Simple Algebraic Data Types

Prev: Products and Coproducts Next: Functors

Prev: Products and Coproducts Next: Functors

Exercises

1. Show the isomorphism between Maybe a and Either () a.
2. Here’s a sum type defined in Haskell:

data Shape = Circle Float
           | Rect Float Float

When we want to define a function like area that acts on a Shape, we do it by pattern matching on the two constructors:

area :: Shape -> Float
area (Circle r) = pi * r * r
area (Rect d h) = d * h

Implement Shape in C++ or Java as an interface and create two classes: Circle and Rect. Implement area as a virtual function.

3. Continuing with the previous example: We can easily add a new function circ that calculates the circumference of a Shape. We can do it without touching the definition of Shape:

circ :: Shape -> Float
circ (Circle r) = 2.0 * pi * r
circ (Rect d h) = 2.0 * (d + h)

Add circ to your C++ or Java implementation. What parts of the original code did you have to touch?

4. Continuing further: Add a new shape, Square, to Shape and make all the necessary updates. What code did you have to touch in Haskell vs. C++ or Java? (Even if you’re not a Haskell programmer, the modifications should be pretty obvious.)
5. Show that $a+a=2\times a$ holds for types (up to isomorphism). Remember that $2$ corresponds to Bool, according to our translation table.