C. McCann cam at uptoisomorphism.net
Mon Nov 15 14:19:59 EST 2010

```On Mon, Nov 15, 2010 at 12:43 PM, Ling Yang <lyang at cs.stanford.edu> wrote:
> Specifically: There are some DSLs that can be largely expressed as monads,
> that inherently play nicely with expressions on non-monadic values.
> We'd like to use the functions that already work on the non-monadic
> values for monadic values without calls to liftM all over the place.

It's worth noting that using liftM is possibly the worst possible way
to do this, aesthetically speaking. To start with, liftM is just fmap
can (and should) also be an instance of Functor, and if the instances
aren't buggy, then liftM f = (>>= return . f) = fmap f.

operator synonym for fmap, found in Control.Applicative, I believe.

> The probability monad is a good example.
>
[snip]
> I'm interested in shortening the description of 'test', as it is
> really just a 'formal addition' of random variables. One can use liftM
> for that:
>
> test = liftM2 (+) (coin 0.5) (coin 0.5)

Also on the subject of Control.Applicative, note that independent
probabilities like this don't actually require a monad, merely the
ability to lift currying into the underlying functor, which is what
Applicative provides. The operator ((<*>) :: f (a -> b) -> f a -> f b)
is convenient for writing such expressions, e.g.:

test = (+) <\$> coin 0.5 <*> coin 0.5

Monads are only required for lifting control flow into the functor,
which in this case amounts to conditional probability. You would not,
for example, be able to easily use simple lifted functions to write
"roll a 6-sided die, flip a coin as many times as the die shows, then
count how many flips were heads".

> I think a good question as a starting point is whether it's possible
> to do this 'monadic instance transformation' for any typeclass, and
> whether or not we were lucky to have been able to instance Num so
> easily (as Num, Fractional can just be seen as algebras over some base
> type plus a coercion function, making them unusually easy to lift if
> most typeclasses actually don't fit this description).

Part of the reason Num was so easy is that all the functions produce
values whose type is the class parameter. Your Num instance could
almost be completely generic for any ((Applicative f, Num a) => f a),
except that Num demands instances of Eq and Show, neither of which can
be blindly lifted the way the numeric operations can.

I imagine it should be fairly obvious why you can't write a
non-trivial generic instance (Show a) => Show (M a) that would work
for any possible monad M--you'd need a function (show :: M a ->
String) which is impossible for abstract types like IO, as well as
function types like the State monad. The same applies to (==), of
course. Trivial instances are always possible, e.g. show _ = "[not
showable]", but then you don't get sensible behavior when a
non-trivial instance does exist, such  as for Maybe or [].

> Note that if we consider this in a 'monadification' context, where we
> are making some choice for each lifted function, treating it as
> entering, exiting, or computing in the monad, instancing the typeclass
> leads to very few choices for each: the monadic versions of +, -, *
> must be obtained with "liftM2",the monadic versions of negate, abs,
> signum must be obtained with "liftM", and the monadic version of
> fromInteger must be obtained with "return . "

Again, this is pretty much the motivation and purpose of
Control.Applicative. Depending on how you want to look at it, the
underlying concept is either lifting multi-argument functions into the
functor step by step, or lifting tuples into the functor, e.g. (f a, f
b) -> f (a, b); the equivalence is recovered using fmap with either
(curry id) or (uncurry id).

Note that things do get more complicated if you have to deal with the
full monadic structure, but since you're lifting functions that have
no knowledge of the functor whatsoever they pretty much have to be
independent of it.

> I suppose I'm basically suggesting that the 'next step' is to somehow
> do this calculation of types on real type values, and use an inductive
> programming tool like Djinn to realize the type signatures. I think
> the general programming technique this is getting at is an orthogonal
> version of LISP style where one goes back and forth between types and
> functions, rather than data and code. I would also appreciate any
> pointers to works in that area.

Well, I don't think there's any good way to do this in Haskell
directly, in general. There's a GHC extension that can automatically
derive Functor for many types, but nothing to automatically derive
Applicative as far as I know (other than in trivial cases with newtype
deriving)--I suspect due to Applicative instances being far less often
uniquely determined than for Functor. And while a fully generic
instance can be written and used for any Applicative and Num, the
impossibility of sensible instances for Show and Eq, combined with the
context-blind nature of Haskell's instance resolution, means that it
can't be written directly in full generality. It would, however, be
fairly trivial to manufacture instance declarations for specific types
using some sort of preprocessor, assuming Show/Eq instances have been
written manually or by creating trivial ones.

Anyway, you may want to read the paper that introduced Applicative,
since that seems to describe the subset of generic lifted functions
you're after: http://www.soi.city.ac.uk/~ross/papers/Applicative.html

If for some reason you'd rather continue listening to me talk about
it, I wrote an extended ode to Applicative on Stack Overflow some time