Operational-style Applicative, Alternative, Monad and MonadPlus, using free monads.
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free-operational.cabal

README.md

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free-operational

A reconstruction of Heinrich Apfelmus's operational package, but:

  1. Built with free monads (using Edward Kmett's free and kan-extensions packages). All the program types in this package can be translated to the corresponding free types and back (using Data.Functor.Yoneda.Contravariant)

  2. Applicative, Alternative and MonadPlus variants of operational's Program type. The Applicative and Alternative program types, in particular, allow for easy static analysis.

Example: Applicative version of Reader

{-# LANGUAGE GADTs, RankNTypes, ScopedTypeVariables #-}

import Control.Applicative.Operational

type Reader r a = ProgramAp (ReaderI r) a

data ReaderI r a where
    Ask :: ReaderI r r

ask :: Reader r r
ask = singleton Ask

runReader :: forall r a. Reader r a -> r -> a
runReader = interpretAp evalI
    where evalI :: forall x. ReaderI r x -> r -> x
          evalI Ask = id

Static analysis example: count how many times ask is used in an applicative Reader program.

countAsk :: forall r a. Reader r a -> Int
countAsk = count . viewAp
    where count :: forall x. ProgramViewAp (ReaderI r) x -> Int
          count (Pure _) = 0
          count (Ask :<**> k) = succ (count k)

Since this Reader language only has one instruction, we can cheat and make this even shorter:

countAsk :: forall r a. Reader r a -> Int
countAsk = length . filter isAsk . instructions
    where isAsk (AnyInstr Ask) = True

Example: Toy Alternative parsers

Simple Alternative parser combinators:

{-# LANGUAGE GADTs, RankNTypes, ScopedTypeVariables #-}

import Control.Applicative
import Control.Alternative.Operational
import Control.Monad
import Control.Monad.Trans.State
import Data.Functor.Compose (Compose(..))
import Data.Traversable
import Data.Maybe (listToMaybe)

data ParserI a where
    Symbol :: Char -> ParserI Char

char :: Operational ParserI f => Char -> f Char
char = singleton . Symbol

string :: (Operational ParserI f, Applicative f) => String -> f String
string = traverse char

oneOf :: (Operational ParserI f, Alternative f) => String -> f Char
oneOf = foldr (<|>) empty . map char

-- | Example parser: match parentheses and count depth.
parens :: ProgramAlt ParserI Int
parens = pure 0  <|>  fmap (+1) (char '(' *> parens <* char ')')

Example "syntactic" interpreter, pattern matching on the view type:

runParser :: ProgramAlt ParserI a -> String -> Maybe a
runParser = fmap listToMaybe . eval . viewAlt
    where
      eval :: ProgramViewAlt ParserI a -> String -> [a]
      eval (Pure a) [] = pure a
      eval (Pure a) _  = empty
      eval (Symbol c :<**> k) [] = empty
      eval (Symbol c :<**> k) (x:xs) 
          | c == x    = pure c <**> eval k xs
          | otherwise = empty
      eval (Many ps) str = fmap asum (sequenceA (map eval ps)) str

asum :: Alternative f => [f a] -> f a
asum = foldr (<|>) empty

Example "denotational" interpreter:

runParser' :: ProgramAlt ParserI a -> String -> Maybe a
runParser' = (firstSuccess .) . runStateT . interpretAlt evalParserI
    where firstSuccess [] = Nothing
          firstSuccess ((a,""):_) = Just a
          firstSuccess (_:xs) = firstSuccess xs

evalParserI :: ParserI a -> StateT String [] a
evalParserI (Symbol c) = 
    do str <- get
       case str of
         x:xs | c == x -> put xs >> return c
         otherwise     -> mzero

Simple static analysis example: enumerate the strings accepted by a (non-degenerate) parser.

enumerate :: ProgramAlt ParserI a -> [String]
enumerate = go [showString ""] . viewAlt
    where
      go :: [ShowS] -> ProgramViewAlt ParserI a -> [String]
      go strs (Pure a) = map ($"") strs
      go strs (Symbol c :<**> k) = go (map (.(showChar c)) strs) k
      go strs (Many ps) = interleave $ map (go strs) ps

interleave :: [[a]] -> [a]
interleave = foldr interleave2 []
    where
      interleave2 :: [a] -> [a] -> [a]
      interleave2 [] ys = ys
      interleave2 (x:xs) ys = x : interleave2 ys xs

Example, using parens from above:

>>> take 5 $ enumerate parens
["","()","(())","((()))","(((())))"]

Another toy static analysis example: optimize a (non-degenerate) parser by merging on common prefixes.

optimize :: ProgramAlt ParserI a -> ProgramAlt ParserI a
optimize = compileAlt . merge . viewAlt

merge :: ProgramViewAlt ParserI a -> ProgramViewAlt ParserI a
merge p@(Pure _) = p
merge (Symbol a :<**> k) = Symbol a :<**> merge k
merge (Many ps) = Many (mergeMany ps)

mergeMany :: [ProgramViewAlt ParserI a] -> [ProgramViewAlt ParserI a]
mergeMany = foldr step [] . map merge
    where step (Pure a) ps = Pure a : ps
          step (Symbol a :<**> l) ((Symbol b :<**> r) : ps) =
               case a `compare` b of
                 EQ -> (Symbol a :<**> Many (mergeMany [l, r])) : ps
                 LT -> (Symbol a :<**> l) : (Symbol b :<**> r) : ps
                 GT -> (Symbol b :<**> r) : (Symbol a :<**> l) : ps
          step (Symbol a :<**> l) ps = (Symbol a :<**> l) : ps
          step (Many ps) ps' = mergeMany (mergeMany ps ++ ps')

tokens :: [String] -> ProgramAlt ParserI String 
tokens = asum . map string

example = ["abactor", "abacus", "abaft", "abaisance", "abaissed", "abalone"]

describe :: forall a. ProgramAlt ParserI a -> Description
describe = eval . viewAlt
    where eval :: forall x. ProgramViewAlt ParserI x -> Description
          eval (Pure _) = Ok
          eval (Symbol c :<**> k) = c :> (eval k)
          eval (Many ps) = OneOf (map eval ps)

data Description = Ok
                 | Char :> Description
                 | OneOf [Description] 
                   deriving Show

>>> describe $ tokens example
OneOf ['a' :> ('b' :> ('a' :> ('c' :> ('t' :> ('o' :> ('r' :> Ok)))))),
       OneOf ['a' :> ('b' :> ('a' :> ('c' :> ('u' :> ('s' :> Ok))))),
              OneOf ['a' :> ('b' :> ('a' :> ('f' :> ('t' :> Ok)))),
                     OneOf ['a' :> ('b' :> ('a' :> ('i' :> ('s' :> ('a' :> ('n' :> ('c' :> ('e' :> Ok)))))))),
                            OneOf ['a' :> ('b' :> ('a' :> ('i' :> ('s' :> ('s' :> ('e' :> ('d' :> Ok))))))),
                                   'a' :> ('b' :> ('a' :> ('l' :> ('o' :> ('n' :> ('e' :> Ok))))))]]]]]

>>> describe $ optimize (tokens example)
'a' :> ('b' :> ('a' :> OneOf ['c' :> OneOf ['t' :> ('o' :> ('r' :> Ok)),
                                            'u' :> ('s' :> Ok)],
                              OneOf ['f' :> ('t' :> Ok),
                                     OneOf ['i' :> ('s' :> OneOf ['a' :> ('n' :> ('c' :> ('e' :> Ok))),
                                                                  's' :> ('e' :> ('d' :> Ok))]),
                                            'l' :> ('o' :> ('n' :> ('e' :> Ok)))]]]))

Sums of instruction sets

Control.Operational.Instruction reexports Data.Functor.Coproduct, which is rather useful in the context of this library:

import Control.Operational.Instruction

-- | An alternative parser instruction set, and an evaluation.
data StringI a where
    String :: String -> StringI String

evalStringI :: StringI a -> StateT String [] a
evalStringI (String "") = return ""
evalStringI (String str) = 
    do str' <- get
       case str `stripPrefix` str' of
         Nothing -> mzero
         Just suffix -> put suffix >> return str

-- | If we know how to interpret two instruction sets at the same
-- type, we know how to interpret their union.
runStringP :: ProgramAlt (Coproduct ParserI StringI) a
           -> String
           -> [(a, String)]
runStringP = runStateT . interpretAlt (coproduct evalParserI evalStringI)

References

  1. http://stackoverflow.com/questions/14263363/is-operational-really-isomorphic-to-a-free-monad
  2. http://www.reddit.com/r/haskell/comments/17a33g/free_functors_the_reason_free_and_operational_are/
  3. http://gergo.erdi.hu/blog/2012-12-01-static_analysis_with_applicatives/
  4. http://paolocapriotti.com/blog/2013/04/03/free-applicative-functors/
  5. http://web.jaguarpaw.co.uk/~tom/blog/2012/09/09/towards-free-applicatives.html