[Haskell-beginners] Re: Thinking about monads

[Ertugrul Soeylemez]

Hello Michael,

you're basically trying to give monads a better name.  Don't forget that
they are an abstract structure, which means that trying to find
intuitions or motivations is the same as trying to find real world
examples.

Monads are not a way to pass the result of some function to another
function.  We've got function composition for that.  Short-circuiting is
a feature of particular monads like Maybe, and as you said, you could
well do without them.  In Haskell, monads are an abstract combinator
class, which brings you mainly two advantages:  Generalization of
functionality and impure operations encapsulated in pure objects.

Those objects are the monadic values, which can be interpreted in a
number of ways.  I like to interpret them as computations, and this is
probably the most common interpretation.  Have a look at this:

  x :: Maybe Integer
  x = Just 3

As you already know, Maybe is a monad.  Just 3 is not a value right
away.  It is a computation, which results in 3.  The result of
computations may have a certain structure.  Maybe allows lack of a
result (the Nothing computation), which adds structure to the result.
The list type is also a monad.  It adds structure in that it allows
arbitrarily many results.  Monads also allow you to combine computations
(pass the result of one computation to another) in a structure-specific
manner.  In the Maybe monad, this is the short-circuiting you mentioned,
if there is no result.  In the list monad, each of the results is passed
and all individual results are collected in a larger result list
(non-determinism).

The big feature is that you can write code, which disregards this
structure.  This is what I referred to as generalization of
functionality.  Have a look at the sequence function from Control.Monad:

  sequence :: [m a] -> m [a]

Its type already suggests, what it does.  It takes a list of
computations and gives a computation, which results in the list of the
corresponding results.  Example:

  sequence [Just 3, Just 4, Just 5]
  = Just [3, 4, 5]

  sequence [[1,2], [3,4,5]]
  = [[1,3], [1,4], [1,5], [2,3], [2,4], [2,5]]

To understand the second example, view the list monad as a way to encode
non-determinism.  I've written more about this in section 11 of my
monads tutorial [1].

As said, the big advantage here is that you can write the sequence
function in a way, which completely disregards the underlying structure
implemented through the actual monad, such that you don't need to
rewrite it for each monad:

  sequence [] = return []
  sequence (c:cs)
    = c >>= \r ->
      sequence cs >>= \rs ->
      return (r:rs)

or by using the mapM function:

  sequence = mapM id

So Haskell monads help you to generalize functionality in the same way
as group theory and category theory help you to generalize proofs.

I hope, this helps.


Greets,
Ertugrul.


[1] http://ertes.de/articles/monads.html#section-11


Michael Mossey <mpm at alumni.caltech.edu> wrote:

> 
> 
> Ertugrul Soeylemez wrote:
> > Michael Mossey <mpm at alumni.caltech.edu> wrote:
> > 
> >> I'm getting a better grasp on monads, I think. My original problem, I
> >> think, was that I was still thinking imperatively. So when I saw this:
> >>
> >> class Monad m where
> >>
> >>     (>>=) :: m a -> (a -> m b) -> m b
> >>
> >> I didn't understand the "big deal". I thought, okay so you "do" something with the 
> >> function (a -> m b) and you "arrive" at m b.  [...]
> > 
> > Think of f being a computation, in which something is missing.  It takes
> > this something through a parameter.  So f is actually a function, which
> > takes a value and results in a computation.  Another intuition:  f is a
> > parametric computation.  Now if c0 is a computation, then
> > 
> >   c0 >>= f
> > 
> > is another computation built by feeding the result of c0 to f.  More
> > graphically you've plugged the result cable of c0 to the input port of
> > f.
> > 
> > As a real world example, consider a computation, which prints a value x:
> > 
> >   print x
> > 
> > That x has to come from somewhere, so this should actually be a function
> > of some value x:
> > 
> >   \x -> print x
> > 
> > If that x should come from another computation, then (>>=) comes into
> > play.  You can pass the result of one computation to the above one.  For
> > example, if x comes from the result of getChar, you can write:
> > 
> >   getChar >>= \x -> print x
> > 
> > or simply:
> > 
> >   getChar >>= print
> > 
> 
> Well, here are my thoughts. I know what you write is the way monads are introduced in 
> most of the texts I've seen, but to the eyes of an imperative programmer, nothing 
> "special" is going on. Let's give an example (but replace getChar by something 
> deterministic). When I see
> 
> thing1 >>= thing2
> 
> I think to myself, this is basically the same as:
> 
> (Example A)
> f input = result'
>    where result = thing1 input
>          result' = thing2 result
> 
> But it's not the same, because certain problems arise. What's special about monads is 
> the way they are used and the particular problem they are trying to solve. For 
> example, here are some problems we need to solve:
> 
> (1) how do you pass state from one function to the next in the most elegant way 
> (avoiding the need to make complicated data types and having the ability to hide 
> implementation details)
> 
> (2) how do you deal with errors? how do you "return early" from a set of computations 
> that have hit a wall?
> 
> I confess I have not read any chapters on monads themselves, but I have finished 
> Chapter 10 of Real World Haskell, which is mostly about motivating monads and 
> implementing something very close to them. They use an operator they call ==>, which 
> is nearly identical to >>=.
> 
> I see one answer to (1). Something like
> 
> (Example B)
> f input =
>     thing1 input >>= \result ->
>     thing2 result >>= \result' ->
>     return (result, result')
> 
> separates the idea of the state we passing "down the chain" from the results we get. 
> I'll rewrite example (A) above, to be more explicit about what we are trying to do:
> 
> (Example C)
> f state = (result, result')
>    where (result, state')   = thing1 state
>          (result', state'') = thing2 result state'
> 
> 
> In example (B), the results are naturally available because they are arguments to 
> functions, and all functions further down the chain are nested within them.
> 
> Now about problem (2)? The way the >>= operator is defined, it allows 
> "short-circuiting" any remaining functions after we get a bad result. If the state is 
> Maybe or Either, we can define >>= such that a result of Nothing or Left causes all 
> remaining functions to be skipped. We could do this without monads, but it would look 
> very ugly.
> 
> As a beginner, I'm not trying to lecture anyone, but putting down my thoughts so I 
> can get feedback. I feel there's no way to "understand" monads without understanding 
> the motivation of the problem we are trying to solve, or without seeing specific 
> implementations. Chapter 10 of Real World Haskell provides a lot of motivation by 
> showing early awkward attempts to solve these problems.
> 
> Regards,
> Mike



-- 
nightmare = unsafePerformIO (getWrongWife >>= sex)
http://blog.ertes.de/