Set type (from Data.Set) cannot be made an instance of Monad because of an Ord constraint on its desired binding operation:returnSet :: a -> Set a
returnSet = singleton
bindSet :: Ord b => Set a -> (a -> Set b) -> Set b
bindSet sa k = unions (map k (toList sa))However, despite being the classic example, in some ways it's not a very good example, because the constraint appears only on the second type parameter of
bindSet, not on the first type parameter, nor on returnSet.Another example of the problem also arises in the context of embedded domain-specific languages. When constructing a deep embedding of a computation that will later be compiled, it is often necessary to restrict the involved types to those that can be reified to the target language. For example:
data EDSL :: * -> * where
Value :: Reifiable a => a -> EDSL a
...
Return :: Reifiable a => a -> EDSL a
Bind :: (Reifiable a, Reifiable b) =>
EDSL a -> (a -> EDSL b) -> EDSL bWhile we can construct a computation using
Return and Bind, we cannot declare a Monad instance using those constructors because of the Reifiable class constraint.(Note: if you want to try out the code in this post, you'll need the following:
{-# LANGUAGE GADTs, MultiParamTypeClasses, KindSignatures,
ConstraintKinds, TypeFamilies, RankNTypes,
InstanceSigs, ScopedTypeVariables #-}
import GHC.Exts (Constraint)
import Data.Set hiding (map))
Restricted Type Classes
There have been numerous solutions proposed to address this problem. John Hughes suggested extending Haskell with Restricted Data Types: data types with attached class constraints. In the same paper, Hughes also suggested defining Restricted Type Classes: type classes that take a constraint as a parameter and impose it on all polymorphic type variables in the class methods. This latter approach was simulated several times (by Oleg Kiselyov and Ganesh Sittampalam, amongst others), before the constraint-kinds extension made it possible to encode it directly:
class RMonad (c :: * -> Constraint) (m :: * -> *) where
return :: c a => a -> m a
(>>=) :: (c a, c b) => m a -> (a -> m b) -> m bIt is then straightforward to define instances that require class constraints:
instance RMonad Reifiable EDSL where
return = Return
(>>=) = BindHowever, restricted type classes are new type classes: using them doesn't allow compatibility with existing type classes. If restricted type classes were already used everywhere instead of the original type classes then there would be no problem, but this is not the case. A variant of restricted type classes (suggested by Orchard and Schrijvers is to use an associated type function with a default instance:
class Monad (m :: * -> *) where
type Con m (a :: *) :: Constraint
type Con m a = ()
return :: Con m a => a -> m a
(>>=) :: (Con m a, Con m b) => m a -> (a -> m b) -> m b
instance Monad EDSL where
type Con EDSL a = Reifiable a
return = Return
(>>=) = Bindm would be broken, as the unknown constraint Con m will need to be satisfied.Normality can be Constraining
If we don't want to modify the type class, then the alternative is to modify the data type. Specifically, we need to modify it in such a way that we can declare the type-class instance we want, but such that the operations of that type class will correspond to the operations we desired on the original data type. For monads, one way to do this is to use continuations, as demonstrated by Persson et al. An alternative (and, in our opinion, more intuitive) way to achieve the same effect is to construct a deep embedding of the computation, and restructure it into a normal form. The normal form we use is the same one used by Unimo and Operational, and consists of a sequence of right-nested
>>=s terminating with a return:
The first argument to each
>>= is a value of the original data type, which we will call primitive operations (a.k.a. "non-proper morphisms", "effect basis", or "instructions sets").The key feature of the normal form is that every type either appears as a type parameter on a primitive operation, or appears as the top-level type parameter of the computation. Consequently, if we enforce that all primitives have constrained type parameters, then only the top-level type parameter can remain unconstrained (which is easy to deal with, as we will show later). We can represent this using the following deep embedding:
data NM :: (* -> Constraint) -> (* -> *) -> * -> * where
Return :: a -> NM c t a
Bind :: c x => t x -> (x -> NM c t a) -> NM c t aThe
t parameter is the type of the primitive operations (e.g. Set), and c is the class constraint (e.g. Ord).We can define a
Monad instance for this deep embedding, which applies the monad laws to restructure the computation into the normal form during construction (just like the Operational package.)instance Monad (NM c t) where
return :: a -> NM c t a
return = Return
(>>=) :: NM c t a -> (a -> NM c t b) -> NM c t b
(Return a) >>= k = k a -- left identity
(Bind ta h) >>= k = Bind ta (\ a -> h a >>= k) -- associativityPrimitive operations can be lifted into the
NM type by applying the remaining monad law:liftNM :: c a => t a -> NM c t a
liftNM ta = Bind ta Return -- right identityNotice that only primitive operations with constrained type parameters can be lifted, thereby preventing any unconstrained types infiltrating the computation.
Once a computation has been constructed, it can then be interpreted in whatever way is desired. In many cases (e.g. the
Set monad), we want to interpret it as the same type as the primitive operations. This can be achieved by the following lowering function, which takes interpretations for return and >>= as arguments.lowerNM :: forall a c t. (a -> t a) ->
(forall x. c x => t x -> (x -> t a) -> t a) -> NM c t a -> t a
lowerNM ret bind = lowerNM'
where
lowerNM' :: NM c t a -> t a
lowerNM' (Return a) = ret a
lowerNM' (Bind tx k) = bind tx (lowerNM' . k)Because the top-level type parameter of the computation is visible, we can (crucially) also constrain that type. For example, we can lower a monadic
Set computation as follows:lowerSet :: Ord a => NM Ord Set a -> Set a
lowerSet = lowerNM singleton bindSetThis approach is essentially how the AsMonad transformer from the RMonad library is implemented.
The idea of defining a deep embedding of a normal form that only contains constrained types is not specific to monads, but can be applied to any type class with a normal form such that all types appears as parameters on primitive operations, or as a top-level type parameter. We've just written a paper about this, which is available online along with accompanying code. The code for our principal solution is also available on Hackage.
