Difference between revisions of "Partibles for composing monads"
(Extra links, added arrow example) 
m (Corrected indenting in Arrow instance) 

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(***) :: Partible p => A p b c > A p b' c' > A p (b, b') (c, c') 
(***) :: Partible p => A p b c > A p b' c' > A p (b, b') (c, c') 

f' *** g' = \ c' u > let !(u1:u2:u3:_) = parts u in 
f' *** g' = \ c' u > let !(u1:u2:u3:_) = parts 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 
where 

unit x u = case part u of !_ > x 
unit x u = case part u of !_ > x 
Revision as of 22:19, 17 July 2021
Having praised monads to the hilt, let me level one criticism. Monads tend to be an allornothing 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.
 How to Declare an Imperative, Philip Wadler.
_{ } Assuming the partible types being used are appropriately defined, then:
instance Partible a => Monad ((>) a) where
return x = \ u > case part u of !_ > x
m >>= k = \ u > case part u of
(u1, u2) > case m u1 of !x > k x u2
m >> w = \ u > case part u of
(u1, u2) > case m u1 of !_ > w u2
fail s = \ u > case part u of !_ > error s
Furthermore:
instance (Partible a, Monad ((>) a)) => MonadFix ((>) a) where
mfix m = \ u > yet (\ x > m x u)
instance (Partible a, Monad ((>) a), Partible b, Monad ((>) b)) => MonadCommute ((>) a) ((>) b) where
mcommute g = \ v u > g u v
instance (Monad m, Partible b, Monad ((>) b)) => MonadCommute m ((>) b) where
mcommute m = \ v > liftM ($ v) m
where:
yet :: (a > a) > a yet f = f (yet f) class Monad m => MonadFix m where mfix :: (a > m a) > m a class (Monad m1, Monad m2) => MonadCommute m1 m2 where mcommute :: m1 (m2 a) > m2 (m1 a)
Using partible types to define monadic ones can enable an intermediate approach to the use of effects, in contrast to the allornothing proposition of only using the monadic interface. In addition, if the definitions for such monadic types are visible (e.g. as type synonyms), this may also allow the manipulation of control in ways beyond what the monadic interface provides.
Composing arrows
Partible types can also be used to define arrow types:
type A p b c = (p > b) > (p > c)
arr :: Partible p => (b > c) > A p b c
arr f = \ c' u > f $! c' u
infixr 3 ***
(***) :: Partible p => A p b c > A p b' c' > A p (b, b') (c, c')
f' *** g' = \ c' u > let !(u1:u2:u3:_) = parts 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 = case part u of !_ > x
(...though most will probably opt for the convenience of the associated Kleisli
type).
See also:
 Plainly partible
 Partible laws
 Burtonstyle nondeterminism
 MonadFix
 Prelude extensions
 Bang patterns
Atravers 04:31, 10 April 2018 (UTC)