Output/Input

Regarding IO a, Haskell's monadic I/O type:

Some operations are primitive actions, corresponding to conventional I/O operations. Special operations (methods in the class Monad, see Section 6.3.6) sequentially compose actions, corresponding to sequencing operators (such as the semicolon) in imperative languages.

The Haskell 2010 Report, (page 107 of 329).

So for I/O, the monadic interface merely provides an abstract way to sequence its actions. However there is another, more direct approach to sequencing:

Control.Parallel.pseq :: a -> b -> b

(as opposed to the non-sequential Prelude.seq.) That means a more direct way of preserving referential transparency is also needed. For simple teletype I/O:

data OI
partOI  :: OI -> (OI, OI)
getChar :: OI -> Char
putChar :: Char -> OI -> ()

where:

OI isn't an ordinary Haskell type - ordinary Haskell types represent values without (externally-visible) side-effects, hence OI being abstract.

The action partOI is needed because each OI value can only be used once.

The action getChar obtains the the next character of input.

The function putChar expects a character, and returns an action which will output the given character.

Now for a few other I/O interfaces - if seq was actually sequential:

monad

type M a   =  OI -> a

unit       :: a -> M a
unit x     =  \ u -> let !_ = partOI u in x 

bind       :: M a -> (a -> M b) -> M b
bind m k   =  \ u -> let !(u1, u2) = partOI u in
                     let !x = m u1 in
                     let !y = k x u2 in
                     y

getcharM   :: M Char
getcharM   =  getChar

putcharM   :: Char -> M () 
putcharM   =  putChar

comonad:

type C a         =  (OI, a)

extract          :: C a -> a
extract (u, x)   =  let !_ = partOI u in x

duplicate        :: C a -> C (C a)
duplicate (u, x) =  let !(u1, u2) = partOI u in
                    (u2, (u1, x))

extend           :: (C a -> b) -> C a -> C b
extend h (u, x)  =  let !(u1, u2) = partOI u in
                    let !y        = h (u1, x) in
                    (u2, y)

getcharC         :: C () -> Char
getcharC (u, ()) =  getChar u

putcharC         :: C Char -> ()
putcharC (u, c)  =  putChar c u

arrow:

type A b c   =  (OI -> b) -> (OI -> c)

arr          :: (b -> c) -> A b c
arr f        =  \ c' u -> let !x = c' u in f x

both         :: A b c -> A b' c' -> A (b, b') (c, c')
f' `both` g' =  \ c' u -> let !(u1:u2:u3:_) = partsOI u in
                          let !(x, x')      = c' u1 in
                          let !y            = f' (unit x) u2 in
                          let !y'           = g' (unit x') u3 in
                          (y, y')
                where
                  unit x u = let !_ = partOI u in x

getcharA     :: A () Char
getcharA     =  \ c' u -> let !(u1, u2) = partOI u in
                          let !_        = c' u1 in
                          let !ch       = getChar u2 in
                          ch     

putcharA     :: A Char ()
putcharA     =  \ c' u -> let !(u1, u2) = partOI u in
                          let !ch       = c' u1 in
                          let !z        = putChar ch u2 in
                          z

The OI interface can also be used to implement I/O models used in earlier versions of Haskell:

dialogues :

runD :: ([Response] -> [Request]) -> OI -> ()
runD d u = foldr (\ (!_) -> id) () $ yet $ \ l -> zipWith respond (d l) (partsOI u)

yet :: (a -> a) -> a
yet f = f (yet f)

respond :: Request -> OI -> Response
respond Getq     u = let !c = getChar u in Getp c
respond (Putq c) u = let !_ = putChar c u in Putp

data Request  = Getq | Putq Char
data Response = Getp Char | Putp

continuations:

type Answer = OI -> ()

runK :: Answer -> OI -> ()
runK a u = a u

doneK :: Answer
doneK = \ u -> let !_ = partOI u in ()

getcharK :: (Char -> Answer) -> Answer
getcharK k   = \ u -> let !(u1, u2) = partOI u in
                      let !c        = getChar u1 in
                      let !a        = k c in
                      a u2

putcharK :: Char -> Answer -> Answer
putcharK c a = \ u -> let !(u1, u2) = partOI u in
                      let !_        = putChar c u1 in
                      a u2

...and even that ~~world~~ state-passing style used in GHC, and by Clean, Single-Assignment C and as part of the I/O model used for the verification of interactive programs in CakeML, remembering that OI values can only be used once:

newtype World = W OI

getcharL :: World -> (Char, World)
getcharL (W u) = let !(u1, u2) = partOI u in
                 let !c = getChar u1 in
                 (c, W u2)

putcharL :: Char -> World -> World
putcharL c (W u) = let !(u1, u2) = partOI u in
                   let !_ = putChar c u1 in
                   W u2

(Rewriting those examples to use pseq is left as an exercise.)