[Haskell-cafe] Re: Can we come out of a monad?

[Dan Doel]

On Saturday 31 July 2010 8:13:37 am Ertugrul Soeylemez wrote:
> I agree to some extent, but only to some.  Mostly the problem of people
> is that they are trying to understand "monads" as opposed to specific
> instances.  It's better to learn "the IO monad", "state monads", "the
> list monad", "the Maybe monad", "the Parser monad", etc.

I think there are 'easy' answers to "what are monads," too, at least in the 
way they tend to appear in Haskell. But, the easiness may well depend on 
having background that isn't common in computer programming.

Some of it is, though. "Embedded domain-specific language" is a buzz phrase 
these days, so it's probably safe to assume most folks are familiar with the 
idea. From that starting point, one might ask how to approach EDSLs from a 
more mathematical perspective, and making use of the type system. We might be 
led to the following:

1) We want to distinguish 'programs written in the EDSL' via types somehow. It 
may not make sense to use EDSL operations just anywhere in the overall 
program.

2) Algebra looks promising for talking about languages. Our DSLs will probably 
have some base operations, which we'll combine to make our programs. So, our 
EDSL type above should probably be related to algebraic theories somehow.

Once we've decided on the above, well, monads are a way in category theory of 
talking about algebraic theories. So it stands to reason that a lot of the 
EDSLs we're interested in will be monads. And so, by talking about monads in 
general, we can construct operations that make sense in and on arbitrary EDSLs 
(like, say, sequence = stick together several expressions).

And that covers a lot of what monads are used for in Haskell.

  'Maybe a' designates expressions in a language with failure
  'Either e a' designates expressions with a throw operation
  'State s a' allows get and put
  'IO a' has most of the features in imperative languages.
  etc.

So the 'easy' answer is that (embedded) languages tend to be algebraic 
theories, and monads are a way of talking about those. Of course, that general 
answer may still be pretty meaningless if you don't know what algebraic 
theories are, so it's still probably good to look at specific monads.

-- Dan