Patrick LeBoutillier patrick.leboutillier at gmail.com
Tue Feb 9 21:35:45 EST 2010

```Daniel,

As usual, thanks a lot for this enlightening response.

Patrick

On Tue, Feb 9, 2010 at 6:31 PM, Daniel Fischer <daniel.is.fischer at web.de> wrote:
> Am Dienstag 09 Februar 2010 23:07:55 schrieb Patrick LeBoutillier:
>> Daniel,
>>
>> > Sure. If you don't mind that the mutations come in a different order,
>> > one thing that works wonders is "sequence",
>> >
>> > sequence :: Monad m => [m a] -> m [a]
>> >
>> > In particular, for m = [], sequence :: [[a]] -> [[a]]. Then, knowing
>> > what sequence does, we can write
>> >
>> >
>> > generateAll :: String -> [String]
>> > generateAll word = sequence (map f word)
>> >     where
>> >     f c = case lookup c leat of
>> >                Just r  -> [c,r]
>> >                Nothing -> [c]
>>
>> That's very nice!
>
> Thanks. But from a clean-code-higher-level perspective, it's even nicer
> with your
>
>> -- Returns a list of possible characters for c
>> mutateLetter :: Char -> [Char]
>> mutateLetter c = c : (maybeToList \$ lookup c leet)
>
> (here is a point where it would be even nicer if lookup had the type
>
> lookup :: (Eq a, MonadPlus m) => a -> [(a,b)] -> m b
> ). The performance-junkie in me would want to look at the core to make sure
> the maybeToList is eliminated by the compiler, though.
>
>>
>> One question though: In the docs sequence is described as:
>>
>>   "Evaluate each action in the sequence from left to right, and
>> collect the results."
>>
>> How is one supposed to deduce what the behavior will be for the list
>> monad (besides looking at the source)?
>
> Given its polymorphic type
>
> sequence :: Monad m => [m a] -> m [a]
>
> , what can sequence do?
>
> For
> sequence []
> , there's really only one possibility (not involving undefined/error), so
> sequence [] = return []
>
> Okay, that was the trivial part, now what can be done with nonempty lists?
> It could ignore the input and return [] in any case, but that wouldn't be
> useful at all, so we can discard that possibility. What could be usefully
> done with
>
> sequence (m1:ms) ?
>
> It has to do something with m1 and something with ms, then combine the
> results to a list of [a], which it returns.
> What can it do with m1? Since all that sequence knows about m1 is the type
> (Monad m => m a), it can't do anything but what's provided by that
> constraint. Basically, it can only put it on the left of a (>>=). There's
> on decision to be made, shall it be
>
> sequence (m1:ms) = m1 >>= \x -> something with x and ms
>
> or
>
> something with ms >>= \xs -> (m1 >>= \x -> something with x and xs)
> ?
> And what can it do with the tail of the list, ms? Why, sequence it of
> course.
> So it's either
>
> sequence (m1:ms) = m1 >>= \x -> (sequence ms >>= \xs -> return (fun x xs))
> {-
> sequence (m1:ms) = do
>    x <- m1
>    xs <- sequence ms
>    return (fun x xs)
> -}
> or
>
> sequence (m1:ms) = sequence ms >>= \xs -> (m1 >>= \x -> return (fun x xs))
> {-
> sequence (m1:ms) = do
>    xs <- sequence ms
>    x <- m1
>    return (fun x xs)
> -}
>
> where
>
> fun :: forall a. a -> [a] -> [a]
>
> Now there's a lot of nonsense you could use for 'fun',
>
> fun x xs = reverse (x:xs)
> fun x xs = x:xs ++ [x,x,x]
> fun x xs = front ++ x:back where (front,back) = splitAt 17 xs
> ...
> , but the most prominent function of type forall a. a -> [a] -> [a] is the
> only one to be reasonably expected here, so
>
> fun x xs = x : xs
>
> and the only question that remains is in which order things are chained.
> That is answered by the docs, left to right, so
>
> sequence [] = return []
> sequence (m1:ms) = m1 >>= \x -> sequence ms >>= \xs -> return (x:xs)
> {-
> sequence (m1:ms) = do
>    x <- m1
>    xs <- sequence ms
>    return (x:xs)
>
> sequence (m1:ms) = m1 >>= \x -> liftM (x :) (sequence ms)
> -}
> (or equivalent).
> Now you need to know how (>>=) is defined for [], namely
>
> ys >>= f = concatMap f ys.
>
> The short answer is, you can't deduce it wihout knowing the Monad instance
> for [], and if you know that well enough to not be confused by "evaluate
> the action" (which takes time), it's fairly straightforward.
>
>>
>>
>> Patrick
>
>

--
=====================
Patrick LeBoutillier