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
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