Haskell's (<-) in Terms of the Natural Transformations of Monad

Question

So I'm playing around with the hasbolt module in GHCi and I had a curiosity about some desugaring. I've been connecting to a Neo4j database by creating a pipe as follows

ghci> pipe <- connect $ def {credentials}

and that works just fine. However, I'm wondering what the type of the (<-) operator is (GHCi won't tell me). Most desugaring explanations describe that

do x <- a
   return x

desugars to

a >>= (\x -> return x)

but what about just the line x <- a? It doesn't help me to add in the return because I want pipe :: Pipe not pipe :: Control.Monad.IO.Class.MonadIO m => m Pipe, but (>>=) :: Monad m => m a -> (a -> m b) -> m b so trying to desugar using bind and return/pure doesn't work without it.

Ideally it seems like it'd be best to just make a Comonad instance to enable using extract :: Monad m => m a -> a as pipe = extract $ connect $ def {creds} but it bugs me that I don't understand (<-).

Another oddity is that, treating (<-) as haskell function, it's first argument is an out-of-scope variable, but that wouldn't mean that

(<-) :: a -> m b -> b

because not just anything can be used as a free variable. For instance, you couldn't bind the pipe to a Num type or a Bool. The variable has to be a "String"ish thing, except it never is actually a String; and you definitely can't try actually binding to a String. So it seems as if it isn't a haskell function in the usual sense (unless there is a class of functions that take values from the free variable namespace... unlikely). So what is (<-) exactly? Can it be replaced entirely by using extract? Is that the best way to desugar/circumvent it?

Answer 1

I'm wondering what the type of the (<-) operator is ...

<- doesn't have a type, it's part of the syntax of do notation, which as you know is converted to sequences of >>= and return during a process called desugaring.

but what about just the line x <- a ...?

That's a syntax error in normal haskell code and the compiler would complain. The reason the line:

ghci> pipe <- connect $ def {credentials}

works in ghci is that the repl is a sort of do block; you can think of each entry as a line in your main function (it's a bit more hairy than that, but that's a good approximation). That's why you need (until recently) to say let foo = bar in ghci to declare a binding as well.

Answer 2

Ideally it seems like it'd be best to just make a Comonad instance to enable using extract :: Monad m => m a -> a as pipe = extract $ connect $ def {creds} but it bugs me that I don't understand (<-).

Comonad has nothing to do with Monads. In fact, most Monads don't have any valid Comonad instance. Consider the [] Monad:

instance Monad [a] where
  return x = [x]
  xs >>= f = concat (map f xs)

If we try to write a Comonad instance, we can't define extract :: m a -> a

instance Comonad [a] where
  extract (x:_) = x
  extract [] = ???

This tells us something interesting about Monads, namely that we can't write a general function with the type Monad m => m a -> a. In other words, we can't "extract" a value from a Monad without additional knowledge about it.

So how does the do-notation syntax do {x <- [1,2,3]; return [x,x]} work?

Since <- is actually just syntax sugar, just like how [1,2,3] actually means 1 : 2 : 3 : [], the above expression actually means [1,2,3] >>= (\x -> return [x,x]), which in turn evaluates to concat (map (\x -> [[x,x]]) [1,2,3])), which comes out to [1,1,2,2,3,3].

Notice how the arrow transformed into a >>= and a lambda. This uses only built-in (in the typeclass) Monad functions, so it works for any Monad in general.

We can pretend to extract a value by using (>>=) :: Monad m => m a -> (a -> m b) -> m b and working with the "extracted" a inside the function we provide, like in the lambda in the list example above. However, it is impossible to actually get a value out of a Monad in a generic way, which is why the return type of >>= is m b (in the Monad)

Answer 3

So what is (<-) exactly? Can it be replaced entirely by using extract? Is that the best way to desugar/circumvent it?

Note that the do-block <- and extract mean very different things even for types that have both Monad and Comonad instances. For instance, consider non-empty lists. They have instances of both Monad (which is very much like the usual one for lists) and Comonad (with extend/=>> applying a function to all suffixes of the list). If we write a do-block such as...

import qualified Data.List.NonEmpty as N
import Data.List.NonEmpty (NonEmpty(..))
import Data.Function ((&))

alternating :: NonEmpty Integer
alternating = do
    x <- N.fromList [1..6]
    -x :| [x]

... the x in x <- N.fromList [1..6] stands for the elements of the non-empty list; however, this x must be used to build a new list (or, more generally, to set up a new monadic computation). That, as others have explained, reflects how do-notation is desugared. It becomes easier to see if we make the desugared code look like the original one:

alternating :: NonEmpty Integer
alternating =
    N.fromList [1..6] >>= \x ->
    -x :| [x]

GHCi> alternating
-1 :| [1,-2,2,-3,3,-4,4,-5,5,-6,6]

The lines below x <- N.fromList [1..6] in the do-block amount to the body of a lambda. x <- in isolation is therefore akin to a lambda without body, which is not a meaningful thing.

Another important thing to note is that x in the do-block above does not correspond to any one single Integer, but rather to all Integers in the list. That already gives away that <- does not correspond to an extraction function. (With other monads, the x might even correspond to no values at all, as in x <- Nothing or x <- []. See also Lazersmoke's answer.)

On the other hand, extract does extract a single value, with no ifs or buts...

GHCi> extract (N.fromList [1..6])
1

... however, it is really a single value: the tail of the list is discarded. If we want to use the suffixes of the list, we need extend/(=>>)...

GHCi> N.fromList [1..6] =>> product =>> sum
1956 :| [1236,516,156,36,6]

If we had a co-do-notation for comonads (cf. this package and the links therein), the example above might get rewritten as something in the vein of:

-- codo introduces a function: x & f = f x
N.fromList [1..6] & codo xs -> do
    ys <- product xs
    sum ys

The statements would correspond to plain values; the bound variables (xs and ys), to comonadic values (in this case, to list suffixes). That is exactly the opposite of what we have with monadic do-blocks. All in all, as far as your question is concerned, switching to comonads just swaps which things we can't refer to outside of the context of a computation.