以 pointfree 风格书写 fx = gxx

Question

I am learning Haskell. I'm sorry for asking a very basic question but I cant seem to find the answer. I have a function f defined by :

f x = g x x

where g is an already defined function of 2 arguments. How do I write this pointfree style? Edit : without using a lambda expression.

Thanks

Answer 1

f can be written with Control.Monad.join :

f = join g

join on the function monad is one of the primitives used when constructing point-free expressions , as it cannot be defined in a point-free style itself (its SKI calculus equivalent, SII — ap id id in Haskell — doesn't type) .

Answer 2

This is known as "W" combinator :

import Control.Monad
import Control.Monad.Instances
import Control.Applicative

f = join g       -- = Wg        (also, join = (id =<<))
  = (g `ap` id)  -- \x -> g x (id x) = SgI
  = (<*> id) g   --                  = CSIg
  = g =<< id     -- \x -> g (id x) x
  = id =<< g     -- \x -> id (g x) x

S,K,I are one basic set of combinators; B,C,K,W are another - you've got to stop somewhere ^{(re: your "no lambda expression" comment)} :

_B = (.)     -- _B f g x = f (g x)     = S(KS)K
_C = flip    -- _C f x y = f y x       = S(S(K(S(KS)K))S)(KK)
_K = const   -- _K x y   = x
_W = join    -- _W f x   = f x x       = CSI = SS(KI) = SS(SK)
_S = ap      -- _S f g x = f x (g x)   = B(B(BW)C)(BB) = B(BW)(BBC)
   = (<*>)                                -- from Control.Applicative
_I = id      -- _I x     = x           = WK = SKK = SKS = SK(...)

{-
Wgx = gxx 
    = SgIx = CSIgx 
           = Sg(KIg)x = SS(KI)gx
    = gx(Kx(gx)) = gx(SKgx) = Sg(SKg)x = SS(SK)gx

-- _W (,) 5 = (5,5)
-- _S _I _I x = x x = _omega x         -- self-application, untypeable
-}

Answer 3

I got here by pure chance, and I want to offer my solution, as nobody has mentioned lifting yet in this thread, not explicitly at least.

This is a solution:

f = liftM2 g id id

How to look at it?

• g has type a -> a -> b , ie it takes two values of some type (the same type for both, otherwise the definition the OP gave of f wouldn't make sense), and gives back another value of some type (not necessarily of the same type as the arguments);

• lift2M g is the lifted version of g and it has type (Monad m) => ma -> ma -> mb : it takes two monadic values , which are each a value in a so-far-unspecified context, and gives back a monadic value ;

• when passing two functions to liftM2 g , the die is cast on what the context of the Monad is: it is that the values are not in there yet, but will eventually be, when the function will receive the arguments it needs; in other words, functions are monads that store their own future values; therefore, lift2M g takes in input two functions (or, the future values of two functions), and gives back the another function (or, its future value ); knowing this, its type is the same as above if you change m to (->) r , or r -> : (r -> a) -> (r -> a) -> (r -> b)

• the two functions that we pass are both id , which promises it'll give back the same value it receives;

• liftM2 g id id is therefore a function of type r -> b that passes its argument to those two id s, which let it unchanged and forward it to g .

In a similar fashion, one can exploit that functions are applicative functors, and use this solution:

f = g <$> id <*> id