Tarmo Uustalu writes:
| We would like to announce our new paper
|
| The essence of dataflow programming
|
| http://cs.ioc.ee/~tarmo/papers/essence.pdf
|
| which describes a novel comonadic foundation of dataflow computing,
| incl. semantics of dataflow languages a la Lucid or Lustre. The
| central point is that comonads structure the context-dependence in
| dataflow paradigms in much the same way as monads organize
| effects. The paper was specifically written for functional
| programmers (as opposed to semanticists).
This is really cool!
For those who haven't read the above paper yet, it describes how to
structure an interpreter for a dataflow language using comonads, similar
to the way you can structure an interpreter for an impure language using
monads. Inspired, I've tried my hand at implementing some of the example
dataflow functions directly in Haskell.
This message is literate Haskell code. It uses the arrow syntax in a few
places; these are just examples, so you may comment them out if you do
not have a recent-enough GHCi or an arrow syntax preprocessor.
{-# OPTIONS -farrows #-}
module Dataflow where
import Prelude hiding (sum)
import Control.Arrow
FIrst, a class for comonads:
class Functor d = Comonad d where
extract :: d a - a
coextend :: (d a - b) - d a - d b
(In the paper, these are counit and cobind.)
We'll also define the injection combinator from Kieburtz's paper[1]:
(.) :: Functor d = d a - b - d b
d . a = fmap (const a) d
As a simple example, the environment comonad:
instance Functor ((,) e) where
fmap f (e,a) = (e, f a)
instance Comonad ((,) e) where
extract (e,a) = a
coextend f d@(e,a) = (e, f d)
This is closely related to the reader monad (in fact they are adjoint).
Given a comonad d, we can also create an arrow Cokleisli d:
newtype Cokleisli d a b = Cokleisli { runCokleisli :: d a - b }
instance Comonad d = Arrow (Cokleisli d) where
arr f = Cokleisli (f . extract)
Cokleisli f Cokleisli g = Cokleisli (g . coextend f)
first (Cokleisli f) = Cokleisli $
\d - (f (fmap fst d), snd (extract d))
Here is something I did not expect to find: you can *apply* cokleisli
arrows.
instance Comonad d = ArrowApply (Cokleisli d) where
app = Cokleisli $
\d - runCokleisli (fst (extract d)) (fmap snd d)
instance Comonad d = ArrowChoice (Cokleisli d) where
left = leftApp
Now, I haven't proven that this implementation of app satisfies the
relevant laws, but assuming it does, it raises some questions. Most of
the papers dealing with arrows state that instances of ArrowApply are
equivalent to monads, but cokleisli arrows allow you to do dataflow
programming, which cannot be done with monads. That may or may not be a
contradiction.
One point to consider is that the type Cokleisli d a b (or d a - b)
is isomorphic to Reader (d a) b (or d a - b), and Reader (d a) is
a monad.
Thus:
instance Functor (Cokleisli d a) where
fmap f (Cokleisli k) = Cokleisli (f . k)
instance Monad (Cokleisli d a) where
return a = Cokleisli (const a)
Cokleisli k = f = Cokleisli $ \d - runCokleisli (f (k d)) d
I don't know whether this is significant or useful.
To describe synchronous dataflow languages (where values can depend on
the past, but not the future), Uustalu and Vene employ the non-empty
list comonad, which I will call History.
data History a = First a | History a : a
infixl 4 :
runHistory :: (History a - b) - [a] - [b]
runHistory f [] = []
runHistory f (a:as) = run (First a) as
where
run az [] = [f az]
run az (a:as) = f az : run (az : a) as
instance Functor History where
fmap f (First a) = First (f a)
fmap f (as : a) = fmap f as : f a
instance Comonad History where
extract (First a) = a
extract (as : a) = a
coextend f d@(First a) = First (f d)
coextend f d@(as : a) = coextend f as : f d
We'll also need a combinator fby, which is short for followed by. In
a dataflow language, you might write:
pos = 0 fby (pos + 1)
Which means that pos is initially zero, and its next value is always the
current value plus one. The fby combinator is easy to define,
fby :: a - History a - a
a0 `fby` First a = a0
a0 `fby` (az : a) = extract az
but defining pos requires recursion.
Thanks to Yampa[2], we know how this sort of thing looks when written in
arrow notation:
type Hist = Cokleisli History
posA :: Hist a Integer
posA = proc _ - do
rec
x - delay 0 - x + 1
returnA - x
We can define 'delay' using 'fby'.
delay :: a - Hist a a
delay a0 = Cokleisli $ \d - a0 `fby` d
Now we just need a instance for ArrowLoop. This was tricky, but I
eventually managed to reverse-engineer the ArrowLoop instance for
Kleisli arrows and come up with a counterpart.
I rely on two combinators. The first, czip, is from the paper and
expresses the ability to merge two
