Advice on writing monadic signatures

Question

Consider the following example functions which both add a random value to the pure input:

addRand1 :: (MonadRandom m) => m (Int -> Int)

addRand2 :: (MonadRandom m) => Int -> m Int -- *can* be written as m (Int -> Int)

It is easy to convert addRand1 into a function with the same signature as addRand2, but not vice versa.

To me, this provides strong evidence that I should write addRand1 over addRand2. In this example, addRand1 has the more truthful/general type, which typically captures important abstractions in Haskell.

While having the "right" signature seems a crucial aspect of functional programming, I also have a lot of practical reasons why addRand2 might be a better signature, even if it could be written with addRand1s signature.

1. With interfaces:

   class FakeMonadRandom m where
     getRandom :: (Random a, Num a) => m a
     getRandomR1 :: (Random a, Num a) => (a,a) -> m a
     getRandomR2 :: (Random a, Num a) => m ((a,a) -> a)
   

   Suddenly getRandomR1 seems "more general" in the sense that it permits more instances (that repeatedly call getRandom until the result is in the range, for example) compared to getRandomR2, which seems to require some sort of reduction technique.

2. addRand2 is easier to write/read:

   addRand1 :: (MonadRandom m) => m (Int -> Int)
   addRand1 = do
     x <- getRandom
     return (+x) -- in general requires `return $ \a -> ...`
   
   addRand2 :: (MonadRandom m) => Int -> m Int
   addRand2 a = (a+) <$> getRandom
   

3. addRand2 is easier to use:

   foo :: (MonadRandom m) => m Int
   foo = do
     x <- addRand1 <*> (pure 3) -- ugly syntax
     f <- addRand1              -- or a two step process: sequence the function, then apply it
     x' <- addRand2 3           -- easy!
     return $ (f 3) + x + x'
   

4. addRand2 is harder to misuse: consider getRandomR :: (MonadRandom m, Random a) => (a,a) -> m a. For a given range, we can sample repeatedly and get different results, which is probably what we intend. However, if we instead have getRandomR :: (MonadRandom m, Random a) => m ((a,a) -> a), we might be tempted to write

   do
     f <- getRandomR
     return $ replicate 20 $ f (-10,10)
   

   but will be very surprised by the result!

I'm feeling very conflicted about how to write monadic code. The "version 2" seems better in many cases, but I recently came across an example where the "version 1" signature was required.*

What sort of factors should influence my design decisions w.r.t. monadic signatures? Is there some way to reconcile apparently conflicting goals of "general signatures" and "natural, clean, easy-to-use, hard-to-misuse syntax"?

_{*: I wrote a function foo :: a -> m b, which worked just fine for (literally) many years. When I tried to incorporate this into a new application (DSL with HOAS), I discovered I was unable to, until I realized that foo could be rewritten to have the signature m (a -> b). Suddenly my new application was possible.}

Answer 1

This depends on multiple factors:

• What signatures are actually possible (here both).
• What signatures are convenient to use.
• Or more generally, if you want to have the most general interface or dually the most general implementations.

The key for understanding the difference between Int -> m Int and m (Int -> Int) is that in the former case, the effect (m ...) can depend on the input argument. For example, if m is IO, you could have a function that launches n missiles, where n is the function argument. On the other hand, the effect in m (Int -> Int) doesn't depend on anything - the effect doesn't "see" the argument of the function it returns.

Coming back you your case: You take a pure input, generate a random number and add it to the input. We can see that the effect (generating the random number) doesn't depend on the input. This is why we can have signature m (Int -> Int). If the task were instead to generate n random numbers, for example, signature Int -> m [Int] would work, but m (Int -> [Int]) wouldn't.

Regarding usability, Int -> m Int is more common in the monadic context, because most monadic combinators expect signatures of the form a -> b -> ... -> m r. For example, you'd normally write

getRandom >>= addRand2

or

addRand2 =<< getRandom

to add a random number to another random number.

Signatures like m (Int -> Int) are less common for monads, but often used with applicative functors (each monad is also an applicative functor), where effects can't depend on parameters. In particular, operator <*> would work nicely here:

addRand1 <*> getRandom

---

Regarding generality, the signature influences how difficult is to use or to implement it. As you observed, addRand1 is more general from the caller's perspective - it can always convert it to addRand2 if needed. On the other hand, addRand2 is less general, therefore easier to implement. In your case it doesn't really apply, but in some cases it might happen that it's possible to implement signature like m (Int -> Int), but not Int -> m Int. This is reflected in the type hierarchy - monads are more specific then applicative functors, which means they give more power to their user, but are "harder" to implement - every monad is an applicative, but not every applicative can be made into a monad.

Answer 2

It is easy to convert addRand1 into a function with the same signature as addRand2, but not vice versa.

Ahem.

-- | Adds a random value to its input
addRand2 :: MonadRandom m => Int -> m Int
addRand2 x = fmap (+x) getRand

-- | Returns a function which adds a (randomly chosen) fixed value to its input
addRand1 :: MonadRandom m => m (Int -> Int)
addRand1 = fmap (+) (addRand2 0)

How come this works? Well, addRand1's job is to come up with a randomly chosen value and partially apply + to it. Adding a random number to a dummy value is a perfectly good way of coming up with a random number!

I think you might be confused about the quantifiers in @chi's statement. He didn't say

For all monads m and types a and b, you can't convert a -> m b to m (a -> b)

∀ m a b. ¬ ∃ f. f :: Monad m => (a -> m b) -> m (a -> b)

He said

You can't convert a -> m b to m (a -> b) for all monads m and types a and b.

¬ ∃ f. f :: ∀ m a b. Monad m => (a -> m b) -> m (a -> b)

Nothing stops you from writing (a -> m b) -> m (a -> b) for a particular monad m and pair of types a and b.