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A type for disjoint (indexed) sums over types

Illustration

We illustrate here the Haskell approach to either types.

The corresponding polymorphic type type constructor is defined as follows:

data Either a b = Left a | Right b

Thus, a value of an either type is either of one type or another and the choice is also conveyed by the constructors Left versus Right. One typical application scenario is error handling where one argument type models error messages (e.g., String) and the other argument type models successful results. In this instance, either types generalize maybe types.

Another typical application scenario is mixed-type computations. For instance, assume that we have some mathematical operations that may return both Int and Float. Here is a corresponding either type:

type IntOrFloat = Either Int Float

As an example of a function that needs to manipulate values of the either type, consider the following function that extracts a Float by applying the conversion fromIntegral if given an Int:

asFloat :: IntOrFloat -> Float
asFloat (Left x) = fromIntegral x
asFloat (Right x) = x

For instance:

> asFloat (Left 42)
42.0
> asFloat (Right 42.0)
42.0

Because case discrimination on an either type is so common, there is even (in Haskell) a standard higher-order function by which the same conversion can be expressed more concisely:

asFloat :: IntOrFloat -> Float
asFloat = either fromIntegral id

Specific either types can also be expressed by other means than a designated type constructor for such types. For instance, in functional programming with algebraic data types, a specific type can be declared for a given sum. For instance, the sum over Int and Float could also be declared like this:

data IntOrFloat = Int Int | Float Float

(We reuse type names as constructor symbols here, which is possible in Haskell, as these are separate namespaces.) The earlier conversion function is now to be defined by ordinary case discrimination over a (non-polymorphic) algebraic data type:

asFloat :: IntOrFloat -> Float
asFloat (Int x) = fromIntegral x
asFloat (Float x) = x

The advantage of the either type constructor is that it captures universally (polymorphically) the notion of disjoint (labeled) sum. Clearly, sums with more than two cases can be expressed by nested applications of the type constructor.

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