It's a bird… It's a plane… It's a free monad!

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20 March 2019 |

|The subject of free monads is resurfacing of late, as it does from time to time. What prompted this write-up is that Sandy Maguire followed up his post with a discussion about an impredicative encoding (aka final encoding) of free monads:

```
newtype Freer f a = Freer (forall m. Monad m => (forall t. f t -> m t) -> m a)
```

That is: given a monad `m`

, and an interpretation of my operations
(`f`

) in that monad, I can build an `m a`

.

As it happens, capabilities-as-type-classes, as embodied by the capability library, are essentially the same thing. Let me explain.

As far as I know, the subject of impredicative encoding of free monads was first tackled, as many good things, by Russell O'Connor, who calls them van Laarhoven free monads. His blog post is a fairly mathy read. But the key bit is this:

```
-- (ops m) is required to be isomorphic to (Π n. i_n -> m j_n)
newtype VLMonad ops a = VLMonad { runVLMonad :: forall m. Monad m => ops m -> m a }
```

Where `Π n. i_n -> m j_n`

is mathematician's way of saying `forall n. i n -> m (j n)`

. Up to a very small difference (rename `i`

to `f`

and pick the identity for `j`

), this is indeed the same type. O'Connor
then proves that this type is isomorphic to the usual presentation of
free monads.

```
data Free f a = Return a | Free (f (Free f a))
```

The comment, in Russell O'Connor's snippet, is crucial in the
proof. Without it you can't establish the isomorphism between
`VLMonad`

and the traditional `Free`

monad.

That's because not all effects can be represented in free monads. The prime example is exception handlers. You can make a function

```
handle :: Free MyEffect a -> Free MyEffect a -> Free MyEffect a
```

But it would have the property that `(handle s f) >>= k = handle (s >>= k) (f >>= k)`

.
That is: exceptions raised after exiting the handler
would still be caught by the handler. It is not a useless function,
but it is not an exception handler. This phenomenon is a property of
the free monad construction. In the impredicative encoding, it can be
thought of as a consequence of the fact that in `forall a. f a -> m a`

, `f`

cannot refer to `m`

.

So, while being isomorphic to `Free`

is a nice theoretical property,
Russell O'Connor's phrasing presents us with an opportunity: if we
simply drop the comment restricting the form of `ops`

, we get a less
constrained free monad type which supports more effects. Therefore, we
won't pay much attention to it at all in the rest of the post.

Functions in `VLMonad`

will look, as functions in a monad do, like

```
somefunction :: A -> VLMonad Ops B
```

Where `Ops`

represent the possible effects. For instance, if you need
a state effect, you would define `Ops`

as

```
data Ops m = Ops
{ put :: Int -> m ()
, get :: m Int
}
```

But, after all, `VLMonad`

is simply a newtype: we could very well
inline its definition.

```
somefunction :: A -> forall m. Monad m => Ops m -> m B
```

The ordering of types and argument is not too idiomatic, though. So let's rearrange them a little:

```
somefunction :: Monad m => Ops m -> A -> m B
```

If this may look like a familiar style of structuring effect, it is. See for instance the style advertised by Éric Torreborre in a recent blog post. It's not really so much an alternative to free monads as a different presentation of free monads (albeit slightly more liberal, if, as per previous section, we ignore the restriction from O'Connor's comment).

Personally, I find it rather tiresome to explicitly carry around the
capabilities (the `Ops`

thing) at every function call. I'd rather keep
my function arguments for the program logic, and leave all the
plumbing to the monad. Therefore, I turn `Ops`

into a type class, and
move it “left of the fat arrow”: really in Haskell `A -> B`

and `A => B`

mean the same thing, they only differ in whose responsibility it is
to pass the arguments around.

```
somefunction :: (Monad m, Ops m) => A -> m B
```

The definition of `Ops`

for a state effect would, then, become

```
class Ops m where
put :: Int -> m ()
get :: m Int
```

This is precisely the style of programming supported by the capability library (see also our announcement blog post).

Free monads, capabilities-as-records and capabilities-as-type-classes, are, essentially, three different flavours of the same thing (with free monads technically being the more restrictive of the three, as it can't have exception handling effects).

Choosing between the three is, ultimately, a matter of taste. I really like capabilities-as-type-classes because it pushes the boilerplate outside of the domain logic.

At the end of the day, what really matters is the core
idea shared by these
three approaches: *capabilities
should be expressed in terms of the domain logic, not in terms of
implementation details*.