[Haskell-cafe] Polyvariadic functions operating with a monoid

Kevin Jardine wrote:

instead of passing around lists of values with these related types, I created a polyvariadic function polyToString... I finally figured out how to do this, but it was a bit harder to figure this out than I expected, and I was wondering if it might be possible to create a small utility library to help other developers do this.

It seems to me that in the general case, we would be dealing with a Monoid rather than a list of strings. We could have a toMonoid function and then return

polyToMonoid value1 value2 ... valueN =

(toMonoid value1) `mappend` (toMonoid value2) 'mappend' ... (toMonoid valueN)

So I tried writing the following code but GHC said it had undecidable instances. Generally speaking, we should not be afraid of undecidable instances: it is a sufficient criterion for terminating type checking but it is not a necessary one. A longer argument can be found at http://okmij.org/ftp/Haskell/types.html#undecidable-inst-defense

However, the posted code has deeper problems, I'm afraid. First, let us look at the case of Strings:

class PolyVariadic p where polyToMonoid' :: String -> p

instance PolyVariadic String where polyToMonoid' acc = acc

instance (Show a, PolyVariadic r) => PolyVariadic (a->r) where polyToMonoid' acc = \a -> polyToMonoid' (acc ++ show a)

polyToMonoid :: PolyVariadic p => p polyToMonoid = polyToMonoid' mempty

test1 = putStrLn $ polyToMonoid True () (Just (5::Int))

*M> test1 True()Just 5 Modulo the TypeSynonymInstances extension, it is Haskell98. If we now generalize it to arbitrary monoids rather than a mere String, we face several problems. First of all, if we re-write the first instance as

instance Monoid r => PolyVariadic r where polyToMonoid' acc = acc

we make it overlap with the second instance: the type variable 'r' may be instantiated to the arrow type a->r'. Now we need a more problematic overlapping instances extension. The problem is deeper however: an arrow type could possibly be an instance of Monoid (for example, functions of the type Int->Int form a monoid with mempty=id, mappend=(.)). If polyToMonoid appears in the context requiring a function type, how could type checker choose the instance of Polyvariadic? The second problem with the posted code

class Monoidable a where toMonoid :: Monoid r => a -> r

is that toMonoid has too `strong' a signature. Suppose we have an instance

instance Monoidable String where toMonoid = \str -> ???

It means that no matter which monoid the programmer may give to us, we promise to inject a string into it. We have no idea about the details of the monoid. It means that the only thing we could do (short of divergence) is to return mempty. That is not too useful. We have little choice but to parametrise Monoidable as well as Polyvariadic with the type of the monoid. To avoid overlapping and disambiguate the contexts, we use the newtype trick. Here is the complete code. It turns out, no undecidable instances are needed.

{-# LANGUAGE TypeSynonymInstances #-} {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-}

module M where

import Data.Monoid

newtype WMonoid m = WMonoid{unwrap :: m}

class Monoid m => Monoidable a m where toMonoid :: a -> m

class Monoid m => PolyVariadic m p where polyToMonoid :: m -> p

instance Monoid m => PolyVariadic m (WMonoid m) where polyToMonoid acc = WMonoid acc

instance (Monoidable a m, PolyVariadic m r) => PolyVariadic m (a->r) where polyToMonoid acc = \a -> polyToMonoid (acc `mappend` toMonoid a)

instance Show a => Monoidable a String where toMonoid = show

test2 = putStrLn $ unwrap $ polyToMonoid "" True () (Just (5::Int))

The remaining problem is how to tell polyToMonoid which monoid we want. It seems simpler just to pass the appropriately specialized mempty method as the first argument, as shown in test2. Granted, a more elegant solution would be a parametrized module (functor) like those in Agda or ML: module type PolyM = functor(M:: sig type m val mempty :: m val mappend :: m -> m -> m end) = struct class Monoidable a where toMonoid :: a -> m class PolyVariadic p where polyToMonoid :: m -> p .etc end The shown solution is essentially the encoding of the above functor.