[Haskell-cafe] IO is not a monad
Yitzchak Gale
gale at sefer.org
Wed Feb 7 07:39:20 EST 2007
Just for the record, I think this completes the
requirements of my challenge. Please comment!
Is this correct?
Thanks.
> 1. Find a way to model strictness/laziness properties
> of Haskell functions in a category in a way that is
> reasonably rich.
We use HaskL, the category of Haskell types, Haskell
functions, and strict composition:
f .! g = f `seq` g `seq` (f . g)
Let undef = \_ -> undefined. A function f is strict iff
f .! undef = undef, lazy iff f .! undef /= undef, and
convergent iff f .! g /= undef for all g /= undef.
We consider only functors for which fmap is a
morphism.
A functor preserves strictness iff fmap is strict.
A functor preserves laziness iff fmap is convergent.
Note that with these definitions, undefined is lazy.
> 2. Map monads in that category to Haskell, and
> see what we get.
Assume that return /= undef, and that >>= is convergent
in its second argument.
The monad laws are:
1. (>>= return) = id
2. (>>= f) . return = f
3. (>>= g) . (>>= f) = (>>= (>>= g) . f)
4. >>= is strict in its second argument.
> 3. Compare that to the traditional concept of
> a monad in Haskell.
As long as we are careful to use the points-free
version, the laws are the same as the traditional
monad laws. In particular, we can use the usual
composition for these laws. But we must add the
strictness law.
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