{-# OPTIONS_GHC -fno-warn-unused-do-bind #-}
+{-# LANGUAGE PatternSynonyms #-}
-module Lambda where
+-- |
+-- Module : Lambda
+-- Copyright : Tomáš Musil 2014
+-- License : BSD-3
+--
+-- Maintainer : tomik.musil@gmail.com
+-- Stability : experimental
+--
+-- This is a toy λ-calculus implementation.
+
+module Lambda
+ ( -- * Types
+ VarName
+ , Term(..)
+ -- * Parsing terms
+ , parseTerm
+ , tRead
+ -- * Reduction
+ , reduce
+ ) where
+
import Data.Text as T
import Data.Attoparsec.Text
import Control.Applicative
+import Control.Monad.State
+
+-- $setup
+-- >>> import Test.QuickCheck
+-- >>> import Control.Applicative
+-- >>> let aTerm 0 = liftA (Var . ("x" ++) . show) (arbitrary :: Gen Int)
+-- >>> let aTerm n = oneof [aTerm 0, liftA (Lambda "x") $ aTerm (n - 1), liftA2 App (aTerm (n `div` 2)) (aTerm (n `div` 2))]
+-- >>> instance Arbitrary Term where arbitrary = sized aTerm
+
+cY :: Term
+cY = tRead "λf.(λx.f (x x)) (λx.f (x x))"
+
+cI :: Term
+cI = tRead "λx.x"
+
+cK :: Term
+cK = tRead "λx y.x"
type VarName = String
-data Term = Var VarName | Lambda VarName Term | App Term Term
+
+-- |
+-- >>> print $ Lambda "x" (Var "x")
+-- (λx.x)
+
+data Term = EmptyT | Var VarName | Lambda VarName Term | App Term Term deriving (Eq)
+
+pattern RedEx x t s = App (Lambda x t) s
+pattern AppApp a b c = App a (App b c)
+pattern EmLambda x y t = Lambda x (Lambda y t)
+
instance Show Term where
show (Var x) = x
- show (Lambda x t) = "\\" ++ x ++ "." ++ show t
- show (App t r) = "(" ++ show t ++ " " ++ show r ++ ")"
+ show (EmLambda x y t) = show (Lambda (x ++ " " ++ y) t)
+ show (Lambda x t) = "(λ" ++ x ++ "." ++ show t ++ ")"
+ show (AppApp a b c) = show a ++ " " ++ braced (App b c)
+ show (App t r) = show t ++ " " ++ show r
+
+braced :: Term -> String
+braced t = "(" ++ show t ++ ")"
+
+-- |
+-- prop> t == tRead (show (t :: Term))
---instance Read Term where
tRead :: String -> Term
tRead s = case parseOnly (parseTerm <* endOfInput) (T.pack s) of
(Right t) -> t
parseVar :: Parser Term
parseVar = do
- x <- many1 letter
+ x <- many1 (letter <|> digit)
return $! Var x
parseLambda :: Parser Term
parseLambda = do
- char '\\'
- vars <- many1 (parseVar <* char ' ')
+ char '\\' <|> char 'λ'
+ vars <- sepBy1 parseVar (char ' ')
char '.'
t <- parseTerm
return $! createLambda vars t
parseApp :: Parser Term
parseApp = do
+ aps <- sepBy1 (parseBraces <|> parseLambda <|> parseVar) (char ' ')
+ return $! createApp aps
+
+createApp :: [Term] -> Term
+createApp [t] = t
+createApp (t:ts:tss) = createApp (App t ts : tss)
+createApp [] = error "empty createApp"
+
+parseBraces :: Parser Term
+parseBraces = do
char '('
t <- parseTerm
- char ' '
- r <- parseTerm
char ')'
- return $! App t r
+ return t
parseTerm :: Parser Term
-parseTerm = parseVar <|> parseLambda <|> parseApp
+parseTerm = parseApp <|>
+ parseBraces <|>
+ parseLambda <|>
+ parseVar
-------------------------------------------------
| otherwise = Lambda x (substitute a b t)
substitute a b (App t u) = App (substitute a b t) (substitute a b u)
+-- | Reduce λ-term
+--
+-- >>> reduce $ tRead "(\\x.x x) (g f)"
+-- g f (g f)
+
reduce :: Term -> Term
reduce (Var x) = Var x
reduce (Lambda x t) = Lambda x (reduce t)
reduce (App t u) = app (reduce t) u
where app (Lambda x v) w = reduce $ substitute x w v
app a b = App a (reduce b)
+
+data Strategy = Eager | Lazy
+
+reduceStep :: (Monad m) => Term -> m Term
+reduceStep (RedEx x s t) = return $ substitute x t s
+reduceStep t = return $ t
+
+traversPost :: (Monad m) => (Term -> m Term) -> Term -> m Term
+traversPost f (App t u) = do
+ nt <- traversPost f t
+ nu <- traversPost f u
+ case App nt nu of
+ l@(RedEx _ _ _) -> traversPost f =<< f l
+ r -> return r
+traversPost f (Lambda x t) = return . Lambda x =<< traversPost f t
+traversPost f (Var x) = return $ (Var x)
+
+data Z = R Term Z | L Z Term | ZL VarName Z | E
+data D = Up | Down
+data Zip = Zip Z Term
+
+move (App l r, c, Down) = (l, L c r, Down)
+move (Lambda x t, c, Down) = (t, ZL x c, Down)
+move (Var x, c, Down) = (Var x, c, Up)
+move (t, L c r, Up) = (r, R t c, Down)
+move (t, R l c, Up) = (App l t, c, Up)
+move (t, ZL x c, Up) = (Lambda x t, c, Up)
+move (t, E, Up) = (t, E, Up)
+
+unmove (t, L c r, Down) = (App t r, c, Down)
+unmove x = x
+
+getF (t, _, _) = t
+
+travPost :: (Monad m) => (Term -> m Term) -> Term -> m Term
+travPost fnc term = tr fnc (term, E, Down)
+ where
+ tr f (t@(RedEx _ _ _), c, Up) = do
+ nt <- f t
+ tr f $ (nt, c, Down)
+ tr f (t, E, Up) = return t
+ tr f (t, c, Up) = tr f $ move (t, c, Up)
+ tr f (t, c, Down) = tr f $ move (t, c, Down)
+
+travPre :: (Monad m) => (Term -> m Term) -> Term -> m Term
+travPre fnc term = tr fnc (term, E, Down)
+ where
+ tr f (t@(RedEx _ _ _), c, Down) = do
+ nt <- f t
+ tr f $ unmove (nt, c, Down)
+ tr f (t, E, Up) = return t
+ tr f (t, c, Up) = tr f $ move (t, c, Up)
+ tr f (t, c, Down) = tr f $ move (t, c, Down)
+
+printT :: Term -> IO Term
+printT t = do
+ print t
+ return t
+
+-- |
+--
+-- >>> toNormalForm Eager 100 cI
+-- Just (λx.x)
+--
+-- >>> toNormalForm Eager 100 $ App cI cI
+-- Just (λx.x)
+--
+-- >>> toNormalForm Eager 1000 $ (App (App cK cI) cY)
+-- Nothing
+--
+-- >>> toNormalForm Lazy 1000 $ (App (App cK cI) cY)
+-- Just (λx.x)
+--
+-- prop> (\ t u -> t == u || t == Nothing || u == Nothing) (toNormalForm Lazy 1000 x) (toNormalForm Eager 1000 x)
+
+
+toNormalForm :: Strategy -> Int -> Term -> Maybe Term
+toNormalForm Eager n = flip evalStateT 0 . travPost (cnt >=> short n >=> reduceStep)
+toNormalForm Lazy n = flip evalStateT 0 . travPre (cnt >=> short n >=> reduceStep)
+
+cnt :: (Monad m) => Term -> StateT Int m Term
+cnt t@(RedEx _ _ _) = do
+ modify (+ 1)
+ return t
+cnt t = return t
+
+short :: Int -> Term -> StateT Int Maybe Term
+short max t = do
+ n <- get
+ if n > max
+ then lift Nothing
+ else return t
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