[Haskell-cafe] What is MonadPlus good for?

[Josef Svenningsson]

On Sun, 13 Feb 2005 19:08:26 -0500, ajb at spamcop.net <ajb at spamcop.net> wrote:
> Quoting Josef Svenningsson <josef.svenningsson at gmail.com>:
> 
> > I think it's unfair to the monad transformers to simply say that they
> > don't obey the law. The interesting thing is whether they *preserve*
> > the law. A monad transformer T preserves a law if given a monad M
> > which obeys the law holds then the monad T M obeys the law.
> 
> The law in question is that mzero is a right-zero for bind.  How can an
> underlying monad be said to "obey" this law if it doesn't support mzero?
> 
You're of course absolutely right that it doesn't make sense to talk
about mzero being a right-identity for bind if the monad doesn't
support mzero. I should have been more clear. Let me have another try
at explaining myself.

Let's consider a specific monad transformer, say (ReaderT r). What I
hope to convince you of is that (ReaderT r) cannot be said break the
mzero-is-right-identity-for-bind law. Now, if we look at the MonadPlus
instance for (ReaderT r) it looks like this:
\begin{code}
instance (MonadPlus m) => MonadPlus (ReaderT r m) where
        mzero       = ReaderT $ \_ -> mzero
        m `mplus` n = ReaderT $ \r -> runReaderT m r `mplus` runReaderT n r
\end{code}
This important thing to note here is that the above instance
declaration relies on an underlying monad m with mzero and mplus. If
we try (and indeed succeed) to prove that (ReaderT r m) satisfies the
mzero-is-right-identity-for-bind law we will see that the proof depend
crucially on the fact that m also obeys the law. This is the best that
the monad transformer can do, namely to preserve the law.

You claimed that monad transformers break the
mzero-is-right-identity-for-bind law because they can be applied to
IO. I say, it's not the monad transformers fault. They cannot possibly
be expected to repair the law if they are given a faulty monad.

I hope this makes things a little clearer.

/Josef