The Comonad.Reader

Today I'd like to talk about free monads.

The free monad of a functor is a monad that is uniquely determined by the functor (up to isomorphism, etc), given by:

 
data Free f a = Roll (f (Free f a)) | Return a
-- newtype Free f a = Free { unfree :: Either a (f (Free f a))) }
 

Usually the above is written up using a newtype around a sum (Either) so you can write it using nice point-free style, but I think this makes for clearer introduction this way.

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You may recall the definition for an exponential functor from my previous entry, which can also be viewed, I suppose, as functors in the
category of right-invertible functions in Haskell.

 
class ExpFunctor f where
	xmap :: (a -> b) -> (b -> a) -> f a -> f b
 

Clarifying the above, an instance of ExpFunctor should satisfy the slightly generalized version of the Functor laws from Control.Monad:

 
xmap id id = id
xmap f g . xmap f' g' = xmap (f . f') (g' . g)
 

Since we like to apply xmap to a pair of functions such that f . g = id, as in the Fegaras/Sheard case, we get:

 
xmap f g . xmap g f
	= xmap (f . g) (f . g) -- by second xmap law
        = xmap id id 	       -- by f . g = id
	= id		       -- by first xmap law
 

In any event, what I thought what I'd do today is note that Wouter Swierstra's Data Types a la Carte approach works over exponential functors.

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I have been trying out various representations for higher-order abstract syntax (HOAS) in Haskell, with an eye towards seeing what I can actually use to get real work done and I have run into a few unexpected headaches, and a couple of neat observations. That said, I should probably start by explaining some terminology.

Encoding a language that binds variables in higher order abstract syntax generally involves constructing an abstract data type that contains functions. A functor for representing expressions from Berendregdt's lambda cube in HOAS goes something like (ignoring any consolidation of binders and sorts)

 
data F a
    = Lam a (a -> a)
    | Pi a (a -> a)
    | App a a
    | Star
    | Box
 

There are a number of mathematical functors that are not instances has Haskell's Functor class, such as the above.

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Updated my little type-level 2s and 16s complement integer library to be ghc 6.6 friendly and uploaded it to hackage based on popular (er.. ok, well, singular) demand.

O.K. it was more of a polite request, but I did update it.

Recently Eric Kidd and Dan Piponi have used a bit of type hackery by Oleg Kiselyov and -fno-implicit-prelude to build some interesting restricted monads, like the Wadler Set and Bag monads.

There is another interesting monad variation - a parameterized monad - where the monad carries around an additional parameter at the type level such as a type-level set of effects. One really good example of this is the separation logic monad in Hoare Type Theory. The pre- and post- conditions can be viewed as the parameter carried around on that monad. Wadler and Thiemann, Jean-Christophe Filliâtre and others have explore this notion for encoding effects.

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If we take a look at the Haskell (.) operator:


(.) :: (a -> b) -> (e -> a) -> e -> b


and take a moment to reflect on the type of fmap


fmap :: Functor f => (a -> b) -> f a -> f b


and the unnamed Reader monad from Control.Monad.Reader


instance Functor ((->) r)


we see that fmap applied to the Reader functor rederives (.).

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