# Type Classes

Typeclasses in Haskell are a means of defining the behaviour associated with a type separately from that type's definition. Whereas, say, in Java, you'd define the behaviour as part of the type's definition -- i.e. in an interface, abstract class or concrete class -- Haskell keeps these two things separate.

There are a number of typeclasses already defined in Haskell's base package. The relationship between these is illustrated in the Remarks section below.

# Eq

All basic datatypes (like Int, String, Eq a => [a]) from Prelude except for functions and IO have instances of Eq. If a type instantiates Eq it means that we know how to compare two values for value or structural equality.

> 3 == 2 
False
> 3 == 3
True

# Required methods

  • (==) :: Eq a => a -> a -> Boolean or (/=) :: Eq a => a -> a -> Boolean (if only one is implemented, the other defaults to the negation of the defined one)

# Defines

  • (==) :: Eq a => a -> a -> Boolean
  • (/=) :: Eq a => a -> a -> Boolean

# Direct superclasses

None

# Notable subclasses

# Ord

Types instantiating Ord include, e.g., Int, String, and [a] (for types a where there's an Ord a instance). If a type instantiates Ord it means that we know a “natural” ordering of values of that type. Note, there are often many possible choices of the “natural” ordering of a type and Ord forces us to favor one.

Ord provides the standard (<=), (<), (>), (>=) operators but interestingly defines them all using a custom algebraic data type

data Ordering = LT | EQ | GT

compare :: Ord a => a -> a -> Ordering

# Required methods

  • compare :: Ord a => a -> a -> Ordering or (<=) :: Ord a => a -> a -> Boolean (the standard’s default compare method uses (<=) in its implementation)

# Defines

  • compare :: Ord a => a -> a -> Ordering
  • (<=) :: Ord a => a -> a -> Boolean
  • (<) :: Ord a => a -> a -> Boolean
  • (>=) :: Ord a => a -> a -> Boolean
  • (>) :: Ord a => a -> a -> Boolean
  • min :: Ord a => a -> a -> a
  • max :: Ord a => a -> a -> a

# Direct superclasses

# Monoid

Types instantiating Monoid include lists, numbers, and functions with Monoid return values, among others. To instantiate Monoid a type must support an associative binary operation (mappend or (<>)) which combines its values, and have a special "zero" value (mempty) such that combining a value with it does not change that value:

mempty  <>  x == x
x <>  mempty  == x

x <> (y <> z) == (x <> y) <> z

Intuitively, Monoid types are "list-like" in that they support appending values together. Alternatively, Monoid types can be thought of as sequences of values for which we care about the order but not the grouping. For instance, a binary tree is a Monoid, but using the Monoid operations we cannot witness its branching structure, only a traversal of its values (see Foldable and Traversable).

# Required methods

  • mempty :: Monoid m => m
  • mappend :: Monoid m => m -> m -> m

# Direct superclasses

None

# Num

The most general class for number types, more precisely for rings, i.e. numbers that can be added and subtracted and multiplied in the usual sense, but not necessarily divided.

This class contains both integral types (Int, Integer, Word32 etc.) and fractional types (Double, Rational, also complex numbers etc.). In case of finite types, the semantics are generally understood as modular arithmetic, i.e. with over- and underflow.

Note that the rules for the numerical classes are much less strictly obeyed than the monad or monoid laws, or those for equality comparison. In particular, floating-point numbers generally obey laws only in a approximate sense.

# The methods

  • `fromInteger :: Num a => Integer -> a`. convert an integer to the general number type (wrapping around the range, if necessary). Haskell [number literals](http://stackoverflow.com/documentation/haskell/369/overloaded-literals/1243/integer-numeral) can be understood as a monomorphic `Integer` literal with the general conversion around it, so you can use the literal `5` in both an `Int` context and a `Complex Double` setting.
  • `(+) :: Num a => a -> a -> a`. Standard addition, generally understood as associative and commutative, i.e.,
      a + (b + c)(a + b) + c
      a + bb + a
    
    
  • `(-) :: Num a => a -> a -> a`. Subtraction, which is the inverse of addition:
      (a - b) + b(a + b) - ba
    
    
  • `(*) :: Num a => a -> a -> a`. Multiplication, an associative operation that's distributive over addition:
      a * (b * c)(a * b) * c
      a * (b + c)a * b + a * c
    
    

    for the most common instances, multiplication is also commutative, but this is definitely not a requirement.

  • negate :: Num a => a -> a. The full name of the unary negation operator. -1 is syntactic sugar for negate 1.

      -anegate a0 - a
    
    
  • `abs :: Num a => a -> a`. The absolute-value function always gives a non-negative result of the same magnitude
      abs (-a)abs a
      abs (abs a)abs a
    
    

    abs a ≡ 0 should only happen if a ≡ 0. For real types it's clear what non-negative means: you always have abs a >= 0. Complex etc. types don't have a well-defined ordering, however the result of abs should always lie in the real subset (i.e. give a number that could also be written as a single number literal without negation).

  • signum :: Num a => a -> a. The sign function, according to the name, yields only -1 or 1, depending on the sign of the argument. Actually, that's only true for nonzero real numbers; in general signum is better understood as the normalising function:

      abs (signum a)1   -- unless a≡0
      signum a * abs aa -- This is required to be true for all Num instances
    
    

    Note that section 6.4.4 of the Haskell 2010 Report explicitly requires this last equality to hold for any valid Num instance.

  • Some libraries, notably linear and hmatrix, have a much laxer understanding of what the Num class is for: they treat it just as a way to overload the arithmetic operators. While this is pretty straightforward for + and -, it already becomes troublesome with * and more so with the other methods. For instance, should * mean matrix multiplication or element-wise multiplication?
    It is arguably a bad idea to define such non-number instances; please consider dedicated classes such as VectorSpace.

    In particular, the “negatives” of unsigned types are wrapped around to large positive, e.g. (-4 :: Word32) == 4294967292.

    This is widely not fulfilled: vector types do not have a real subset. The controversial Num-instances for such types generally define abs and signum element-wise, which mathematically speaking doesn't really make sense.

    # Maybe and the Functor Class

    In Haskell, data types can have arguments just like functions. Take the Maybe type for example.

    Maybe is a very useful type which allows us to represent the idea of failure, or the possiblity thereof. In other words, if there is a possibility that a computation will fail, we use the Maybe type there. Maybe acts kind of like a wrapper for other types, giving them additional functionality.

    Its actual declaration is fairly simple.

    Maybe a = Just a | Nothing
    
    

    What this tells is that a Maybe comes in two forms, a Just, which represents success, and a Nothing, which represents failure. Just takes one argument which determines the type of the Maybe, and Nothing takes none. For example, the value Just "foo" will have type Maybe String, which is a string type wrapped with the additional Maybe functionality. The value Nothing has type Maybe a where a can be any type.

    This idea of wrapping types to give them additional functionality is a very useful one, and is applicable to more than just Maybe. Other examples include the Either, IO and list types, each providing different functionality. However, there are some actions and abilities which are common to all of these wrapper types. The most notable of these is the ability to modify the encapsulated value.

    It is common to think of these kinds of types as boxes which can have values placed in them. Different boxes hold different values and do different things, but none are useful without being able to access the contents within.

    To encapsulate this idea, Haskell comes with a standard typeclass, named Functor. It is defined as follows.

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

    As can be seen, the class has a single function, fmap, of two arguments. The first argument is a function from one type, a, to another, b. The second argument is a functor (wrapper type) containing a value of type a. It returns a functor (wrapper type) containing a value of type b.

    In simple terms, fmap takes a function and applies to the value inside of a functor. It is the only function necessary for a type to be a member of the Functor class, but it is extremely useful. Functions operating on functors that have more specific applications can be found in the Applicative and Monad typeclasses.

    # Type class inheritance: Ord type class

    Haskell supports a notion of class extension. For example, the class Ord inherits all of the operations in Eq, but in addition has a compare function that returns an Ordering between values. Ord may also contain the common order comparison operators, as well as a min method and a max method.

    The => notation has the same meaning as it does in a function signature and requires type a to implement Eq, in order to implement Ord.

    data Ordering = EQ | LT | GT
    
    class Eq a => Ord a where
        compare :: Ord a => a -> a -> Ordering
        (<)     :: Ord a => a -> a -> Bool
        (<=)    :: Ord a => a -> a -> Bool
        (>)     :: Ord a => a -> a -> Bool
        (>=)    :: Ord a => a -> a -> Bool
        min     :: Ord a => a -> a -> a
        max     :: Ord a => a -> a -> a
    
    

    All of the methods following compare can be derived from it in a number of ways:

    x < y   = compare x y == LT
    x <= y  = x < y || x == y -- Note the use of (==) inherited from Eq
    x > y   = not (x <= y)
    x >= y  = not (x < y)
    
    min x y = case compare x y of
                   EQ -> x
                   LT -> x
                   GT -> y
    
    max x y = case compare x y of
                   EQ -> x
                   LT -> y
                   GT -> x
    
    

    Type classes that themselves extend Ord must implement at least either the compare method or the (<=) method themselves, which builds up the directed inheritance lattice.

    # Remarks

    The following diagram taken from the Typeclassopedia article shows the relationship between the various typeclasses in Haskell.

    Relationships among standard Haskell type classes, Figure 1 as published in Typeclassopedia.