Difference between revisions of "Partibles for composing monads"

Revision as of 04:22, 27 April 2021

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.

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 all-or-nothing 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.

Difference between revisions of "Partibles for composing monads"

Revision as of 04:22, 27 April 2021

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+<i>
-[[Category:Proposals]]
-<blockquote><tt>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.</tt></blockquote>
+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.
+</i>
-* ''How to Declare an Imperative'', Philip Wadler.
+* <tt>[https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.91.3579&rep=rep1&type=pdf How to Declare an Imperative], Philip Wadler.</tt>
+<sub> </sub>
-Some initial definitions:
+Assuming the partible types being used are appropriately defined, then:
 <haskell>
-        class Partible a where
+instance Partible a => Monad ((->) a) where
-            part  :: a -> (a, a)
+    return x = \ u -> case part u of !_ -> x
-            parts :: a -> [a]
-             -- Minimal complete definition: part or parts
+    m >>= k  = \ u -> case part u of
-            part u  = case parts u of u1:u2:_ -> (u1, u2)
+                        (u1, u2) -> case m u1 of !x -> k x u2
-            parts u = case part u of (u1, u2) -> u1 : parts u2
-        instance Partible a => Monad ((->) a) where
+    m >> w   = \ u -> case part u of
-            return x = \ u -> part u `seq` x
+                        (u1, u2) -> case m u1 of !_ -> w u2
-            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
+    fail s   = \ u -> case part u of !_ -> error s
-        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 () (fix (\ l -> zipWith respond (d l) (parts u)))
-         -- fix f = f (fix f)
-        instance Partible a => MonadFix ((->) a) where
-            mfix m = \ u -> fix (\ 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
-            part (Fresh g u) = case part u of (u1, u2) -> (Fresh g u1, Fresh g u2)
-        fresh :: Fresh a -> [a]
-        fresh u = [ g v | Fresh g v <- parts u ]
-        instance Functor Fresh where
-            fmap f (Fresh g u) = Fresh (f . g) u
-         -- one more primitive
-        primGensym :: OI -> Int
-        supplyInts :: OI -> Fresh Int
-        supplyInts = \ u -> afresh primGensym u
-        instance (Partible a, Partible b) => Partible (a, b) where
-            part (u, v) = case (part u, part v) of
-                            ((u1, u2), (v1, v2)) -> ((u1, v1), (u2, v2))
-        instance (Partible a, Partible b) => Partible (Either a b) where
-            part (Left u)  = case part u of (u1, u2) -> (Left u1, Left u2)
-            part (Right v) = case part v of (v1, v2) -> (Right v1, Right v2)
-        data Some a = Only a | More a (Some a)
-        instance Partible a => Partible (Some a) where
-            part (Only u)    = case part u of
-                                 (u1, u2) -> (Only u1, Only u2)
-            part (More u us) = case part u of
-                                 (u1, u2) ->
-                                   case part us of
-                                     (us1, us2) -> (More u1 us1, More u2 us2)
-        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
-         -- ...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
 </haskell>
+Furthermore:
+<haskell>
-So what qualifies as being partible?
+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
-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 <code>supplyInts</code>.
+    mcommute g = \ v u -> g u v
+instance (Monad m, Partible b, Monad ((->) b)) => MonadCommute m ((->) b) where
-If its violation causes a runtime error, the use-once property of partible values aids in maintaining referential transparency in the effectful segments of a program; using another example from Wadler's paper minimally rewritten in Haskell syntax using <code>OI</code> values:
+    mcommute m = \ v -> liftM ($ v) m
-<haskell>
-                let
-                  x = (primPutChar 'h' u `seq` primPutChar 'a' u)
-                in x `seq` x
 </haskell>
+where:
+:{|
-would trigger the error; the working version being:
+|<haskell>
+yet   :: (a -> a) -> a
+yet f =  f (yet f)
+class Monad m => MonadFix m where
-<haskell>
-                let
+    mfix :: (a -> m a) -> m a
-                  x = (\ v -> case part v of
-                                (v1, v2) -> primPutChar 'h' v1 `seq` primPutChar 'a' v2)
-                in
-                  case part u of
-                    (u1, u2) -> x u1 `seq` x u2
-</haskell>
+class (Monad m1, Monad m2) => MonadCommute m1 m2 where
+    mcommute :: m1 (m2 a) -> m2 (m1 a)
-...rather tedious, if it weren't for Haskell's standard monadic methods:
-<haskell>
-                let
-                  x = primPutChar 'h' >> primPutChar 'a'
-                in x >> x
 </haskell>
+|}
+Using partible types to define monadic ones can enable an intermediate approach to the use of effects, in contrast to the ''all-or-nothing 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.
+----
-Higher-order functions allows the manipulation of control e.g. <code>Prelude.until</code> in Haskell. As the definition of <code>runDialogue</code> 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.
+See also:
+* [[Plainly partible]]
+* [[Partible laws]]
+* [[Burton-style nondeterminism]]
+* [[MonadFix]]
+* [[Prelude extensions]]
+* [https://downloads.haskell.org/~ghc/7.8.4/docs/html/users_guide/bang-patterns.html Bang patterns]
-Other references and articles:
-* [https://maartenfokkinga.github.io/utwente/mmf2001c.pdf ''An alternative approach to I/O''], Maarten Fokkinga and Jan Kuper.
+[[User:Atravers|Atravers]] 04:31, 10 April 2018 (UTC)
-* ''Functional Pearl: On generating unique names'', Lennart Augustsson, Mikael Rittri and Dan Synek.
+[[Category:Monad]]
-* [http://conal.net/blog/posts/can-functional-programming-be-liberated-from-the-von-neumann-paradigm ''Can functional programming be liberated from the von Neumann paradigm?''], Conal Elliott.
+[[Category:Proposals]]
-* [http://xenon.stanford.edu/~blynn/haskell/io.html ''Haskell - A Problem With I/O''], Ben Lynn.
-* [http://www.alsonkemp.com/haskell/reflections-on-leaving-haskell ''Reflections on leaving Haskell''], Alson Kemp.
-* [https://www.slideshare.net/nobsun/lazy-io ''Non-Imperative Functional Programming''], Nobuo Yamashita.
-* [http://h2.jaguarpaw.co.uk/posts/mtl-style-for-free ''MTL style for free''], Tom Ellis.
-* ''Functional I/O Using System Tokens'', Lennart Augustsson.
-* ''I/O Trees and Interactive Lazy Functional Programming'', Samuel A. Rebelsky.
-Thank you to those who commented on early drafts of this document.
-[[User:Atravers|Atravers]] 04:31, 10 April 2018 (UTC)