Haskell function application and currying

Question

I am always interested in learning new languages, a fact that keeps me on my toes and makes me (I believe) a better programmer. My attempts at conquering Haskell come and go - t

Answer 1

(f . g) x = f (g x)

This is true. You concluded from that that

(f . g) x y = f (g x y)

must also be true, but that is not the case. In fact, the following is true:

(f . g) x y = f (g x) y

which is not the same.

Why is this true? Well (f . g) x y is the same as ((f . g) x) y and since we know that (f . g) x = f (g x) we can reduce that to (f (g x)) y, which is again the same as f (g x) y.

So (concatMap . ins) 1 [[2,3]] is equivalent to concatMap (ins 1) [[2,3]]. There is no magic going on here.

Another way to approach this is via the types:

. has the type (b -> c) -> (a -> b) -> a -> c, concatMap has the type (x -> [y]) -> [x] -> [y], ins has the type t -> [t] -> [[t]]. So if we use concatMap as the b -> c argument and ins as the a -> b argument, then a becomes t, b becomes [t] -> [[t]] and c becomes [[t]] -> [[t]] (with x = [t] and y = [t]).

So the type of concatMap . ins is t -> [[t]] -> [[t]], which means a function taking a whatever and a list of lists (of whatevers) and returning a list of lists (of the same type).

Answer 2

I'd like to add my two cents. The question and answer make it sound like . is some magical operator that does strange things with re-arranging function calls. That's not the case. . is just function composition. Here's an implementation in Python:

def dot(f, g):
    def result(arg):
        return f(g(arg))
    return result

It just creates a new function which applies g to an argument, applies f to the result, and returns the result of applying f.

So (concatMap . ins) 1 [[2, 3]] is saying: create a function, concatMap . ins, and apply it to the arguments 1 and [[2, 3]]. When you do concatMap (ins 1 [[2,3]]) you're instead saying, apply the function concatMap to the result of applying ins to 1 and [[2, 3]] - completely different, as you figured out by Haskell's horrendous error message.

UPDATE: To stress this even further. You said that (f . g) x was another syntax for f (g x). This is wrong! . is just a function, as functions can have non-alpha-numeric names (>><, .., etc., could also be function names).

Answer 3

You're overthinking this problem. You can work it all out using simple equational reasoning. Let's try it from scratch:

permute = foldr (concatMap . ins) [[]]

This can be converted trivially to:

permute lst = foldr (concatMap . ins) [[]] lst

concatMap can be defined as:

concatMap f lst = concat (map f lst)

The way foldr works on a list is that (for instance):

-- let lst = [x, y, z]
foldr f init lst
= foldr f init [x, y, z]
= foldr f init (x : y : z : [])
= f x (f y (f z init))

So something like

permute [1, 2, 3]

becomes:

foldr (concatMap . ins) [[]] [1, 2, 3]
= (concatMap . ins) 1 
    ((concatMap . ins) 2
       ((concatMap . ins) 3 [[]]))

Let's work through the first expression:

(concatMap . ins) 3 [[]]
= (\x -> concatMap (ins x)) 3 [[]]  -- definition of (.)
= (concatMap (ins 3)) [[]]
= concatMap (ins 3) [[]]     -- parens are unnecessary
= concat (map (ins 3) [[]])  -- definition of concatMap

Now ins 3 [] == [3], so

map (ins 3) [[]] == (ins 3 []) : []  -- definition of map
= [3] : []
= [[3]]

So our original expression becomes:

foldr (concatMap . ins) [[]] [1, 2, 3]
= (concatMap . ins) 1 
    ((concatMap . ins) 2
       ((concatMap . ins) 3 [[]]))
= (concatMap . ins) 1 
    ((concatMap . ins) 2 [[3]]

Let's work through

(concatMap . ins) 2 [[3]]
= (\x -> concatMap (ins x)) 2 [[3]]
= (concatMap (ins 2)) [[3]]
= concatMap (ins 2) [[3]]     -- parens are unnecessary
= concat (map (ins 2) [[3]])  -- definition of concatMap
= concat (ins 2 [3] : [])
= concat ([[2, 3], [3, 2]] : [])
= concat [[[2, 3], [3, 2]]]
= [[2, 3], [3, 2]]

So our original expression becomes:

foldr (concatMap . ins) [[]] [1, 2, 3]
= (concatMap . ins) 1 [[2, 3], [3, 2]]
= (\x -> concatMap (ins x)) 1 [[2, 3], [3, 2]]
= concatMap (ins 1) [[2, 3], [3, 2]]
= concat (map (ins 1) [[2, 3], [3, 2]])
= concat [ins 1 [2, 3], ins 1 [3, 2]] -- definition of map
= concat [[[1, 2, 3], [2, 1, 3], [2, 3, 1]], 
          [[1, 3, 2], [3, 1, 2], [3, 2, 1]]]  -- defn of ins
= [[1, 2, 3], [2, 1, 3], [2, 3, 1], 
   [1, 3, 2], [3, 1, 2], [3, 2, 1]]

Nothing magical here. I think you may have been confused because it's easy to assume that concatMap = concat . map, but this is not the case. Similarly, it may seem like concatMap f = concat . (map f), but this isn't true either. Equational reasoning will show you why.