Functions are data (and very cool data, too)
Published on December 23, 2009 under the tag haskell
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.
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
class Shape a where inShape :: a -> Point -> Bool
Okay, nice. Now let’s define a
Rectangle, which will be an instance of
data Rectangle = Rectangle Point Point instance Shape Rectangle where Rectangle (x0, y0) (x1, y1)) (x, y) = x0 <= x && x <= x1 && inShape (<= y && y <= y1 y0
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
data Circle = Circle Point Float instance Shape Circle where Circle (x0, y0) r) (x, y) = dx * dx + dy * dy <= r * r inShape (where dx = x0 - x = y0 - y dy
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
>*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'
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
True when this
Point lies in the
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 = ($)inShape
Now we’ll write our rectangle function again.
rectangle :: Point -> Point -> Shape = x0 <= x && x <= x1 && rectangle (x0, y0) (x1, y1) (x, y) <= y && y <= y1 y0
Note that we will use partial function application here – the
(x, y) argument will not be given if we create a rectangle. And our circle function:
circle :: Point -> Float -> Shape = dx * dx + dy * dy <= r * r circle (x0, y0) r (x, y) where dx = x0 - x = y0 - y dy
Now we can solve our problem with
*GoodShape> let ls = [rectangle (10, 10) (11, 16), circle (5, 5) 3] *GoodShape> :t ls ls :: [Point -> Bool] *GoodShape> any (\a -> inShape a (4, 3)) ls True
As we said, the
inShape area is kind of deprecated now, we could write our query as:
>*GoodShape> any ($(4, 3)) ls True
Conclusion? When learning Haskell, it sometimes really helps to forget everything you know from object-oriented programming languages.