[Haskell-cafe] questions about Arrows

[Ben]

Hello --

Three related questions, going from most specific to most general :

1 ) Consider the stream processing arrow which computes a running sum,
with two implementations : first using generic ArrowCircuits (rSum);
second using Automaton (rSum2) :

module Foo where

import Control.Arrow
import Control.Arrow.Operations
import Control.Arrow.Transformer
import Control.Arrow.Transformer.All

rSum :: ArrowCircuit a => a Int Int
rSum = proc x -> do
  rec out <- delay 0 -< out + x
  returnA -< out

rSum2 = Automaton (f 0)
  where f s n = let s' = s + n
                in (s', Automaton (f s'))

runAuto _ [] = []
runAuto (Automaton f) (x:xs) =
  let (y, a) = f x
  in y : runAuto a xs

take 10 $ runAuto rSum [1..]
[0,1,3,6,10,15,21,28,36,45]

take 10 $ runAuto rSum2 [1..]
[1,3,6,10,15,21,28,36,45,55]

Note that the circuit version starts with the initial value zero.

Is there a way to write rSum2 in the general ArrowCircuit form, or
using ArrowLoop?

2) Are the ArrowLoop instances for (->), Kleisli Identity, and
Kleisli ((->) r) all morally equivalent?  (e.g., up to tagging and untagging?)

3) One can define fix in terms of trace and trace in terms of fix.

trace f x = fst $ fix (\(m, z) -> f (x, z))
fix f = trace (\(x, y) -> (f y, f y)) undefined

Does this mean we can translate arbitrary recursive functions into
ArrowLoop equivalents?

Best regards, Ben