Tuple of types distance

The first prerequisite to understand how multimethods work is to know what is the tuple of types distance.

Here is a simple hierarchy, the one that we used in the examples of the multimethods section.

Type distance

Suppose that the type X is a super class (including interface) of Y. The distance between X and Y is defined as the minimum number of edges between them in the type graph. The distance is not defined for types that do not have an inheritance relation.

With our hierarchy, we have:

• distance(A, A) = 0
• distance(I, A) = 1
• distance(I, B) = 1
• distance(B, C) = 1
• distance(I, C) = 2
• distance(I, D) = 2
• distance(A, C) = undefined

Tuple of types distance

We extend naturally the type distance to a tuple of types distance so that we have with our hierarchy:

In dimension 2:

• distance((A, B), (A, B)) = (0, 0)
• distance((A, B), (A, C)) = (0, 1)
• distance((A, B), (D, C)) = (1, 1)
• distance((I, B), (D, C)) = (2, 1)
• distance((A, B), (C, C)) = undefined

In dimension 3:

• distance((A, B, A), (D, C, A)) = (1, 1, 0)
• distance((I, B, B), (D, C, C)) = (2, 1, 1)

Now that we know how to define a distance on tuple of types of various dimensions, we can define comparators.