Subtyping - Examples

Examples

A simple practical example of subtypes is shown in the diagram, right. The type "bird" has three subtypes "duck", "cuckoo" and "ostrich". Conceptually, each of these is a variety of the basic "bird" that inherits many "bird" characteristics but has some specific differences. The UML notation is used in this diagram, with open-headed arrows showing the direction and type of the relationship between the supertype and its subtypes.

As a more practical example, a language might allow floating point values to be used wherever integer values are expected (Float <: Integer), or it might define a generic type Number as a common supertype of integers and the reals. In this second case, we only have Integer <: Number and Float <: Number, but Integer and Float are not subtypes of each other.

Programmers may take advantage of subtyping to write code in a more abstract manner than would be possible without it. Consider the following example:

function max (x as Number, y as Number) is if x < y then return y else return x end

If integer and real are both subtypes of Number, and an operator of comparison with an arbitrary Number is defined for both types, then values of either type can be passed to this function. However, the very possibility of implementing such an operator highly constrains the Number type (for example, one can't compare an integer with a complex number), and actually only comparing integers with integers and reals with reals makes sense. Rewriting this function so that it would only accept 'x' and 'y' of the same type requires bounded polymorphism.

Subtyping in type theory is characterized by the fact that any expression of type A may also be given type B if A<:B; the formal typing rule that codifies this is known as the subsumption rule.