Why are monad transformers different to stacking monads?

Question

In many cases, it isn't clear to me what is to be gained by combining two monads with a transformer rather than using two separate monads. Obviously, using two separate monads is a hassle and can involve do notation inside do notation, but are there cases where it just isn't expressive enough?

One case seems to be StateT on List: combining monads doesn't get you the right type, and if you do obtain the right type via a stack of monads like Bar (where Bar a = (Reader r (List (Writer w (Identity a))), it doesn't do the right thing.

But I'd like a more general and technical understanding of exactly what monad transformers are bringing to the table, when they are and aren't necessary, and why.

To make this question a little more focused:

1. What is an actual example of a monad with no corresponding transformer (this would help illustrate what transformers can do that just stacking monads can't).
2. Are StateT and ContT the only transformers that give a type not equivalent to the composition of them with m, for an underlying monad m (regardless of which order they're composed.)

(I'm not interested in particular implementation details as regards different choices of libraries, but rather the general (and probably Haskell independent) question of what monad transformers/morphisms are adding as an alternative to combining effects by stacking a bunch of monadic type constructors.)

(To give a little context, I'm a linguist who's doing a project to enrich Montague grammar - simply typed lambda calculus for composing word meanings into sentences - with a monad transformer stack. It would be really helpful to understand whether transformers are actually doing anything useful for me.)

Thanks,

Reuben

Answer 1

To answer you question about the difference between Writer w (Maybe a) vs MaybeT (Writer w) a, let's start by taking a look at the definitions:

newtype WriterT w m a = WriterT { runWriterT :: m (a, w) }
type Writer w = WriterT w Identity

newtype MaybeT m a = MaybeT { runMaybeT :: m (Maybe a) }

Using ~~ to mean "structurally similar to" we have:

Writer w (Maybe a)  == WriterT w Identity (Maybe a)
                    ~~ Identity (Maybe a, w)
                    ~~ (Maybe a, w)

MaybeT (Writer w) a ~~ (Writer w) (Maybe a)
                    == Writer w (Maybe a)
                    ... same derivation as above ...
                    ~~ (Maybe a, w)

So in a sense you are correct -- structurally both Writer w (Maybe a) and MaybeT (Writer w) a are the same - both are essentially just a pair of a Maybe value and a w.

The difference is how we treat them as monadic values. The return and >>= class functions do very different things depending on which monad they are part of.

Let's consider the pair (Just 3, []::[String]). Using the association we have derived above here's how that pair would be expressed in both monads:

three_W :: Writer String (Maybe Int)
three_W = return (Just 3)

three_M :: MaybeT (Writer String) Int
three_M = return 3

And here is how we would construct a the pair (Nothing, []):

nutin_W :: Writer String (Maybe Int)
nutin_W = return Nothing

nutin_M :: MaybeT (Writer String) Int
nutin_M = MaybeT (return Nothing)   -- could also use mzero

Now consider this function on pairs:

add1 :: (Maybe Int, String) -> (Maybe Int, String)
add1 (Nothing, w) = (Nothing w)
add1 (Just x, w)  = (Just (x+1), w)

and let's see how we would implement it in the two different monads:

add1_W :: Writer String (Maybe Int) -> Writer String (Maybe Int)
add1_W e = do x <- e
             case x of
               Nothing -> return Nothing
               Just y  -> return (Just (y+1))

add1_M :: MaybeT (Writer String) Int -> MaybeT (Writer String) Int
add1_M e = do x <- e; return (e+1)
  -- also could use: fmap (+1) e

In general you'll see that the code in the MaybeT monad is more concise.

Moreover, semantically the two monads are very different...

MaybeT (Writer w) a is a Writer-action which can fail, and the failure is automatically handled for you. Writer w (Maybe a) is just a Writer action which returns a Maybe. Nothing special happens if that Maybe value turns out to be Nothing. This is exemplified in the add1_W function where we had to perform a case analysis on x.

Another reason to prefer the MaybeT approach is that we can write code which is generic over any monad stack. For instance, the function:

square x = do tell ("computing the square of " ++ show x)
              return (x*x)

can be used unchanged in any monad stack which has a Writer String, e.g.:

WriterT String IO
ReaderT (WriterT String Maybe)
MaybeT (Writer String)
StateT (WriterT String (ReaderT Char IO))
...

But the return value of square does not type check against Writer String (Maybe Int) because square does not return a Maybe.

When you code in Writer String (Maybe Int), you code explicitly reveals the structure of monad making it less generic. This definition of add1_W:

add1_W e = do x <- e 
              return $ do 
                y <- x 
                return $ y + 1

only works in a two-layer monad stack whereas a function like square works in a much more general setting.

Answer 2

What is an actual example of a monad with no corresponding transformer (this would help illustrate what transformers can do that just stacking monads can't).

IO and ST are the canonical examples here.

Are StateT and ContT the only transformers that give a type not equivalent to the composition of them with m, for an underlying monad m (regardless of which order they're composed.)

No, ListT m a is not (isomorphic to) [m a]:

newtype ListT m a =
  ListT { unListT :: m (Maybe (a, ListT m a)) }

Answer 3

To make this question a little more focused:

1. What is an actual example of a monad with no corresponding transformer (this would help illustrate what transformers can do that just stacking monads can't).

There are no known examples of a monad that lacks a transformer, as long as the monad is defined explicitly as a pure lambda-calculus term, with no side effects and no external libraries being used. The Haskell monads such as IO and ST are essentially interfaces to an external library defined by low-level code. Those monads cannot be defined by pure lambda-calculus, and their monad transformers probably do not exist.

Even though there are no known explicit examples of monads without transformers, there is also no known general method or algorithm for obtaining a monad transformer for a given monad. One could define an arbitrarily complicated type constructor that combines products, co-products, and function types, for example like this code in Haskell:

 type D a = Either a ((a -> Bool) -> Maybe a)

One can implement the monad's methods for D and prove that the monad laws hold, but it is far from obvious how to define a transformer for the monad D.

This D may be a contrived and artificial example of a monad, but there might be legitimate cases for using that monad, which is a "free pointed monad on the Search monad on Maybe".

To clarify: A "search monad on n" is the type S n q a = (a -> n q) -> n a where n is another monad and q is a fixed type.

A "free pointed monad on M" is the type P a = Either a (M a) where M is another monad.

In any case, I just want to illustrate the point. I don't think it would be easy for anyone to come up with the monad transformer for D and then to prove that it satisfies the laws of monad transformers. There is no known algorithm that would takes an arbitrary monad's code, such as the code of D, and then generate the code for its transformer.

2. Are StateT and ContT the only transformers that give a type not equivalent to the composition of them with m, for an underlying monad m (regardless of which order they're composed.)

Monad transformers are necessary because stacking two monads is not always a monad. Most "simple" monads, like Reader, Writer, Maybe, etc., stack with other monads in a particular order. But the result of stacking, say, Writer + Reader + Maybe, is a more complicated monad that no longer allows stacking with new monads.

There are several examples of monads that do not stack at all: State, Cont, List, Free monads, the Codensity monad, and a few other, less well known monads, like the "free pointed" monad shown above.

For each of those "non-stacking" monads, one needs to guess the correct monad transformer somehow.

I have studied this question for a while and I have assembled a list of techniques for creating monad transformers, together with full proofs of all laws. There doesn't seem to be any system to creating a monad transformer for a specific monad. I even found a couple of monads that have two inequivalent transformers.

Generally, monad transformers can be classified in 6 different families:

1. Functor composition in one or another order: EitherT, WriterT, ReaderT and a generalization of Reader to a special class of monads, called "rigid" monads. An example of a "rigid" monad is Q a = (H a) -> a where H is an arbitrary (but fixed) contravariant functor.

2. The "adjunction recipe": StateT, ContT, CodensityT, SearchT, which gives transformers that are not functorial.

3. The "recursive recipe": ListT, FreeT

4. Cartesian product of monads: If M and N are monads then their Cartesian product, type P a = (M a, N a) is also a monad whose transformer is the Cartesian product of transformers.

5. The free pointed monad: P a = Either a (M a) where M is another monad. For that monad, the transformer's type is m (Either a (MT m a)) where MT is the monad M's transformer.

6. Monad stacks, that is, monads obtained by applying one or more monad transformers to some other monad. A monad stack's transformer is build via a special recipe that uses all the transformers of the individual monads in the stack.

There may be monads that do not fit into any of these cases, but I have seen no examples so far.

Details and proofs of these constructions of monad transformers are in my draft book here https://github.com/winitzki/sofp