I recently encountered a Haskell function which apparently requires "higher-rank" type constraints. I worked around the issue with some type trickery. I'd like to share the problem and the workaround, and especially get feedback on whether there's a better solution.

I did say type trickery:

`{-# LANGUAGE`

GADTs

, FlexibleContexts

, RankNTypes

, ScopedTypeVariables #-}

Suppose we're implementing an in-place sorting algorithm. We'll use the `ST`

monad, so that we can expose a pure interface. Assuming that we only want to sort primitive types, we will use the unboxed `STUArray`

type, which should improve space and time usage. Actually, it's totally unwarranted premature optimization for the sake of the problem. :)

`import Control.Monad.ST`

import Data.Array.ST

The monadic part of this code is simple enough. I've left out the part where we actually, you know, sort the list. Consult your bedside copy of CLRS.

`sortM`

:: forall a s.

(Ord a, MArray (STUArray s) a (ST s))

=> [a]

-> ST s [a]

sortM xs = do

arr <- newListArray (1, length xs) xs

:: ST s (STUArray s Int a)

-- do some in-place sorting here

getElems arr

Note that `sortM`

has a constraint `MArray (STUArray s) a (ST s)`

. All we're really saying is that `a`

is a type primitive enough that `STUArray`

can unbox it. The rest of the constraint just connects the concrete world of `STUArray`

with the generic `MArray`

interface.

Note also that we give an explicit type on our call to `newListArray`

. Otherwise, there's nothing to constrain what sort of array we get, and GHC will complain of ambiguity. Furthermore, we need to unify this `a`

and `s`

with the `a`

and `s`

from the signature on `sortM`

. We do that using the `ScopedTypeVariables`

extension and the explicit `forall`

. (If you turn on `UnicodeSyntax`

you can write it as `∀`

; how cool is that?)

Good enough so far. Now, the point of using `ST`

was to expose a pure interface, right?

`sortP_1 :: (Ord a) => [a] -> [a]`

sortP_1 xs = runST (sortM xs)

Suddenly GHC is very unhappy:

```
Could not deduce (MArray (STUArray s) a (ST s)) from the context ()
arising from a use of `sortM' at code.hs:21:20-27
Possible fix:
add (MArray (STUArray s) a (ST s)) to the context of
the polymorphic type `forall s. ST s a'
or add an instance declaration for (MArray (STUArray s) a (ST s))
In the first argument of `runST', namely `(sortM xs)'
In the expression: runST (sortM xs)
In the definition of `sortP_1': sortP_1 xs = runST (sortM xs)
```

Okay, fair enough. We've given no reason for GHC to believe that `a`

is an `STUArray`

-compatible type. Let's copy over the constraint from `sortM`

:

`sortP_2`

:: (Ord a, MArray (STUArray s) a (ST s))

=> [a] -> [a]

sortP_2 xs = runST (sortM xs)

Still no good:

```
Could not deduce (MArray (STUArray s1) a (ST s1))
from the context ()
arising from a use of `sortM' at code.hs:27:20-27
Possible fix:
add (MArray (STUArray s1) a (ST s1)) to the context of
the polymorphic type `forall s. ST s a'
or add an instance declaration for (MArray (STUArray s1) a (ST s1))
In the first argument of `runST', namely `(sortM xs)'
In the expression: runST (sortM xs)
In the definition of `sortP_2': sortP_2 xs = runST (sortM xs)
```

Note that GHC gives an error about the variable `s1`

, not `s`

. This is a valuable hint. Our caller (notionally) chooses one `s`

when she provides evidence to satisfy the constraint `MArray (STUArray s) a (ST s)`

. Then `runST`

(notionally) gets to choose a different `s`

because of its higher-rank type.

What to do? It's useless for our caller to pick some particular `s`

and provide evidence. We need that evidence *for all* `s`

:

`sortP_3`

:: (Ord a, forall s. MArray (STUArray s) a (ST s))

=> [a] -> [a]

sortP_3 xs = runST (sortM xs)

Unfortunately, GHC does not support this "higher-rank constraint":

```
code.hs:32:13: malformed class assertion
```

Well... the `->`

of functions and the `=>`

of constraints are not as different as they may appear. A Haskell compiler can implement type classes by translating constraints to explicit "dictionary" arguments. Let's do the same thing ourselves:

`data Evidence s e where`

Evidence

:: (MArray (STUArray s) e (ST s))

=> Evidence s e

We don't quite need to reproduce the `MArray`

methods in our "dictionary". (That's good, because some of them are hidden. A strange class indeed.) It's enough to write a GADT constructor with this constrained type. The constructor will implicitly capture the constraint evidence (such as a dictionary) from its call-site.

Now we need to express the universal quantification on `s`

:

`data ElemType e = ElemType (forall s. Evidence s e)`

A (non-⊥) value of type `ElemType e`

is evidence that `e`

is a suitable element type for `STUArray`

. That's all we wanted the caller to say in the first place!

Now we can pass the appropriate evidence:

`sortP_4`

:: forall a.

(Ord a)

=> ElemType a

-> [a] -> [a]

sortP_4 (ElemType Evidence) xs = runST (sortM xs)

Not quite:

```
Couldn't match expected type `forall s. Evidence s a'
against inferred type `Evidence s e'
In the pattern: Evidence
In the pattern: ElemType Evidence
In the definition of `sortP_4':
sortP_4 (ElemType Evidence) xs = runST (sortM xs)
```

GHC still wants some coaxing to match up this `s`

with the other `s`

. We push the pattern-matching of `Evidence`

inside `runST`

:

`sortP_5`

:: forall a.

(Ord a)

=> ElemType a

-> [a] -> [a]

sortP_5 (ElemType e) xs = runST (f e) where

f :: Evidence s a -> ST s [a]

f Evidence = sortM xs

And...

```
[1 of 1] Compiling Main ( code.hs, interpreted )
Ok, modules loaded: Main.
*Main>
```

A glorious sight! Then, a moment of concern: is `sortP_5`

actually usable? Does `ElemType`

have non-⊥ values? Let's check:

```
*Main> sortP_5 (ElemType Evidence) ([3,4] :: [Int])
[3,4]
```

Nice. It's sorted and everything. :) As expected, this works only for unboxable types:

```
*Main> sortP_5 (ElemType Evidence) ["foo","bar"]
<interactive>:1:18:
Could not deduce (MArray (STUArray s) [Char] (ST s))
from the context ()
arising from a use of `Evidence' at <interactive>:1:18-25
```

As usual, this interesting exercise came from a question on the `#haskell`

IRC channel. That's why it's one of my favorite places, even if they do specialize in making GHC cry.

Thanks for sharing.I found a lot of interesting information here. A really good post, very thankful and hopeful that you will write many more posts like this one.

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