Compose

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`Compose` is a nice little module which shows off some of the features of the various monads around.

The task is to write a function `compose :: [a -> a] -> (a -> a)`, which should take a list of functions and chain them together: a value would be fed into the first, which then produces a new value, which is fed into the third, and so on. I.e. `compose [(*2), (+1)] 3 = 8`.

This page illustrates the solution in different monads. Most are a bit of a joke; you'd probably only ever use the first solution presented, but nevertheless the nice features of the various monads are demonstrated.

The Sane Solution

```compose :: [a -> a] -> a -> a
compose fs v = foldl (flip (.)) id fs \$ v
```

This one's easy. We fold over the list using composition as our combinator to join items. We flip it, though, because we want a value to be fed into the first function in the list first, but `f . g` applies `g` before `f`.

We use `id` as the starting value, as `id` is an identity for composition.

Using `State`

```composeState :: [a -> a] -> a -> a
composeState = execState . mapM modify
```

Here we use but a single feature of the very powerful `State` monad. To understand how this code works, consider it in the less eta-reduced form:

```composeState fs v = execState (mapM modify fs) v
```

`mapM` iterates over the list of functions, applying `modify` to each one. If we were to expand a list after it had been mapped over in this way,

```fs = mapM modify [(*2), (+1), \n -> (n - 5) * 4]
-- fs is entirely equivalent to the following do-block:
fs' = do modify (*2)
modify (+1)
modify (\n -> (n - 5) * 4)
```

In other words, we obtain a stateful computation that modifies the state with the first function in the list, then the second, and so on.

Using `Reader`

```composeReader :: [a -> a] -> a -> a
composeReader fs v = runReader (compose' fs) v
where compose' []     = ask
compose' (f:fs) = local f (compose' fs)
```

The `Reader` monad is a bit like `State`, but not as general. It doesn't allow you to permanently change the environment, so that spoils our previous approach of chaining `modify` calls together in a do-block.

However, there is a useful little function `local` which allows us to temporarily modify the environment for a given computation. We can use this to recurse on our list: we modify the environment by the first function in the list for the computation `compose' fs`, which is the recursion on the rest of the list. Again, we could expand out:

```fs = compose' [(*2), (+1), \n -> (n - 5) * 4]
-- again, this is entirely equivalent to the following:
fs' = local (*2) \$
local (+1) \$
local (\n -> (n - 5) * 4) ask
```

`ask` is a simple function that just returns the current environment.

Our tactic in words here is to create a computation (the result of the `compose'` call) which modifies the environment for another computation which further modifies the environment for another computation which modifies the environment yet further... until finally we hit the end of our list and we just return the environment, which has had all the functions in our list applied to it.

Once this composition has been built up, we run it, starting off with an environment of the starting value.

Using `Writer`

```composeWriter :: [a -> a] -> a -> a
composeWriter fs v = (execWriter \$ compose' fs) v
where compose' []     = return id
compose' (f:fs) = censor (. f) (compose' fs)
```

This example comes with a disclaimer. You should never use `Writer` in this way. This example was really just an experimentation to see if `Writer` could be twisted in this way!

Our tactic here is really rather similar to that of the `Reader` example. We build up a computation which is the result of modifying the environment (or, actually, as we're working in `Writer`, we modify the output). `censor`, to quote All About Monad, "...takes a function and a `Writer` and produces a new `Writer` whose output is the same but whose log entry has been modified by the function.". So, we compose each function in turn onto the "log output" (which is actually a chain of composed functions).

Once this computation has been built up, we extract this long composition chain, and apply it to our starting value.

Also interesting to note is that this method is really equivalent to the "Sane method" described above. We just iterate along the list, accumulating a composition chain. The only difference here is that we're using some of `Writer`'s plumbing to make it more subtle.

Using `Cont`

The `Cont` example is so complicated it needs its own helper functions!

```getCC :: MonadCont m => m (m a)
getCC = callCC (\c -> let x = c x in return x)
getCC' :: MonadCont m => a -> m (a, a -> m b)
getCC' x0 = callCC (\c -> let f x = c (x, f) in return (x0, f)
```

These are rather complex if you don't know CPS and the `Cont` monad well already, but essentially they create a GOTO-style checkpoint. You can use a do-block something along these lines:

```do stuff;
boing <- getCC
moreStuff;
boing;
```

And you'll pop back up to `boing`. `getCC'` is similar, but it lets you use a parameter. Again, to exemplify:

```x `modulo` m = (`runContT` return) \$ do (u, jump) <- getCC' x
lift \$ print u
case u of
_ | u < 0     -> jump (u + m)
| u >= m    -> jump (u - m)
| otherwise -> return u
```

This take `m`-sized chunks off `u` (which starts off being `x`) until `u` is in range. Right, on to the actual `compose` itself:

```composeCont :: [a -> a] -> a -> a
composeCont fs = runCont compose' id
where compose' = do ((gs,f), jump) <- getCC' (fs,id)
case gs of
[]      -> return f
(g:gs') -> jump (gs', g . f)
```

This is similar to the `modulo` loop above, but uses a pair as the passed parameter. It walks down the list, accumulating the composition chain. Instead of building a composition chain, it is possible to act on the value directly, a bit like with the `State` example:

```composeCont :: [a -> a] -> a -> a
composeCont fs x = runCont compose' id
where compose' = do ((gs,y), jump) <- getCC' (fs,x)
case gs of
[]      -> return y
(g:gs') -> jump (gs', g y)
```

Thanks to Cale Gibbard for providing this example.

References

Mainly see All About Monads, specifically chapter two, which has overviews and examples for all the major monads.

The whole code

In case you wish to run this code, here it is in its entirety:

```-- Thread a value through a list of function applications

module Compose where

import Control.Monad.Writer
import Control.Monad.Reader
import Control.Monad.State
import Control.Monad.Cont

compose :: [a -> a] -> a -> a
compose fs v = foldl (flip (.)) id fs \$ v

composeState :: [a -> a] -> a -> a
composeState = execState . mapM modify

composeReader :: [a -> a] -> a -> a
composeReader fs v = runReader (compose' fs) v
where compose' []     = ask
compose' (f:fs) = local f (compose' fs)

composeWriter :: [a -> a] -> a -> a
composeWriter fs v = (execWriter \$ compose' fs) v
where compose' []     = return id
compose' (f:fs) = censor (. f) (compose' fs)

getCC :: MonadCont m => m (m a)
getCC = callCC (\c -> let x = c x in return x)
getCC' :: MonadCont m => a -> m (a, a -> m b)
getCC' x0 = callCC (\c -> let f x = c (x, f) in return (x0, f))

composeCont :: [a -> a] -> a -> a
composeCont fs = runCont compose' id
where compose' = do ((gs,f), jump) <- getCC' (fs,id)
case gs of
[]      -> return f
(g:gs') -> jump (gs',g . f)

main = do let fs = [(+1), (*2), \n -> (n - 5) * 4]
v  = 3
putStrLn \$ "compose: "       ++ (show \$ compose       fs v)
putStrLn \$ "compostState: "  ++ (show \$ composeState  fs v)
putStrLn \$ "composeReader: " ++ (show \$ composeReader fs v)
putStrLn \$ "composeWriter: " ++ (show \$ composeWriter fs v)
putStrLn \$ "composeCont: "   ++ (show \$ composeCont   fs v)

{-
*Compose> main
compose: 12
compostState: 12
composeReader: 12
composeWriter: 12
composeCont: 12
-}
```