peterv bf3 at telenet.be
Thu Aug 2 07:01:05 EDT 2007

```IMHO although this is a great explanation of what a monad is (as far as I understand it), for newbies I think it helps a lot to first look at the specific cases where monads are used and why they are invented, as explained in http://sigfpe.blogspot.com/2006/08/you-could-have-invented-monads-and.html and then looking at http://haskell.org/haskellwiki/IO_inside for the IO thingy.

However, one thing which I find annoying is that a "classic pure" function cannot evaluate an IO function unless you use unsafePerformIO; one must "promote" (?demote) the pure function into the IO monad.

For example, as an exercise I tried to convert the Monte Carlo experiment as demonstrated in http://mitpress.mit.edu/sicp/full-text/sicp/book/node53.html into Haskell (warning: newbie code ahead, should be much nicer when I once become a real Haskeller in a million years ;-)

In the code below I had to change the monteCarlo1 into a completely different monteCarlo2 in order to use the IO random facility. Okay, this is "expected behavior", but for an imperative programmer, this is quite a shock!

Any ways of "promoting" such a pure function into the monadic one automatically? I tried playing with "liftM", without succes.

import Data.Ratio
import Data.List
import System.Random

-- Monte Carlo using "pure" functions
monteCarlo1 :: Integral n => n -> (a -> (a,Bool)) -> a -> Ratio n
monteCarlo1 trials experiment startValue = trialsPassed % trials
where trialsPassed = genericLength \$ filter id outcomes
outcomes = snd \$ unzip \$ genericTake trials \$ iterate (experiment . fst) (experiment startValue)

cesaroTest1 :: StdGen -> (StdGen, Bool)
cesaroTest1 gen0 = (gen2, (gcd rand1 rand2) == 1)
where (rand1, gen1) = random gen0
(rand2, gen2) = random gen1

estimatePi1 trials = sqrt \$ 6 / (fromRational \$ monteCarlo1 trials cesaroTest1 (mkStdGen 0))

-- Monte Carlo using monadic IO
-- no genericReplicateM, so must use Int
monteCarlo2 :: Int -> IO Bool -> IO (Ratio Integer)
monteCarlo2 trials experiment = do
outcomes <- replicateM trials experiment
return \$ fromIntegral (genericLength \$ filter id outcomes) % fromIntegral trials

cesaroTest2 = do
rand1 <- getStdRandom random
rand2 <- getStdRandom random
return \$ (gcd rand1 rand2) == 1

estimatePi2 trials = do
mc <- monteCarlo2 trials cesaroTest2
return \$ sqrt \$ 6 / (fromRational mc)

main = let pi1 = estimatePi1 50000 in
do pi2 <- estimatePi2 50000
putStrLn (show pi1) >> putStrLn (show pi2)

-----Original Message-----
Sent: Thursday, August 02, 2007 12:18 AM

a Monad is a type constructor with two operations, implementing
a standard interface and following a few simple rules.

the Monad type class tells you the interface (what operations
you've got, and their types), the Monad laws tell you what all
types implementing that interface should have in common.

the monadic interface gives you two operations, one to throw
things into a monad thing (return), and one to chain two monad
things together (>>=). the chaining explicitly caters for information
flowing from the first to the second parameter of (>>=).

thrown together in that way: whatever it is the monad does,
anything just thrown into it will take no part in that action,
and whichever way you use that chaining operation, the
structure of chaining is irrelevant, only the ordering of chained

there are usually other ways to create 'primitive' monadic things,
which can be combined into complex monadic structures using
the operations from the Monad interface.

there is usually a way to interpret monadic structures built in
this way (a 'run' operation of some kind).

that's it, i think?-)

claus

examples include:

- i/o: primitive monadic things are basic i/o operations,
the 'run' operation is outside the language, applied to
'Main.main', and interprets (abstract) IO monad structures
sequentially, starting with the leftmost innermost i/o
operation in the structure and applying the second
argument of (>>=) to the result of executing the first.

- []: primitive monadic things are lists, the 'run' operation
is the identity, ie, the lists are directly exposed as data
structures, return creates a singleton list, (>>=) applies
its second argument to each element of its first argument
and concatenates the results (concatMap).

- State: primitive monadic things are operations on a state
type, returning a result and a state; return returns its
parameter, passing its input state unchanged, (>>=) applies
its first parameter to the input state, applies its second
parameter to the result value and result state of the first.
'run' is runState and applies a (possibly) complex monadic
thing to an input state, returning a result and a (modified)
state.

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