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Clearing away the smoke and mirrors

The implementation in GHC uses the following one:

type IO a  =  World -> (a, World)

An IO computation is a function that (logically) takes the state of the world, and returns a modified world as well as the return value. Of course, GHC does not actually pass the world around; instead, it passes a dummy “token,” to ensure proper sequencing of actions in the presence of lazy evaluation, and performs input and output as actual side effects!

A History of Haskell: Being Lazy With Class, Paul Hudak, John Hughes, Simon Peyton Jones and Philip Wadler.

...so what starts out as an I/O action of type:

World -> (a, World)

is changed by GHC to approximately:

() -> (a, ())

As the returned unit-value () contains no useful information, that type can be simplified further:

() -> a

Why "approximately"? Because "logically" a function in Haskell has no observable effects.

Previously seen

The type () -> a (or variations of it) have appeared elsewhere - examples include:

The use of λ, and in particular (to avoid an irrelevant bound variable) of λ() , to delay and possibly avoid evaluation is exploited repeatedly in our model of ALGOL 60. A function that requires an argument-list of length zero is called a none-adic function.

(\ () -> ) :: () -> a
abstype 'a Job = JOB of unit -> 'a

data Job a = JOB (() -> a)

A value of type Obs 𝜏 is called an observer. Such a value observes (i.e. views or inspects) a state and returns a value of type 𝜏. [...] An observer type Obs 𝜏 may be viewed as an implicit function space from the set of states to the type 𝜏.

type Obs tau = State -> tau
  • page 15 of Non-Imperative Functional Programming by Nobuo Yamashita:
type a :-> b = OI a -> b
data Time_ a = GetCurrentTime (UTCTime -> a)
data IO a = IO (() -> a)

[...] The type Id can be hidden by the synonym data type

:: Create a  :==  Id -> a

type Create a = Id -> a

An early implementation of Fran represented behaviors as implied in the formal semantics:

data Behavior a = Behavior (Time -> a)

The type 'a io is represented by a function expecting a dummy argument of type unit and returning a value of type 'a.

type 'a io = unit -> a

type Io a = () -> a

But I can already tell you why we cannot follow other languages and use simply X or () -> X.

newtype OI a = forall o i. OI (FFI o i) o (i -> a) deriving Functor

type Oi a = forall i . i -> a
class IO[A](run: () => A)

class Io a where run :: () -> a

Let's say you want to implement IO in SML :

structure Io : MONAD =
  type 'a t = unit -> 'a

type T a = () -> a
newtype IO a = IO { runIO :: () -> a }
newtype Supply r a = Supply { runSupply :: r -> a }

As long as we have its special case IO c = () ~> c, we can represent (up to isomorphism) […] a ~> c […]

Of these, it is the implementation of OI a in Yamashita's oi package which is most interesting as its values are monousal - once used, their contents remain constant. This single-use property also appears in the implementation of the abstract decision type described by F. Warren Burton in Nondeterminism with Referential Transparency in Functional Programming Languages.

IO, redefined

Based on these and other observations, a reasonable distillment of these examples would be OI -> a, which then implies:

type IO a = OI -> a

Using Burton's pseudodata approach:

 -- abstract; single-use I/O-access mediator
data Exterior
getchar :: Exterior -> Char
putchar :: Char -> Exterior -> ()

 -- from section 2 of Burton's paper
data Tree a = Node { contents :: a,
                     left     :: Tree a,
                     right    :: Tree a }

 -- utility definitions
type OI  =  Tree Exterior

getChar' :: OI -> Char
getChar' =  getchar . contents

putChar' :: Char -> OI -> ()
putChar' c = putchar c . contents

part     :: OI -> (OI, OI)
parts    :: OI -> [OI]

part t   =  (left t, right t)
parts t  =  let !(t1, t2) = part t in
            t1 : parts t2

Of course, in an actual implementation OI would be abstract like World, and for similar reasons. This permits a simpler implementation for OI and its values, instead of being based on (theoretically) infinite structured values like binary trees. That simplicity has benefits for the OI interface, in this case:

data OI
part :: OI -> (OI, OI)
getChar' :: OI -> Char
putChar' :: Char -> OI -> ()

Various questions

  • Is the C language "purely functional"?
  • C isn't "pure" - it allows unrestricted access to observable effects, including those of I/O.
  • C isn't "functional" - it was never intended to be referentially transparent, which severely restricts the ability to use equational reasoning.
  • Is the Haskell language "purely functional"?
Haskell is not a purely functional language, but is often described as being referentially transparent.
  • Can functional programming be liberated from the von Neumann paradigm?
That remains an open research problem.
  • Can a language be "purely functional" or "denotative"?
Conditionally, yes - the condition being the language is restricted in what domains it can be used in:
  • If a language is free of observable effects, including those of I/O, then the only other place where those effects can reside is within its implementation.
  • There is no bound on the ways in which observable effects can be usefully combined, leading to a similarly-unlimited variety of imperative computations.
  • A finite implementation cannot possibly accommodate all of those computations, so a subset of them must be chosen. This restricts the implementation and language to those domains supported by the chosen computations.
  • Why do our programs need to read input and write output?
Because programs are usually written for practical purposes, such as implementing domain-specific little languages like Dhall.

See also