Arrows that are also Functors
Edward Kmett
ekmett at gmail.com
Tue Apr 26 21:19:53 CEST 2011
On Tue, Apr 19, 2011 at 11:48 PM, Tyson Whitehead <twhitehead at gmail.com>wrote:
> On April 19, 2011 23:22:12 Tyson Whitehead wrote:
> > ArrowLoop from MonadFix
> >
> > loop' f = fst' .' loop'' (f .' arr' (second snd))
> > where loop'' f = mfix (\y -> f .' arr' (,y))
>
> BTW haskellers, I've been wondering if mfix would better be defined as
>
> mfix' :: (m a -> m a) -> m a
>
> where "mfix' f = mfix (f . pure)" for the computational monads. The
> advantage
> being you can give a useful definition for structural monads as well.
Note: This does not generalize the signature of mfix, it only overlaps
slightly, as not every monad m permits the extraction of the value a
injected (consider Cont r), so you necessarily change the meaning or
obliterate a number of instances.
Recall the main motivation for mfix was to support Erkoek and Launchbury's
recursive do:
http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=mfix+recursive+do
http://www.haskell.org/haskellwiki/MonadFix
This necessitates 4 laws for mfix, which don't translate nicely.
- mfix (return . h) = return (fix h)
- mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)
- if
h
is strict,
mfix (liftM h . f) = liftM h (mfix (f . h))
- mfix (\x -> mfix (\y -> f x y)) = mfix (\x -> f x x)
The other commonly proposed mfix replacement is to define it once, as guided
by the types, but while this works for fix and the the comonadic equivalent,
it doesn't generate a useful mfix for recursive do either.
-Edward
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