maanantai 20. helmikuuta 2012

The State Monad Challenge

I call monads like IO and STM "action monads", because each monadic value, such as getName :: IO String is actually an action that will yield the value when executed. In this posting I'll throw you a challenge: how would you implement the State monad, as described below.
Generally, a Monad can be constructed as below.
  1. Introduce your data type like data State
  2. Introduce the typeclass instance instance Monad State
  3. Implement the rest of the fucking Monad
Obviously the hard part is the third one. But now..

The State Monad Challenge

The challenge is to implement the State monad, which will allow you to write algorithms with the benefits of mutable state without having actual side-effects or mutable state. Um? Ok, let's say it allows you to read and write into a single variable, while actually carrying the current state in the stack. The function that use State don't have to worry about this though. You should be able to write stuff like
sum :: Num a => [a] -> a
sum nums = runState 0 $ do forM_ nums $ \num -> do 
                                acc <- readState
                                writeState (acc + num)
That's a sum algorithm using state. Not a very smart one, but serves the purpose of showing what you can do with State. It's important to note that the signature of sum indicates that it's pure, even though the implementation uses State internally.
We can refactor it to two pieces to separate the pure from the stateful:
sum nums = runState 0 $ add nums >> readState
add :: Num t => [s] -> State s ()
add nums = forM_ nums $ \num -> do acc <- readState
                                   writeState (acc + num)
From here we can imply a few things:
  1. State has two type parameters. Let's call them s (for state) and a (for return value)
  2. There are functions readState and writeState for reading and writing to the state variable
  3. There's a function runState for running a calculation with a given start state.
So let's start with
{-# LANGUAGE MultiParamTypeClasses #-}
module State(State, runState, readState, writeState) where

data State s a

instance Monad (State s) where
  return x = undefined
  action1 >>= action2 = undefined

runState :: s -> State s a -> a
runState st0 action = undefined

readState :: State s s
readState = undefined

writeState :: s -> State s ()
writeState x = undefined
Wow wow wow?! Lot's of stuff here? Well, that's just the interface for the State Monad. As you can see, I've allowed MultiParamTypeClasses there and declared that State s is a Monad. The tricky parts are coming up next. You should decide
  1. What kind of data the "monadic value" of State should encapsulate, i.e. what is the data constructor of State
  2. How will you implement the monad methods return and >>=
  3. How will you then implement runStatewriteState and readState.
As a tip, I suggest you start by considering what a single "state action" must be able to do, which is something like "given a starting state, it will return an end state and a return value"..