# Partibles for composing monads

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Having praised monads to the hilt, let me level one criticism. Monads tend to be
an all-or-nothing proposition. If you discover that you need interaction deep within
your program, you must rewrite that segment to use a monad. If you discover
that you need two sorts of interaction, you tend to make a single monad support
both sorts. It seems to me that instead we should be able to move smoothly from
no monads (no interactions) to one monad (a single form of interaction) to many
monads (several independent forms of interactions). How to achieve this remains a
challenge for the future.

Some sample definitions:

```        class Partible a where
part  :: a -> (a, a)
parts :: a -> [a]

-- Minimal complete definition: part or parts
part u  = case parts u of u1:u2:_ -> (u1, u2)
parts u = case part u of (u1, u2) -> u1 : parts u2

instance Partible a => Monad ((->) a) where
return x = \ u -> part u `seq` x
m >>= k  = \ u -> case part u of (u1, u2) -> (\ x -> x `seq` k x u2) (m u1)
m >> w   = \ u -> case part u of (u1, u2) -> m u1 `seq` w u2
fail s   = \ u -> part u `seq` error s

data OI  -- abstract
primPartOI :: OI -> (OI, OI)  -- primitive
-- type IO a = OI -> a

instance Partible OI where part = primPartOI

-- more primitives
primGetChar :: OI -> Char
primPutChar :: Char -> OI -> ()

-- copy 'n' paste from Wadler's paper
type Dialogue = [Response] -> [Request]
data Request  = Getq | Putq Char
data Response = Getp Char | Putp

respond :: Request -> OI -> Response
respond Getq     = primGetChar >>= return . Getp
respond (Putq c) = primPutChar c >> return Putp

runDialogue :: Dialogue -> OI -> ()
runDialogue d =
\ u -> foldr seq () (yet (\ l -> zipWith respond (d l) (parts u)))

instance Partible a => MonadFix ((->) a) where
mfix m = \ u -> yet (\ x -> m x u)

-- to be made into an abstract data type...
data Fresh a = Fresh (OI -> a) OI

afresh :: (OI -> a) -> OI -> Fresh a
afresh g u = Fresh g u

instance Partible (Fresh a) where
parts (Fresh g u) = [ Fresh g v | v <- parts u ]

fresh :: Fresh a -> [a]
fresh (Fresh g u) = [ g v | v <- parts u ]

instance Functor Fresh where
fmap f (Fresh g u) = Fresh (f . g) u

-- another primitive
primGensym :: OI -> Int

supplyInts :: OI -> Fresh Int
supplyInts = \ u -> afresh primGensym u

-- another abstract data type
data Throw e
curb  :: (Throw e -> a) -> (e -> OI -> a) -> OI -> a
catch :: (Throw e -> a) -> (e -> Throw e -> a) -> Throw e -> a
throw :: e -> Throw e -> a

partThrow :: Throw e -> (Throw e, Throw e)
instance Partible (Throw e) where part = partThrow

instance (Partible a, Partible b) => Partible (a, b) where
parts (u, v) = zipWith (,) (parts u) (parts v)

instance (Partible a, Partible b) => Partible (Either a b) where
parts (Left u)  = map Left (parts u)
parts (Right v) = map Right (parts v)

data Some a = Only a | More a (Some a)

instance Partible a => Partible (Some a) where
parts (Only u)    = map Only (parts u)
parts (More u us) = zipWith More (parts u) (parts us)

type M1 a = (Fresh Int, OI) -> a
type M2 a = Either (Fresh a) OI -> a
type M3 a = Some (Either (Fresh Char) (Fresh Int)) -> a
type M4 a = (Throw IOException, Some (Either Float OI)) -> a
-- ...whatever suits the purpose

class (Monad m1, Monad m2) => MonadCommute m1 m2 where
mcommute :: m1 (m2 a) -> m2 (m1 a)

instance (Partible a, Partible b) => MonadCommute ((->) a) ((->) b) where
mcommute m = \ v u -> m u v
```

So what qualifies as being partible?

A partible value can be used only once to generate new values that can be used for the same purpose. Think of a very large sheet of paper - new sheets can be made from it, other sheets can be made from those, etc, with the original sheet no longer in existence. Unlike paper sheets, partible values are intended to have no limits e.g. the result of applying `supplyInts`.

If its violation causes a runtime error, the use-once property of partible values can help to maintain referential transparency in the effectful segments of a program; using another example from Wadler's paper minimally rewritten in Haskell syntax using `OI` values:

```         \ u -> let
x = (primPutChar 'h' u `seq` primPutChar 'a' u)
in x `seq` x
```

would trigger the error; the working version being:

```                let
x = (\ v -> case part v of
(v1, v2) -> primPutChar 'h' v1 `seq` primPutChar 'a' v2)
in
\ u -> case part u of
(u1, u2) -> x u1 `seq` x u2
```

...rather tedious, if it weren't for Haskell's standard monadic methods:

```                let
x = primPutChar 'h' >> primPutChar 'a'
in x >> x
```

Higher-order functions allows the manipulation of control e.g. `Prelude.until` in Haskell. As the definition of `runDialogue` shows, monadic types with visible definitions based on types of partible values may also allow the manipulation of control in ways beyond what the standard monadic methods provide.

The patches for an initial implementation in GHC are available:

Other references and articles:

• Functional Pearl: On generating unique names, Lennart Augustsson, Mikael Rittri and Dan Synek.
• Non-Imperative Functional Programming, Nobuo Yamashita.
• Functional I/O Using System Tokens, Lennart Augustsson.
• I/O Trees and Interactive Lazy Functional Programming, Samuel A. Rebelsky.
• Arborescent data structures and lazy evaluation: A new approach to numerical problems, Manuel Carcenac.

See also:

Thank you to those who commented on early drafts of this document.

Atravers 04:31, 10 April 2018 (UTC)