instance of Monoid must first be an instance of Semigroup (as well as implementing the Monoid method mempty).

This means that if you encounter a type that is an instance of Monoid, then it will be an instance of Semigroup and so you can use the method <>. For this reason, this is often called inheritance , although the relationship to inheritance in other languages is not direct.

Constraint implication (instances)

instance Eq a => Eq [a] where
    ls1 == ls2 = ...

Read this as saying: for any type a, if a is an instance of Eq, then [a] is also an instance of Eq.

Similarly:

instance Num a => Monoid (Sum a)

This states that if a is an instance of Num (as are e.g. Int and Double) then Sum a (or concretely, Sum Int or Sum Int) are instances of Monoid.

This allows Haskell's type checker to make potentially quite complex deductions. For example:

compareLists :: Ord a => ([a], [a]) -> Bool
compareLists (x, y) = x == y

Haskell knows that == can be called on x and y. How?

  1. It knows that x and y both have type [a]
  2. It knows that a is an instance of Ord (from the type signature)
  3. It knows that Ord a implies Eq a
  4. It knows that Eq a implies Eq [a]

Note

Libraries like lens use the ability of the type checker to make these deductions in sophisticated ways.

Typeclass error messages

repl example
> import Data.List

> data Piece = Bishop | Knight deriving Eq

> sort [Knight, Bishop]

"No instance for (Ord Piece) arising from a use of ‘sort’..."

The error is raised because sort has type sort :: Ord a => [a] -> [a], which means that it expects as input, a list of values of a type which is an instance of the Ord class.

Using typeclasses

Warning

It is recommended that you avoid creating your own type classes unless it is entirely necessary. This is because:

  1. There is usually a solution to a problem which doesn't require typeclasses.
  2. It is easy to create a typeclass that is badly designed.

Instead, rely on existing type classes from libraries.

Constraints float up

repl example
5 :: Num a => a

> :t 5
5 :: Num a => a
> :t [5, 4]
[5, 4] :: Num a => [a] -- (1)!

> :t (True, 4)
(True, 4) :: Num b => (Bool, b)

-- with custom type
> data Square a = Empty | Piece a
> :t Piece 4
Piece 4 :: Num a => Square a
  1. Not [forall a . Num a => a] which is a very different type, beyond the scope of this guide.

Similarly, if a function which requires a constraint is used as part of a larger program, that constraint floats to the top:

complexFunction :: Eq a => a -> ...
complexFunction x = let y = notEqual (x, x) in ...

Because notEqual is called on (x, x), x must be of a type that is an instance of Eq.

The compiler will reason in this way, even if you don't write down the type signature yourself.

Tip

A common difficulty that you may encounter is that you don't know what instance of a typeclass is being invoked:

-- first example
{-# LANGUAGE OverloadedStrings #-}
import Data.Text
example :: Text
example = "hello" `append` mempty

In this case, you know the type of mempty, which is mempty :: forall a. Monoid a => a. However, you do not know which instance of Monoid is being used when mempty is called. Mousing over mempty in VSCode will reveal that the instance is Text.

You can then look up Text on Hackage and find the source, which gives the definition of mempty for Text.

Type class recursion

Type class instances may use the very method they are defining in the definition.

instance Eq (Int, Int) where
    (x, y) == (x', y') = (x == x') && (y == y')

Note

Here, == on the right hand side of the definition is the Eq method for Int, but on the right hand side, it is the method for (Int, Int).

repl example
> (4,3) == (4,3)
True

Type classes over others kinds

In the typeclass Eq, types which are instances are normal types like Int, Bool, or Either Int Bool, i.e. types with kind *.

Note

Accordingly, a type signature like 

(==) :: Eq a => a -> a -> Bool

can be more explicitly stated as:

(==) :: forall (a :: *). Eq a => a -> a -> Bool

However, for many important typeclasses, the types which are instances have other kinds, like * -> *. For instance:

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

This means that [] or Maybe are candidates to be instances of Functor, but Int, Bool, (or even [Int]) are not.

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