[Haskell-cafe] A question about "monad laws"

[Jonathan Cast]

On 11 Feb 2008, at 5:33 AM, Deokjae Lee wrote:

> Tutorials about monad mention the "monad axioms" or "monad laws". The
> tutorial "All About Monads" says that "It is up to the programmer to
> ensure that any Monad instance he creates satisfies the monad laws".
>
> The following is one of the laws.
>
> (x >>= f) >>= g == x >>= (\v -> f v >>= g)
>
> However, this seems to me a kind of mathematical identity.

What do you mean by identity?  It's easy enough to violate:

newtype Gen a = Gen (StdGen -> a)
runGen :: Gen a -> StdGen -> a
runGen (Gen f) = f
instance Monad Gen where
   return x = Gen (const x)
   a >>= f = Gen (\ r -> let (r1, r2) = split r in runGen (f (runGen  
a r1)) r2) [1]

split returns two generators independent of each other and of its  
argument; thus, this `monad' violates all *three* of the monad laws,  
in the sense of equality.  So, for example, the program

do
   x <- a
   return x

denotes a random variable independent of (and hence distinct from)  
a.  In general, you want some kind of backstop `equality' (e.g., that  
the monad laws hold in the sense of identical distribution) when you  
violate them; otherwise, you will violate the expectations your users  
have from the syntax of the do notation.

jcc

[1] Test.QuickCheck