Conal asked me to write some thoughts on the `IO`

monad.
He has not seen a proof that `IO`

is a monad and was wondering if it really is.
I do not think I can prove that `IO`

is a monad, since I do not know exactly what `IO`

is, but I think I can provide some evidence that `IO`

is a monad by giving a partial model of `IO`

that is a monad.

Wait a minute! Look up at the title. Doesn’t the title say that I/O is not a monad?
Well, I want to distinguish between I/O that is model of input and output, and the abstract data type constructor `IO`

.

The story you are about to read is a fib, but it is short. The names are made up, but the problems are real. It starts on Sunday, March 1^{st} 1992. The Haskell 1.2 report was released. Back then, Haskell had the following mechanism for I/O

type Dialogue = [Response] -> [Request] main :: Dialogue

One’s `main`

program would create a lazy list of system requests.
`Request`

was an algebraic data type that had a series of constructors such
as

data Request = ReadChan String | AppendChan String String | GetArgs ... stdin, stdout :: String stdin = "stdin" stdout = "stdout"

For each request made, the run time system would preform some I/O effect and return a `Response`

. The `main`

function could access this response by reading it from the lazy list parameter. The possible responses were also an algebraic data type.

data Response = Success | Str String | StrList String | Bn Bin | Failure IOError

There are some advantages of this model of I/O. Because the type of `main`

is
a concrete data type, it is easier to understand what the model of I/O is in Haskell 1.2.
However this model of I/O is actually pretty bad.
When writing your code in Haskell 1.2, you need to make sure that you always consume
a `Response`

for each `Request`

you make.
If you ever become out of sync, problems arise.
If you do not consume a `Response`

after you make `Request`

you will end up
reading the wrong response for the next `Request`

.
If you consume a `Response`

before making your `Request`

, I expect your code
would deadlock.

By most accounts, this was a pretty awful way of interacting with I/O.
The problem is that nothing in the type of `Dialogue`

enforces that the first `n` + 1 elements of the `Requests`

list should depend on exactly the first `n` elements of the `Response`

list. Fortunately, we can make a new data type that captures this invariant.

data Dialogue1 = Stop | Go Request (Response -> Dialogue1)

Our new `Dialogue1`

type now exactly captures the notion of a stream transformer
from `[Response]`

to `[Request]`

that satisfies our invariant. Notice that
the data structure for `Dialogue`

is a tree holding a `Request`

at each node
and has branch for each `Response`

.

We can write an interpreter from `Dialogue1`

to `Dialogue`

intepret1 :: Dialogue1 -> Dialogue intepret1 Stop _ = [] intepret1 (Go req cont) resps = req:rest resps where rest (hresp:tresp) = intepret1 (cont hresp) tresp

We can do even better.
Certain requests only produce certain responses, so we can refine our `Dialogue1`

data type to only respond appropriately to requests by adding a constructor for each
`Request`

.

data Dialogue2 = Stop2 | ReadChan2 String (Either IOError String -> Dialogue2) | AppendChan2 String String (Either IOError () -> Dialogue2) | GetArgs2 ( [String] -> Dialogue2) ...

The `IOError`

s could even be refined if you desired.
We have the obvious interpretation from `Dialogue2`

to `Dialogue1`

.

intepret2 :: Dialogue2 -> Dialogue1 intperet2 Stop2 = Stop intperet2 (ReadChan2 s cont) = Go (ReadChan s) rest where rest (Str resp) = cont (Right resp) rest (Failure err) = cont (Left err) intperet2 (AppendChan2 s1 s2 cont) = Go (AppendChan s1 s2) rest where rest Success = cont (Right ()) rest (Failure err) = cont (Left err) intperet2 (GetArgs2 cont) = Go (GetArgs) rest where rest (StrList sl) = cont sl

After this was developed Haskell 1.2.5 was released (not really) with this new `Dialogue2`

type for `main`

and things were much better. It was now impossible
for one’s requests and responses to become out of sync, and you only needed to handle
responses that were possible for your requests.

Notice that there is no monad anywhere.
There are not even any type constructors around.
This is why I/O is not a monad.
**A model of I/O is a data type, not a data type constructor**.

What we can do is write a data type constructor along with some combinators to make
it easier to build this `Dialogue2`

type.
Let us call this new type constructor `IO`

.

data IO a = StopIO a | ReadChanIO String (Either IOError String -> IO a) | AppendChanIO String String (Either IOError () -> IO a) | GetArgsIO ( [String] -> IO a) ...

All we have done is add a parameter to `Stop`

to hold a polymorphic value.
The resulting data type `IO ()`

, the type of `main`

in Haskell 1.3, is isomorphic to `Dialogue2`

(I will not even bother writing out the isomorphism).

We can write combinators to build `IO a`

out of smaller pieces, with the ultimate
goal of building an `IO ()`

value.

instance Monad IO where return a = StopIO a (StopIO a) >>= f = f a (ReadChanIO s cont) >>= f = ReadChanIO s (\resp -> cont resp >>= f) (AppendChanIO s1 s2 cont) >>= f = AppendChanIO s1 s2 (\resp -> cont resp >>= f) (GetArgsIO cont) >>= f = GetArgsIO (\resp -> cont resp >>= f)

These combinators satisfy the monad laws, so this data type constructor `IO`

is a
monad, even though the type modeling input and output, which is `IO ()`

, is not. In fact, this monad is a free monad. That is why I call this the free monad model of `IO`

.

One might want to identify different values of `IO ()`

(or `Dialogue2`

) as representing equivalent I/O. Such an equivalence relation should lift (naturally?) to an
equivalence relation on `IO a`

and that would produce a non-free monad model of `IO`

. However, `IO`

would still be a monad.

This idea of generalizing data types into monads is useful in general. I have used it in the past when writing an assembler.

newtype AssemblyCodeMonad a = AssemblyCodeMonad (RWS () [Instruction Register] Location a) deriving (Monad, MonadFix) type AssemblyCode = AssemblyCodeMonad ()

Exercise for the reader: Extend `Dialogue2`

to support concurrency.
Then check to see if this can still be generalized to a monad in the same way that we
have done for sequential I/O.