Monads

s.6.3.6, p.90, RR, Class Monad:

class Monad m where
   (>>=)  :: m a -> (a -> m b) -> m b
   (>>)   :: m a -> m b -> m b
   return :: a -> m a
   fail   :: String -> m a

   m >> k  =  m >>= \ _ -> k
   fail s  =  error s

Think of m a as being the (type of) a "step" in a sequence of operations, and of >> as being an operator, like `;' in an imperative language, which connects a sequence of steps together. >>= is like >> except that the former's right operand takes a parameter of type a.

Note that >> is defined in terms of >>= for brevity and consistency; they behave in essentially the same way. Below we see that >>= must be (essentially) associative and that therefore >> must be associative, i.e. p>>(q>>r) = (p>>q)>>r.

". . . Instance of Monad should [obey]":

return a >>= k          = k a
m >>= return            = m
m >>= (\x -> k x >>= h) = (m >>= k)>>= h  --~associative±
fmap f xs               = xs >>= return . f

"... lists, Maybe and IO are all instances of Monad. .... for IO, the fail method invokes error."

	>>=	>>	return	fail
Monad defn	(>>=) :: m a -> (a->m b) -> mb	(>>) :: m a -> m b -> m b	return :: a -> m a	fail :: String -> m a
		m >> k = m >>= \_ -> k --default		fail s = error s --default
instances...
IO	(>>=) =... --~magic	--"--	return =... --~magic	fail s = error s
[] list	m >>= k = concat(map k m)	--"--	return x = [x]	fail s = []
Maybe	(Just x) >>= k = k x Nothing >>= k = Nothing	--"--	return = Just --~success	fail s = Nothing

Example: [m.hs]

module Main where main = putStrLn "-- Monad experiments, 11/2005 --" -- example 1.1 m = list >> (putStrLn.show) -- equiv to... ( ["anna", "carol"] >>= \x -> -- do x <- [..., ...] ["bill", "dave" ] >>= \y -> -- y <- [..., ...] return (x,y) -- return (x,y) ) --[(a,b), (a,d), (c,b), (c,d)] -- example 1.2 m = list >> (putStrLn.show) ( ["anna", "carol"] >>= \x -> concat [ return [x], -- blob with TV ["bill", "dave" ] >>= \y -> -- a date? if x > y then fail "bad vibe" else return [x,y] ] ) --[[a],[a,b],[a,d],[c],[c,d]] -- example 2 m = Maybe >> let sqrt x = if x >= 0 then Just(x**0.5) else Nothing in (putStrLn.show) ( sqrt 4 >>= \x -> sqrt 9 >>= \y -> return (x+y) ) -- Just 5.0 -- example 3 m = IO >> return "print-me" >>= putStrLn >> fail "-- stopping by invoking IO's fail --" -- ------------------------------------------------LA--11/2005--

RR = revised report, CUP, 2003.