Have you every wondered why monads turn out to be the abstraction behind IO? To find an answer we will build a two (very incomplete) models for IO:

• one that is very common in imperative languages, based on a sequence of IO actions
• a recursive one (hence in a functional programming style), that turns out to be monadic.
{-# LANGUAGE GADTs #-}
module MonadicIO where
import           Control.Monad (ap)
import qualified System.IO as IO

Monoidal IO

data IOAction x
= Write  FilePath String
| Return x

Mostly found in imperative languages, where IO is a sequence of operations. And where Return has the semantics of a final statement. There are many algebras that provide a sequences. The most general one we could pick are non associative semigroups (also called magmas). It would have two problems:

• since it’s non associative we could interpret sequences depending on the bracketing, but this is too much freedom for us. We want that all expression build by putting brackets differently have always the same semantics;
• it’s not strictly necessary but, having a unit for the binary operation might be convenient

For that reasons, we will use associative unital magmas, e.g. a monoid. The good choice should be the most general such object, i.e. a free monoid

type MonoidalIO x = [IOAction x]

For a side note: DList is a free monoid, while [] is free in the class of left strict monoids, e.g. monoids satisfying: undefined <> == undefined, but let’s not focus on these differences here …

Let us provide a way to actually run MonoidalIO, since we are in Haskell let us interpret MonoidalIO in the IO monad.

runMonoidalIO :: MonoidalIO x -> IO x
runMonoidalIO (Return x : _)            = return x
runMonoidalIO ((Write path str) : next) =
IO.writeFile path str >> runMonoidalIO next
runMonoidalIO ((Read path) : next)      =
runMonoidalIO [] = error "error: no action"

There is yet another way of organising a sequence of computations. And it is especially compelling in a language with algebraic data types.

In a recursive style we can describe the whole program progression using a single recursive data structure, where each computation carries a continuation.

data MonadicIO x
= WriteM FilePath String (MonadicIO x)
| ReturnM x
instance Functor MonadicIO where
fmap f (ReturnM x)      = ReturnM (f x)
fmap f (ReadM path io)  = ReadM path ((fmap . fmap) f io)
fmap f (WriteM path str io) = WriteM path str (fmap f io)

We can transform any MonoidalIO into MonadicIO.

-- | transform MonoidalIO into MonadicIO
--
fromMonoidalIO :: MonoidalIO x -> MonadicIO x
fromMonoidalIO ((Read path) : next)      = ReadM path (\_ -> fromMonoidalIO next)
fromMonoidalIO ((Write path str) : next) = WriteM path str (fromMonoidalIO next)
fromMonoidalIO (Return x : _)            = ReturnM x
fromMonoidalIO []                        = error "error: no action"

We cannot transform MonadicIO to MonoidalIO, only because we did not provide a way to bind data read from a file in MonoidalIO just for simplicity of the presentation. But the two approaches should be equivalent.

We also need a way to run MonadicIO, again since we are in Haskell we’ll provide a map to IO:

runMonadicIO :: MonadicIO x -> IO x
runMonadicIO (ReturnM x)          = return x
runMonadicIO (WriteM path str io) = IO.writeFile path str >> runMonadicIO io

But this allows only to run expressions of type MonadicIO x, we still need a way to run expressions of type MonadicIO (MonadicIO (... x)). This proves that the MonadicIO can be run (if we’d end up with a functor that would not have a natural transformation to Haskell’s IO we’d be in troubles)

In MonoidalIO we relied on associativity of the list concatenation, a similar requirements is needed here. We want that the end result is independent of how it was build using >>= or equivalently how we join a value of type MonadicIO (MonadicIO (MonadicIO a) into MonadicIO a). If we have an expression of type x :: MonadicIO (MonadicIO (MonadicIO x)) there are two ways of running it, by using of the two maps:

joinMonadicIO :: MonadicIO (MonadicIO x) -> MonadicIO x
joinMonadicIO (ReadM path io)    = ReadM path (joinMonadicIO . io)
assoc1 :: MonadicIO (MonadicIO (MonadicIO x)) -> MonadicIO x
assoc1 = joinMonadicIO . joinMonadicIO

or

assoc2 :: MonadicIO (MonadicIO (MonadicIO x)) -> MonadicIO x
assoc2 = joinMonadicIO . fmap joinMonadicIO

We really want both assoc1 and assoc2 to be equal, what guarantees that the way we build an expression of type MonadicIO x does not matter. This is exactly the associativity law for monads. And indeed MonadicIO is a monad, and joinMonadicIO is its join operator. This is in a tight analogy to the associativity law of monoids in MonoidalIO.

In Haskell we are more accustomed with the monadic bind operator >>= to build a monadic expression of type m b from m a and a continuation a -> m b. There are two ways to build m d from ma :: m a, fab :: a -> m b and fbc :: b -> m c:: c -> m d:

• either ma >>= fab >>= fbc
• or ma >>= (\a -> fab a >>= fbc)

Associativity for >>= tells us that these two are equal. This is equivalent with associativity of join which we expressed above in the form

(join . join == join . fmap join) :: m (m (m a)) -> m a

Note that associativity of >>= bind expresses the associativity of building a monadic expression, while join expresses associativity of assembling it from m (m (m a). These two are equivalent: each of the associativity law implies the other one under the inverse correspondence: ma >>= f = join \$ fmap f ma (e.g. each bind builds up m (m a), but then it joins it into m a); the inverse is join = (>>= id).

instance Applicative MonadicIO where
pure  = ReturnM
(<*>) = ap

return = ReturnM
ReturnM x >>= f          = f x
WriteM path str io >>= f = WriteM path str (io >>= f)
ReadM path io >>= f      = ReadM path (fmap (>>= f) io)

GHC IO

Let me note, GHC’s IO monad is build differently; to give much more flexibility for building IO actions for many different operations supported by many different platforms. In the recursive style we need to built in all the operations that are possible to run. This would be too restrictive for a general purpose language. And also for performance reasons its much better to build IO from native types (e.g. lambdas and types that require MagicHash). This avoids memory overhead of terms of kind Type (to be precise: allocation rate in case of running a recursive type). Haskell abstracts over a state monad, e.g. a type s -> (s, a) (where s is a state of the world), but it is still a monad, and monad laws guarantee that the semantic of an expression is independent of bracketing of bind (and join) operators. It is also a recursive type, though the recursiveness is hidden in the monadic join.

Conclusions

By using a recursive IO, e.g. a usual functional style programming, we end up with a type that satisfies monad laws. The monad associativity guarantees that when we build the expression using do notation (e.g. >>= or join) the bracketing will not change the semantics of an expression.

At last let us point out some benefits of recursive/monadic IO:

• much easier to support the return values of actions, e.g. in MonoidalIO we did not have access to data read from a file. In a functional language, recursive / monadic IO does not require any thing more than lambdas to bind the return value.
• values returned by IO operations are trapped inside the MonoidalIO monad. This gives a clear indication which functions have access to IO and which are pure.
• For a lasy language using a recursive data type it is a compelling choice. In Haskell, the evaluation is guided by pattern matching, every bind (as you can consult above), evaluates just a single layer of a computation.
• Finally, functional programming has a deep connection with the roots of computability and logic, which are closely related to Church’s lambda calculus and recursive functions, through the work of Gödel.