[Haskell-beginners] Cartesian Product in Standard Haskell Libraries

[Jay Sulzberger]

On Mon, 24 Dec 2012, Kim-Ee Yeoh <ky3 at atamo.com> wrote:

>> the result is of type [()] but for a cartesian n-product, you would like
> [[a]]
>
> Right. So what we have here is a product over a count of 0 sets, which is
> isomorphic to the function space that has the null set as domain. The
> latter has exactly one element: the trivial function.
>
> My apologies for misreading what OP wrote:
>
>> This looks to me to be a violation of the rule that the Cartesian
>> product of an empty list of lists is a list with one element in
>> it.
>
> I thought he meant something along the lines of sequence ["","x"].
>
>
> -- Kim-Ee

Thanks, Kim-Ee!

Your answer and also Chaddai's are very helpful.

I hope to post more in this thread.

oo--JS.


>
>
> On Mon, Dec 24, 2012 at 4:42 PM, Chaddaï Fouché <chaddai.fouche at gmail.com>wrote:
>
>> On Mon, Dec 24, 2012 at 8:01 AM, Jay Sulzberger <jays at panix.com> wrote:
>>>
>>>  > sequence []
>>>   []
>>>   it :: [()]
>>>
>>> This looks to me to be a violation of the rule that the Cartesian
>>> product of an empty list of lists is a list with one element in
>>> it.  It looks to be a violation because "[]" looks like a name
>>> for an empty list.  But we also have
>>>
>>>  > length (sequence [])
>>>   1
>>>   it :: Int
>>>
>>> which almost reassures me.
>>>
>>
>> Well the type of the first response is a dead give-away : the result
>> is of type [()] but for a cartesian n-product, you would like [[a]]
>> (with a maybe instantiated to a concrete type) ...
>> What's happening here is that sequence is not "the cartesian
>> n-product" in general, it is only that in the list monad but in
>> "sequence []" there's nothing to indicate that we're in the list
>> monad, so GHC default to the IO monad and unit () so sequence has the
>> type "[IO ()] -> IO [()]" and there's no IO action in the list
>> parameter, so there's nothing in the result list.
>>
>> Try :
>>> sequence [] :: [[Int]]
>> and you should be reassured.
>>
>> --
>> Jedaï
>>
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>