alpha equivalence

[fp.git] / src / Lambda.hs

diff --git a/src/Lambda.hs b/src/Lambda.hs
index f9fbe2b..07d1f4b 100644 (file)
--- a/src/Lambda.hs
+++ b/src/Lambda.hs
@@ -1,20 +1,98 @@
 {-# OPTIONS_GHC -fno-warn-unused-do-bind #-}
 {-# 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.

 
 

-import Data.Text as T
+module Lambda 
+  ( -- * Types
+    VarName
+  , Term(..)
+    -- * Parsing terms
+  , parseTerm
+  , tRead
+    -- * Reduction
+  , reduce
+  , toNormalForm
+  , Strategy(..)
+  ) where
+  
+
+import Data.Text as T hiding (map)

 import Data.Attoparsec.Text
 import Control.Applicative
 import Data.Attoparsec.Text
 import Control.Applicative

+import Control.Monad.State 
+
+-- $setup
+-- >>> import Test.QuickCheck
+-- >>> import Control.Applicative
+-- >>> let aVarName = oneof . map (pure . (:[])) $ ['a'..'e']
+-- >>> let aVar = liftA Var aVarName
+-- >>> let aComb = oneof . map pure $ [cS, cK, cI, cY]
+-- >>> let aTerm 0 = aVar 
+-- >>> let aTerm n = oneof [aVar, aComb, liftA2 Lambda aVarName $ aTerm (n - 1), liftA2 App (aTerm (n `div` 2)) (aTerm (n `div` 2))] 
+-- >>> instance Arbitrary Term where arbitrary = sized aTerm
+--
+-- TODO: shrink Terms
+
+cP :: Term
+cP = tRead "(λa d c.(λa.e) b (λc.d)) ((λa.(λd.a) (λd c.b ((λa.a) a)) (a ((λa.(λd.e) ((λe.(λd b.a) (λa c.(λa a d.(λd.b (λa d.c) e) (λb b.c a (a d (λb d d e a.d (λb b.d))))) ((λb.a) c)) (d ((λc.(λd.a (λe.e)) (c d)) ((λe.b) a))) c (λa.d (e (λe.(λd c.b) a))) (c (b a)) a (λe.(λa b e b a.d) b)) ((λe.b) (λa.b)) ((λe d.b) b) e) b) ((λc c.a e) (λb.(λb.e) a)))) (λe.e) b (λd c e e c a.c)) a)"
+
+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"
+
+cS :: Term
+cS = tRead "λx y z.x z (y z)"

 
 type VarName = String
 
 type VarName = String

-data Term = Var VarName | Lambda VarName Term | App Term Term
+
+-- | 
+-- >>> print $ Lambda "x" (Var "x")
+-- (λx.x)
+
+data Term = Var VarName | Lambda VarName Term | App Term Term deriving (Eq)
+
+varnames :: [VarName]
+varnames = map (:[]) ['a'..'z'] ++ [c : s | s <- varnames, c <- ['a'..'z']]
+
+alphaNorm :: Term -> Term
+alphaNorm t = alpha varnames t
+  where
+    alpha (v:vs) (Lambda x t) = Lambda v . alpha vs $ substitute x (Var v) t
+    alpha vs (App u v) = App (alpha vs u) (alpha vs v)
+    alpha vs (Var x) = Var x
+
+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
 
 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
 tRead :: String -> Term
 tRead s = case parseOnly (parseTerm <* endOfInput) (T.pack s) of
     (Right t) -> t


@@ -22,13 +100,13 @@ tRead s = case parseOnly (parseTerm <* endOfInput) (T.pack s) of
 
 parseVar :: Parser Term
 parseVar = do
 
 parseVar :: Parser Term
 parseVar = do

-  x <- many1 letter
+  x <- many1 (letter <|> digit)

   return $! Var x
 
 parseLambda :: Parser Term
 parseLambda = do
   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
   char '.'
   t <- parseTerm
   return $! createLambda vars t


@@ -40,15 +118,26 @@ createLambda _ _ = error "createLambda failed"
 
 parseApp :: Parser Term
 parseApp = do
 
 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 '('
   t <- parseTerm

-  char ' '
-  r <- parseTerm

   char ')'
   char ')'

-  return $! App t r
+  return t 

 
 parseTerm :: Parser Term
 
 parseTerm :: Parser Term

-parseTerm = parseVar <|> parseLambda <|> parseApp
+parseTerm = parseApp <|>
+            parseBraces <|>
+            parseLambda <|>
+            parseVar

 
 -------------------------------------------------
 
 
 -------------------------------------------------
 


@@ -71,9 +160,96 @@ substitute a b (Lambda x t)
   | otherwise = Lambda x (substitute a b t)
 substitute a b (App t u) = App (substitute a b t) (substitute a b u)
 
   | 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)
 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
+
+data Z = R Term Z | L Z Term | ZL VarName Z | E
+data D = Up | Down
+type TermZipper = (Term, Z, D)
+
+move :: TermZipper -> TermZipper
+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 :: TermZipper -> TermZipper
+unmove (t, L c r, Down) = (App t r, c, Down)
+unmove x = x
+
+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 _ (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 _ (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 100 $ (App (App cK cI) cY)
+-- Nothing
+--
+-- >>> toNormalForm Lazy 100 $ (App (App cK cI) cY)
+-- Just (λx.x)
+--
+-- prop> (\ t u -> t == u || t == Nothing || u == Nothing) (alphaNorm <$> toNormalForm Lazy 1000 x) (alphaNorm <$> 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 maxN t = do
+  n <- get
+  if n > maxN
+    then lift Nothing
+    else return t