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=== <u>Previously seen</u> === |
=== <u>Previously seen</u> === |
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− | The type <code>() -> a</code> (or variations of it) have appeared elsewhere: |
+ | The type <code>() -> a</code> (or variations of it) have appeared elsewhere - examples include: |
* page 2 of 13 in [https://fi.ort.edu.uy/innovaportal/file/20124/1/22-landin_correspondence-between-algol-60-and-churchs-lambda-notation.pdf A Correspondence Between ALGOL 60 and Church's Lambda-Notation: Part I] by Peter Landin: |
* page 2 of 13 in [https://fi.ort.edu.uy/innovaportal/file/20124/1/22-landin_correspondence-between-algol-60-and-churchs-lambda-notation.pdf A Correspondence Between ALGOL 60 and Church's Lambda-Notation: Part I] by Peter Landin: |
Revision as of 07:29, 21 December 2021
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:
- page 2 of 13 in A Correspondence Between ALGOL 60 and Church's Lambda-Notation: Part I by Peter Landin:
The use of
λ
, and in particular (to avoid an irrelevant bound variable) ofλ()
, 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
- page 3 of Assignments for Applicative Languages by Vipin Swarup, Uday S. Reddy and Evan Ireland:
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 type𝜏
. [...] An observer typeObs 𝜏
may be viewed as an implicit function space from the set of states to the type𝜏
.type Obs tau = State -> tau
- page 15 of Non-Imperative Functional Programming by Nobuo Yamashita:
type a :-> b = OI a -> b
- MTL style for free by Tom Ellis:
data Time_ a = GetCurrentTime (UTCTime -> a)
- An impure lazy programming language, also by Tom Ellis:
data IO a = IO (() -> a)
- page 2 of Unique Identifiers in Pure Functional Languages by Péter Diviánszky:
[...] The type
Id
can be hidden by the synonym data type:: Create a :== Id -> a
type Create a = Id -> a
- page 7 of Functional Reactive Animation by Conal Elliott and Paul Hudak:
An early implementation of Fran represented behaviors as implied in the formal semantics:
data Behavior a = Behavior (Time -> a)
- page 26 of How to Declare an Imperative by Philip Wadler:
The type
'a io
is represented by a function expecting a dummy argument of typeunit
and returning a value of type'a
.type 'a io = unit -> a
type Io a = () -> a
- The Haskell I/O Tutorial by Albert Lai:
But I can already tell you why we cannot follow other languages and use simply
X
or() -> X
.
- Free Monads for Less (Part 3 of 3): Yielding IO by Edward Kmett:
newtype OI a = forall o i. OI (FFI o i) o (i -> a) deriving Functor
type Oi a = forall i . i -> a
- page 27 of Purely Functional I/O in Scala by Rúnar Bjarnason:
class IO[A](run: () => A)
class Io a where run :: () -> a
Let's say you want to implement
IO
in SML :structure Io : MONAD = struct type 'a t = unit -> 'a ⋮ end
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 allows for 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"?
- No:
- 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.
- No:
- 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.