[Haskell-cafe] Why Kleisli composition is not in the Monad signature?

[Alberto G. Corona]

The particular case from which the former is a generalization:


*instance Monad m => Monoid  (a -> a) where*
*    mappend = (.)*
*    mempty  = id*
*
*
Here the monoid is defined for the functions within the set of values of
type a. There are no null elements.



2012/10/24 Alberto G. Corona <agocorona at gmail.com>

> What hiders according with my experience, the understanding of this
> generalization are some mistakes. two typical mistakes from my side was to
> consider an arrow as a function, and the consideration of  m  as a kind of
> container, which it is not from the point of view of category theory.
>
> a -> m b
>
> instead of as a container, 'm b' must be considered as the set of elements
> of type b (wrapped within some constructor) plus zero or more null elements
> of the monad 'm', Such are elements like, for example, Nothing, the empty
> list or Left .  so that:
>
> null >>= f= null            and
> f >>= \x -> null= null
>
> in the other side (->) is an arrow of category theory, not a function.That
> means that  there may be weird additional things that functions are not be
> permitted to have. For example, from an element in the set of 'a' may
> depart many arrows to elements of 'b'. This permits the nondeterminism of
> the list monad.
>
> A function like this:
>
> repeatN :: Int -> a -> [a]
>
> can have two interpretations: one functional interpretation, where repeatN
> is a pure function with results in the list container. The other is the
> category interpretation, where 'repeatN n' is a morphism that produces n
> arrows from the set of the Strings to the set of String plus the empty set,
> The list container is a particular case of container that hold the result
> of this nondeterministic morphism (other instantiations of this
> nondeterministic monad would be a set monad or whatever monad build using a
> multielement container. The container type is not the essence. the essence
> it is the nondeterministic nature, which, in haskell practical terms, needs
> a multielement container to hold the multiple results).
>
> so the monadic re-ínterpretation of the repeatN signature is:
>
> repeatN ::Int -> a -> a + {[]}
>
> Here the empty list is the null element of the list monad
>
> in the same way:
> functional signature:  a -> Maybe b
> monadic interpretation:  a -> b + {Nothing}
>
> functional signature:  a -> Either c b
> monadic interpretation:   a -> b + {Left c}
>
> So when i see m b in the context of a monad, I think on nothing more that
> the set of values of type b (wrapped within some constructor) plus some
> null elements (if they exist).
>
> so in essence
>
> a -> m b
>
> is a -> (b + some null elements)
>
> that´s a generalisation of  a -> b
>
> where -> is an arrow, not a function (can return more than one result, can
> return different things on each computation etc)
>
> And this instance of monoid show how kleisly is the composition and return
> is the identity element
>
> *instance Monad m => Monoid  (a -> m a) where*
> *    mappend = (>=>)*
> *    mempty  = return*
> *
> *
>
> According with the above said,   'a -> m a' must be understood as the set
> of monadic endomorphisms in a:
>
> a -> a +{null elements of the monad m}
>
> Which is, in essence, a -> a
>
> 2012/10/16 Simon Thompson <s.j.thompson at kent.ac.uk>
>
>>
>> Not sure I really have anything substantial to contribute, but it's
>> certainly true that if you see
>>
>>   a -> m b
>>
>> as a generalisation of the usual function type, a -> b, then return
>> generalises the identity and
>> kleisli generalises function composition. This makes the types pretty
>> memorable (and often the
>> definitions too).
>>
>> Simon
>>
>>
>> On 16 Oct 2012, at 20:14, David Thomas <davidleothomas at gmail.com> wrote:
>>
>> >> class Monad m where
>> >>  return :: a -> m a
>> >>  kleisli :: (a -> m b) -> (b -> m c) -> (a -> m c)
>>
>> Simon Thompson | Professor of Logic and Computation
>> School of Computing | University of Kent | Canterbury, CT2 7NF, UK
>> s.j.thompson at kent.ac.uk | M +44 7986 085754 | W www.cs.kent.ac.uk/~sjt
>>
>>
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>>
>
>
>
> --
> Alberto.
>



-- 
Alberto.
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