Navigating and modifying ASTs built on the Free monad in Haskell

Question

I'm attempting to structure an AST using the Free monad based on some helpful literature that I've read online.

I have some questions about working with these kinds of ASTs in practice, which I've boiled down to the following example.

Suppose my language allows for the following commands:

{-# LANGUAGE DeriveFunctor #-}

data Command next
  = DisplayChar Char next
  | DisplayString String next
  | Repeat Int (Free Command ()) next
  | Done
  deriving (Eq, Show, Functor)

and I define the Free monad boilerplate manually:

displayChar :: Char -> Free Command ()
displayChar ch = liftF (DisplayChar ch ())

displayString :: String -> Free Command ()
displayString str = liftF (DisplayString str ())

repeat :: Int -> Free Command () -> Free Command ()
repeat times block = liftF (Repeat times block ())

done :: Free Command r
done = liftF Done

which allows me to specify programs like the following:

prog :: Free Command r
prog =
  do displayChar 'A'
     displayString "abc"

     repeat 5 $
       displayChar 'Z'

     displayChar '\n'
     done

Now, I'd like to execute my program, which seems simple enough.

execute :: Free Command r -> IO ()
execute (Free (DisplayChar ch next)) = putChar ch >> execute next
execute (Free (DisplayString str next)) = putStr str >> execute next
execute (Free (Repeat n block next)) = forM_ [1 .. n] (\_ -> execute block) >> execute next
execute (Free Done) = return ()
execute (Pure r) = return ()

and

λ> execute prog
AabcZZZZZ

Okay. That's all nice, but now I want to learn things about my AST, and execute transformations on it. Think like optimizations in a compiler.

Here's a simple one: If a Repeat block only contains DisplayChar commands, then I'd like to replace the whole thing with an appropriate DisplayString. In other words, I'd like to transform repeat 2 (displayChar 'A' >> displayChar 'B') with displayString "ABAB".

Here's my attempt:

optimize c@(Free (Repeat n block next)) =
  if all isJust charsToDisplay then
    let chars = catMaybes charsToDisplay
    in
      displayString (concat $ replicate n chars) >> optimize next
  else
    c >> optimize next
  where
    charsToDisplay = project getDisplayChar block
optimize (Free (DisplayChar ch next)) = displayChar ch >> optimize next
optimize (Free (DisplayString str next)) = displayString str >> optimize next
optimize (Free Done) = done
optimize c@(Pure r) = c

getDisplayChar (Free (DisplayChar ch _)) = Just ch
getDisplayChar _ = Nothing

project :: (Free Command a -> Maybe u) -> Free Command a -> [Maybe u]
project f = maybes
  where
    maybes (Pure a) = []
    maybes c@(Free cmd) =
      let build next = f c : maybes next
      in
        case cmd of
          DisplayChar _ next -> build next
          DisplayString _ next -> build next
          Repeat _ _ next -> build next
          Done -> []

Observing the AST in GHCI shows that this work correctly, and indeed

λ> optimize $ repeat 3 (displayChar 'A' >> displayChar 'B')
Free (DisplayString "ABABAB" (Pure ()))


λ> execute . optimize $ prog
AabcZZZZZ
λ> execute prog
AabcZZZZZ 

But I'm not happy. In my opinion, this code is repetitive. I have to define how to traverse through my AST every time I want to examine it, or define functions like my project that give me a view into it. I have to do this same thing when I want to modify the tree.

So, my question: is this approach my only option? Can I pattern-match on my AST without dealing with tonnes of nesting? Can I traverse the tree in a consistent and generic way (maybe Zippers, or Traversable, or something else)? What approaches are commonly taken here?

The whole file is below:

{-# LANGUAGE DeriveFunctor #-}

module Main where

import Prelude hiding (repeat)

import Control.Monad.Free

import Control.Monad (forM_)
import Data.Maybe (catMaybes, isJust)

main :: IO ()
main = execute prog

prog :: Free Command r
prog =
  do displayChar 'A'
     displayString "abc"

     repeat 5 $
       displayChar 'Z'

     displayChar '\n'
     done

optimize c@(Free (Repeat n block next)) =
  if all isJust charsToDisplay then
    let chars = catMaybes charsToDisplay
    in
      displayString (concat $ replicate n chars) >> optimize next
  else
    c >> optimize next
  where
    charsToDisplay = project getDisplayChar block
optimize (Free (DisplayChar ch next)) = displayChar ch >> optimize next
optimize (Free (DisplayString str next)) = displayString str >> optimize next
optimize (Free Done) = done
optimize c@(Pure r) = c

getDisplayChar (Free (DisplayChar ch _)) = Just ch
getDisplayChar _ = Nothing

project :: (Free Command a -> Maybe u) -> Free Command a -> [Maybe u]
project f = maybes
  where
    maybes (Pure a) = []
    maybes c@(Free cmd) =
      let build next = f c : maybes next
      in
        case cmd of
          DisplayChar _ next -> build next
          DisplayString _ next -> build next
          Repeat _ _ next -> build next
          Done -> []

execute :: Free Command r -> IO ()
execute (Free (DisplayChar ch next)) = putChar ch >> execute next
execute (Free (DisplayString str next)) = putStr str >> execute next
execute (Free (Repeat n block next)) = forM_ [1 .. n] (\_ -> execute block) >> execute next
execute (Free Done) = return ()
execute (Pure r) = return ()

data Command next
  = DisplayChar Char next
  | DisplayString String next
  | Repeat Int (Free Command ()) next
  | Done
  deriving (Eq, Show, Functor)

displayChar :: Char -> Free Command ()
displayChar ch = liftF (DisplayChar ch ())

displayString :: String -> Free Command ()
displayString str = liftF (DisplayString str ())

repeat :: Int -> Free Command () -> Free Command ()
repeat times block = liftF (Repeat times block ())

done :: Free Command r
done = liftF Done

Answer 1

If your issue is with boilerplate, you won't get around it if you use Free! You will always be stuck with an extra constructor on each level.

But on the flip side, if you are using Free, you have a very easy way to generalize recursion over your data structure. You can write this all from scratch, but I used the recursion-schemes package:

import Data.Functor.Foldable 

data (:+:) f g a = L (f a) | R (g a) deriving (Functor, Eq, Ord, Show)

type instance Base (Free f a) = f :+: Const a 
instance (Functor f) => Foldable (Free f a) where 
  project (Free f) = L f 
  project (Pure a) = R (Const a)
instance Functor f => Unfoldable (Free f a) where 
  embed (L f) = Free f
  embed (R (Const a)) = Pure a 
instance Functor f => Unfoldable (Free f a) where 
  embed (L f) = Free f
  embed (R (Const a)) = Pure a 

If you are unfamiliar with this (read the documentation), but basically all you need to know is project takes some data, like Free f a, and "un-nests" it by one level, producing something like (f :+: Const a) (Free f a). Now, you have given regular functions like fmap, Data.Foldable.foldMap, etc, access to the structure of your data, since the argument of the functor is the sub-tree.

Executing is very simple, although not much more concise:

execute :: Free Command r -> IO ()
execute = cata go where 
  go (L (DisplayChar ch next)) = putChar ch >> next
  go (L (DisplayString str next)) = putStr str >> next
  go (L (Repeat n block next)) = forM_ [1 .. n] (const $ execute block) >> next
  go (L Done) = return ()
  go (R _) = return ()

However, simplification becomes much easier. We can define simplification over all datatypes which have Foldable and Unfoldable instances:

reduce :: (Foldable t, Functor (Base t), Unfoldable t) => (t -> Maybe t) -> t -> t 
reduce rule x = let y = embed $ fmap (reduce rule) $ project x in 
  case rule y of 
    Nothing -> y
    Just y' -> y' 

The simplification rule only needs to simplify one level of the AST (namely, the top-most level). Then, if the simplification can apply to the substructure, it will perform it there too. Note that the above reduce works bottom up; you can also have a top down reduction:

reduceTD :: (Foldable t, Functor (Base t), Unfoldable t) => (t -> Maybe t) -> t -> t 
reduceTD rule x = embed $ fmap (reduceTD rule) $ project y
  where y = case rule x of 
              Nothing -> x 
              Just x' -> x' 

Your example simplification rule can be written very simply:

getChrs :: (Command :+: Const ()) (Maybe String) -> Maybe String 
getChrs (L (DisplayChar c n)) = liftA (c:) n
getChrs (L Done) = Just []
getChrs (R _) = Just []
getChrs _ = Nothing 

optimize (Free (Repeat n dc next)) = do 
  chrs <- cata getChrs dc
  return $ Free $ DisplayString (concat $ map (replicate n) chrs) next
optimize _ = Nothing

Because of the way you've defined your datatype, you don't have access to the 2nd arguement of Repeat, so for things like repeat' 5 (repeat' 3 (displayChar 'Z')) >> done, the inner repeat can't be simplified. If this is a situation you expect to deal with, you either change your datatype and accept a lot more boilerplate, or write an exception:

reduceCmd rule (Free (Repeat n c r)) = 
let x = Free (Repeat n (reduceCmd rule c) (reduceCmd rule r)) in 
    case rule x of
      Nothing -> x
      Just x' -> x' 
reduceCmd rule x = embed $ fmap (reduceCmd rule) $ project x 

---

Using recursion-schemes or the like will probably make your code more easily extensible. But it isn't necessary by any means:

execute = iterM go where 
  go (DisplayChar ch next) = putChar ch >> next
  go (DisplayString str next) = putStr str >> next
  go (Repeat n block next) = forM_ [1 .. n] (const $ execute block) >> next
  go Done = return ()

getChrs can't access Pure, and your programs will be of the form Free Command (), so before you apply it, you have to get replace () with Maybe String.

getChrs :: Command (Maybe String) -> Maybe String
getChrs (DisplayChar c n) = liftA (c:) n
getChrs (DisplayString s n) = liftA (s++) n 
getChrs Done = Just []
getChrs _ = Nothing 

optimize :: Free Command a -> Maybe (Free Command a)
optimize (Free (Repeat n dc next)) = do 
  chrs <- iter getChrs $ fmap (const $ Just []) dc
  return $ Free $ DisplayString (concat $ map (replicate n) chrs) next
optimize _ = Nothing

Note that reduce is almost the exact same as before, except for two things: project and embed are replaced with pattern matching on Free and Free, respectively; and you need a separate case for Pure. This should tell you that Foldable and Unfoldable generalize things that "look like" Free.

reduce
  :: Functor f =>
     (Free f a -> Maybe (Free f a)) -> Free f a -> Free f a

reduce rule (Free x) = let y = Free $ fmap (reduce rule) $ x in 
  case rule y of 
    Nothing -> y
    Just y' -> y' 
reduce rule a@(Pure _) = case rule a of 
                           Nothing -> a
                           Just  b -> b 

All the other functions are modified similarly.

Answer 2

Here's my take using syb (as mentioned on Reddit):

{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE DeriveDataTypeable #-}

module Main where

import Prelude hiding (repeat)

import Data.Data

import Control.Monad (forM_)

import Control.Monad.Free
import Control.Monad.Free.TH

import Data.Generics (everywhere, mkT)

data CommandF next = DisplayChar Char next
                   | DisplayString String next
                   | Repeat Int (Free CommandF ()) next
                   | Done
  deriving (Eq, Show, Functor, Data, Typeable)

makeFree ''CommandF

type Command = Free CommandF

execute :: Command () -> IO ()
execute = iterM handle
  where
    handle = \case
        DisplayChar ch next -> putChar ch >> next
        DisplayString str next -> putStr str >> next
        Repeat n block next -> forM_ [1 .. n] (\_ -> execute block) >> next
        Done -> return ()

optimize :: Command () -> Command ()
optimize = optimize' . optimize'
  where
    optimize' = everywhere (mkT inner)

    inner :: Command () -> Command ()
    -- char + char becomes string
    inner (Free (DisplayChar c1 (Free (DisplayChar c2 next)))) = do
        displayString [c1, c2]
        next

    -- char + string becomes string
    inner (Free (DisplayChar c (Free (DisplayString s next)))) = do
        displayString $ c : s
        next

    -- string + string becomes string
    inner (Free (DisplayString s1 (Free (DisplayString s2 next)))) = do
        displayString $ s1 ++ s2
        next

    -- Loop unrolling
    inner f@(Free (Repeat n block next)) | n < 5 = forM_ [1 .. n] (\_ -> block) >> next
                                         | otherwise = f

    inner a = a

prog :: Command ()
prog = do
    displayChar 'a'
    displayChar 'b'
    repeat 1 $ displayChar 'c' >> displayString "def"
    displayChar 'g'
    displayChar 'h'
    repeat 10 $ do
        displayChar 'i'
        displayChar 'j'
        displayString "klm"
    repeat 3 $ displayChar 'n'

main :: IO ()
main = do
    putStrLn "Original program:"
    print prog
    putStrLn "Evaluation of original program:"
    execute prog
    putStrLn "\n"

    let opt = optimize prog
    putStrLn "Optimized program:"
    print opt
    putStrLn "Evaluation of optimized program:"
    execute opt
    putStrLn ""

Output:

$ cabal exec runhaskell ast.hs
Original program:
Free (DisplayChar 'a' (Free (DisplayChar 'b' (Free (Repeat 1 (Free (DisplayChar 'c' (Free (DisplayString "def" (Pure ()))))) (Free (DisplayChar 'g' (Free (DisplayChar 'h' (Free (Repeat 10 (Free (DisplayChar 'i' (Free (DisplayChar 'j' (Free (DisplayString "klm" (Pure ()))))))) (Free (Repeat 3 (Free (DisplayChar 'n' (Pure ()))) (Pure ()))))))))))))))
Evaluation of original program:
abcdefghijklmijklmijklmijklmijklmijklmijklmijklmijklmijklmnnn

Optimized program:
Free (DisplayString "abcdefgh" (Free (Repeat 10 (Free (DisplayString "ijklm" (Pure ()))) (Free (DisplayString "nnn" (Pure ()))))))
Evaluation of optimized program:
abcdefghijklmijklmijklmijklmijklmijklmijklmijklmijklmijklmnnn

It might be possible to get rid of the *Free*s using GHC 7.8 Pattern Synonyms, but for some reason the above code only works using GHC 7.6, the Data instance of Free seems to be missing. Should look into that...