I am in the process of teaching myself Haskell and I was wondering about the following type signatures:
Prelude> :t ($) ($) :: (a -> b) -> a -> b Prelude>
How should I interpret (no pun intended) that?
A semi-similar result is also proving to be puzzling:
Prelude> :t map map :: (a -> b) -> [a] -> [b] Prelude>
Since the type of elem y z is Bool , this matches with the function (&&) x , and thus the type of (&&) x (elem y z) is thus Bool .
In Haskell, every statement is considered as a mathematical expression and the category of this expression is called as a Type. You can say that "Type" is the data type of the expression used at compile time.
If you are using an interactive Haskell prompt (like GHCi) you can type :t <expression> and that will give you the type of an expression. e.g. or e.g.
Well, as said already, $
can be easily understood if you just forget about currying and see it like, say, in C++
template<typename A, typename B> B dollar(std::function<B(A)> f, A x) { return f(x); }
But actually, there is more to this than just applying a function to a value! The apparent similarity between the signatures of $
and map
has in fact a pretty deep category-theory meaning: both are examples of the morphism-action of a functor!
In the category Hask that we work with all the time, objects are types. (That is a bit confusionsome, but don't worry). The morphisms are functions.
The most well-known (endo-)functors are those which have an instance of the eponymous type class. But actually, mathematically, a functor is only something that maps both objects to objects and morphisms to morphisms1. map
(pun intended, I suppose!) is an example: it takes an object (i.e. type) A
and maps it to a type [A]
. And, for any two types A
and B
, it takes a morphism (i.e. function) A -> B
, and maps it to the corresponding list-function of type [A] -> [B]
.
This is just a special case of the functor class signature operation:
fmap :: Functor f => (a->b) -> (f a->f b)
Mathematics doesn't require this fmap
to have a name though. And so there can be also the identity functor, which simply assigns any type to itself. And, every morphism to itself:
($) :: (a->b) -> (a->b)
"Identity" exists obviously more generally, you can also map values of any type to themselves.
id :: a -> a id x = x
And sure enough, a possible implementation is then
($) = id
1Mind, not anything that maps objects and morphisms is a functor... it does need to satisfy the functor laws.
I'll start with map
. The map
function applies an operation to every element in a list. If I had
add3 :: Int -> Int add3 x = x + 3
Then I could apply this to a whole list of Int
s using map
:
> map add3 [1, 2, 3, 4] [4, 5, 6, 7]
So if you look at the type signature
map :: (a -> b) -> [a] -> [b]
You'll see that the first argument is (a -> b)
, which is just a function that takes an a
and returns a b
. The second argument is [a]
, which is a list of values of type a
, and the return type [b]
, a list of values of type b
. So in plain english, the map
function applies a function to each element in a list of values, then returns the those values as a list.
This is what makes map
a higher order function, it takes a function as an argument and does stuff with it. Another way to look at map
is to add some parentheses to the type signature to make it
map :: (a -> b) -> ([a] -> [b])
So you can also think of it as a function that transforms a function from a
to b
into a function from [a]
to [b]
.
The function ($)
has the type
($) :: (a -> b) -> a -> b
And is used like
> add3 $ 1 + 1 5
All it does is take what's to the right, in this case 1 + 1
, and passes it to the function on the left, here add3
. Why is this important? It has a handy fixity, or operator precedence, that makes it equivalent to
> add3 (1 + 1)
So whatever to the right gets essentially wrapped in parentheses before being passed to the left. This just makes it useful for chaining several functions together:
> add3 $ add3 $ add3 $ add3 $ 1 + 1
is nicer than
> add3 (add3 (add3 (add3 (1 + 1))))
because you don't have to close parentheses.
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