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Functions are data (and very cool data, too)

This blog post is a short rant about how people who come from an Object-Oriented programming language (like me) are often using Haskell classes in a bad way.

The problem

As a simple problem, suppose that we have a number of arbitrary shapes, and we want to find out if a certain point lies in any of those shapes. First, let us define a Point as an (x, y) pair.

type Point = (Float, Float)

What we want to do in pseudo-code is basically:

let ls = -- Create a list of shapes
any (\a -> inShape a (4, 3)) ls
-- Get back if (4, 3) is situated in any of the given shapes.

An Object-Oriented approach

When coming from an Object-Oriented background, you would associate a shape with an interface (in Java terminology). Haskell has a construct that looks a lot like interfaces, called classes. We’ll define a class Shape.

class Shape a where
    inShape :: a -> Point -> Bool

Okay, nice. Now let’s define a Rectangle, which will be an instance of Shape.

data Rectangle = Rectangle Point Point
instance Shape Rectangle where
    inShape (Rectangle (x0, y0) (x1, y1)) (x, y) = x0 <= x && x <= x1 &&
                                                   y0 <= y && y <= y1

As you can see, we created a Rectangle datatype, and it’s constructor takes the left-top point and the right-bottom point. Let’s add a Circle.

data Circle = Circle Point Float
instance Shape Circle where
    inShape (Circle (x0, y0) r) (x, y) = dx * dx + dy * dy <= r * r
        where dx = x0 - x
              dy = y0 - y

A Circle takes a center point and a radius. Note that, until now on, we are writing our Haskell program in more or less the same way as we would write it in Java. The real problem arises when we want to declare our list of arbitrary shapes. In ghci:

>*BadShape> let ls = [Rectangle (10, 10) (11, 16), Circle (5, 5) 3]

<interactive>:1:39:
    Couldn't match expected type `Rectangle'
               against inferred type `Circle'

Oh yes, we should’ve thought of that: all elements in a list must be of the same type¹. We’re doing it wrong².

A more functional approach

We need to stop thinking of classes here. Because a Shape is a collection of arbitrary points, we could consider a Shape as a function. This function would, for any Point, return True when this Point lies in the Shape.

type Shape = Point -> Bool

If we want to keep our inShape function, it would simple apply the Shape function now. It is thus quite redundant, but could be defined, for compatibility reasons, as:

inShape :: Shape -> Point -> Bool
inShape = ($)

Now we’ll write our rectangle function again.

rectangle :: Point -> Point -> Shape
rectangle (x0, y0) (x1, y1) (x, y) = x0 <= x && x <= x1 &&
                                     y0 <= y && y <= y1

Note that we will use partial function application here – the (x, y) argument will not be given if we create a rectangle. And our circle function:

circle :: Point -> Float -> Shape
circle (x0, y0) r (x, y) = dx * dx + dy * dy <= r * r
        where dx = x0 - x
              dy = y0 - y

Now we can solve our problem with ghci.

*GoodShape> let ls = [rectangle (10, 10) (11, 16), circle (5, 5) 3]
*GoodShape> :t ls
ls :: [Point -> Bool]
*GoodShape> any (\a -> inShape a (4, 3)) ls
True

As we said, the inShape area is kind of deprecated now, we could write our query as:

>*GoodShape> any ($(4, 3)) ls
True

Conclusion? When learning Haskell, it sometimes really helps to forget everything you know from object-oriented programming languages.