There is one final feature of Haskell's type system that sets it apart from other programming languages. The kind of polymorphism that we have talked about so far is commonly called parametric polymorphism. There is another kind called ad hoc polymorphism, better known as overloading. Here are some examples of ad hoc polymorphism:
Let's start with a simple, but important, example: equality.
There are many types for which we would like equality defined, but
some for which we would not. For example, comparing the equality of
functions is generally considered computationally intractable, whereas
we often want to compare two lists for equality. (The kind of
equality we are referring to here is "value equality," and opposed
to the "pointer equality" found, for example, with Java's ==.
Pointer equality is not referentially transparent, and thus does not
sit well in a purely functional language.) To highlight the issue,
consider this definition of the function elem which tests for
membership in a list:
x `elem`  = False
x `elem` (y:ys) = x==y || (x `elem` ys)
[For the stylistic reason we discussed in Section 3.1, we have chosen to define elem in infix form. == and || are the infix operators for equality and logical or, respectively.]
Intuitively speaking, the type of elem "ought" to be: a->[a]->Bool. But this would imply that == has type a->a->Bool, even though we just said that we don't expect == to be defined for all types.
Furthermore, as we have noted earlier, even if == were defined on all types, comparing two lists for equality is very different from comparing two integers. In this sense, we expect == to be overloaded to carry on these various tasks.
Type classes conveniently solve both of these problems. They
allow us to declare which types are instances of which class,
and to provide definitions of the overloaded operations
associated with a class. For example, let's define a type class
containing an equality operator:
class Eq a where
(==) :: a -> a -> Bool
Here Eq is the name of the class being defined, and == is the single operation in the class. This declaration may be read "a type a is an instance of the class Eq if there is an (overloaded) operation ==, of the appropriate type, defined on it." (Note that == is only defined on pairs of objects of the same type.)
The constraint that a type a must be an instance of the class Eq
is written Eq a. Thus Eq a is not a type expression, but rather
it expresses a constraint on a type, and is called a context.
Contexts are placed at the front of type expressions. For example,
the effect of the above class declaration is to assign the following
type to ==:
(==) :: (Eq a) => a -> a -> Bool
This should be read, "For every type a that is an instance of the class Eq, == has type a->a->Bool". This is the type that would be used for == in the elem example, and indeed the constraint imposed by the context propagates to the principal type for elem:
elem :: (Eq a) => a -> [a] -> Bool
This is read, "For every type a that is an instance of the class Eq, elem has type a->[a]->Bool". This is just what we want---it expresses the fact that elem is not defined on all types, just those for which we know how to compare elements for equality.
So far so good. But how do we specify which types are instances of
the class Eq, and the actual behavior of == on each of those
types? This is done with an instance declaration. For example:
instance Eq Integer where
x == y = x `integerEq` y
The definition of == is called a method. The function integerEq happens to be the primitive function that compares integers for equality, but in general any valid expression is allowed on the right-hand side, just as for any other function definition. The overall declaration is essentially saying: "The type Integer is an instance of the class Eq, and here is the definition of the method corresponding to the operation ==." Given this declaration, we can now compare fixed precision integers for equality using ==. Similarly:
instance Eq Float where
x == y = x `floatEq` y
allows us to compare floating point numbers using ==.
Recursive types such as Tree defined earlier can also be handled:
instance (Eq a) => Eq (Tree a) where
Leaf a == Leaf b = a == b
(Branch l1 r1) == (Branch l2 r2) = (l1==l2) && (r1==r2)
_ == _ = False
Note the context Eq a in the first line---this is necessary because the elements in the leaves (of type a) are compared for equality in the second line. The additional constraint is essentially saying that we can compare trees of a's for equality as long as we know how to compare a's for equality. If the context were omitted from the instance declaration, a static type error would result.
The Haskell Report, especially the Prelude, contains a wealth
of useful examples of type classes.
Indeed, a class Eq is defined
that is slightly larger than the one defined earlier:
class Eq a where
(==), (/=) :: a -> a -> Bool
x /= y = not (x == y)
This is an example of a class with two operations, one for equality, the other for inequality. It also demonstrates the use of a default method, in this case for the inequality operation /=. If a method for a particular operation is omitted in an instance declaration, then the default one defined in the class declaration, if it exists, is used instead. For example, the three instances of Eq defined earlier will work perfectly well with the above class declaration, yielding just the right definition of inequality that we want: the logical negation of equality.
Haskell also supports a notion of class extension. For example,
we may wish to define a class Ord which inherits all of the
operations in Eq, but in addition has a set of comparison operations
and minimum and maximum functions:
class (Eq a) => Ord a where
(<), (<=), (>=), (>) :: a -> a -> Bool
max, min :: a -> a -> a
Note the context in the class declaration. We say that Eq is a superclass of Ord (conversely, Ord is a subclass of Eq), and any type which is an instance of Ord must also be an instance of Eq. (In the next Section we give a fuller definition of Ord taken from the Prelude.)
One benefit of such class inclusions is shorter contexts: a type
expression for a function that uses operations from both the Eq and
Ord classes can use the context (Ord a), rather than
(Eq a, Ord a), since Ord "implies" Eq. More importantly,
methods for subclass operations can assume the existence of methods
for superclass operations. For example, the Ord declaration in the
Standard Prelude contains this default method for (<):
x < y = x <= y && x /= y
As an example of the use of Ord, the principal typing of quicksort defined in Section 2.4.1 is:
quicksort :: (Ord a) => [a] -> [a]
In other words, quicksort only operates on lists of values of ordered types. This typing for quicksort arises because of the use of the comparison operators < and >= in its definition.
Haskell also permits multiple inheritance, since classes may
have more than one superclass. For example, the declaration
class (Eq a, Show a) => C a where ...
creates a class C which inherits operations from both Eq and Show.
Class methods are treated as top level declarations in Haskell. They share the same namespace as ordinary variables; a name cannot be used to denote both a class method and a variable or methods in different classes.
Contexts are also allowed in data declarations; see §4.2.1.
Class methods may have additional class constraints on any type
variable except the one defining the current class. For example, in
class C a where
m :: Show b => a -> b
the method m requires that type b is in class Show. However, the method m could not place any additional class constraints on type a. These would instead have to be part of the context in the class declaration.
So far, we have been using "first-order" types. For example, the type constructor Tree has so far always been paired with an argument, as in Tree Integer (a tree containing Integer values) or Tree a (representing the family of trees containing a values). But Tree by itself is a type constructor, and as such takes a type as an argument and returns a type as a result. There are no values in Haskell that have this type, but such "higher-order" types can be used in class declarations.
To begin, consider the following Functor class (taken from the Prelude):
class Functor f where
fmap :: (a -> b) -> f a -> f b
The fmap function generalizes the map function used previously. Note that the type variable f is applied to other types in f a and f b. Thus we would expect it to be bound to a type such as Tree which can be applied to an argument. An instance of Functor for type Tree would be:
instance Functor Tree where
fmap f (Leaf x) = Leaf (f x)
fmap f (Branch t1 t2) = Branch (fmap f t1) (fmap f t2)
This instance declaration declares that Tree, rather than Tree a, is an instance of Functor. This capability is quite useful, and here demonstrates the ability to describe generic "container" types, allowing functions such as fmap to work uniformly over arbitrary trees, lists, and other data types.
[Type applications are written in the same manner as function applications. The type T a b is parsed as (T a) b. Types such as tuples which use special syntax can be written in an alternative style which allows currying. For functions, (->) is a type constructor; the types f -> g and (->) f g are the same. Similarly, the types [a] and  a are the same. For tuples, the type constructors (as well as the data constructors) are (,), (,,), and so on.]
As we know, the type system detects typing errors in expressions. But what about errors due to malformed type expressions? The expression (+) 1 2 3 results in a type error since (+) takes only two arguments. Similarly, the type Tree Int Int should produce some sort of an error since the Tree type takes only a single argument. So, how does Haskell detect malformed type expressions? The answer is a second type system which ensures the correctness of types! Each type has an associated kind which ensures that the type is used correctly.
Type expressions are classified into different kinds which take one of two possible forms:
Kinds do not appear directly in Haskell programs. The compiler infers kinds before doing type checking without any need for `kind declarations'. Kinds stay in the background of a Haskell program except when an erroneous type signature leads to a kind error. Kinds are simple enough that compilers should be able to provide descriptive error messages when kind conflicts occur. See §4.1.2 and §4.6 for more information about kinds.
"Classes capture common sets of operations. A particular object may be an instance of a class, and will have a method corresponding to each operation. Classes may be arranged hierarchically, forming notions of superclasses and sub classes, and permitting inheritance of operations/methods. A default method may also be associated with an operation."
In contrast to OOP, it should be clear that types are not objects, and in particular there is no notion of an object's or type's internal mutable state. An advantage over some OOP languages is that methods in Haskell are completely type-safe: any attempt to apply a method to a value whose type is not in the required class will be detected at compile time instead of at runtime. In other words, methods are not "looked up" at runtime but are simply passed as higher-order functions.
A different perspective can be gotten by considering the relationship between parametric and ad hoc polymorphism. We have shown how parametric polymorphism is useful in defining families of types by universally quantifying over all types. Sometimes, however, that universal quantification is too broad---we wish to quantify over some smaller set of types, such as those types whose elements can be compared for equality. Type classes can be seen as providing a structured way to do just this. Indeed, we can think of parametric polymorphism as a kind of overloading too! It's just that the overloading occurs implicitly over all types instead of a constrained set of types (i.e. a type class).
The classes used by Haskell are similar to those used in other object-oriented languages such as C++ and Java. However, there are some significant differences: