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plurals

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Id
5189804
data
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0
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0
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3
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2011-03-04T04:49:23.097
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2011-03-04T04:49:23.097
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5
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It's probably impossible without violating the ‘legitimacy’ of HOAS, in the sense that the <code>E => E</code> must be used not just for binding in the object language, but for computation in the meta language. That said, here's a solution in Haskell. It abuses a <code>Literal</code> node to drop in a unique ID for later substitution. Enjoy! <pre><code>import Control.Monad.State -- HOAS representation data Expr = Lam (Expr -> Expr) | App Expr Expr | Lit Integer -- Rotate transformation rot :: Expr -> Expr rot e = case e of Lam f -> descend uniqueID (f (Lit uniqueID)) _ -> e where uniqueID = 1 + maxLit e descend :: Integer -> Expr -> Expr descend i (Lam f) = Lam $ descend i . f descend i e = Lam $ \a -> replace i a e replace :: Integer -> Expr -> Expr -> Expr replace i e (Lam f) = Lam $ replace i e . f replace i e (App e1 e2) = App (replace i e e1) (replace i e e2) replace i e (Lit j) | i == j = e | otherwise = Lit j maxLit :: Expr -> Integer maxLit e = execState (maxLit' e) (-2) where maxLit' (Lam f) = maxLit' (f (Lit 0)) maxLit' (App e1 e2) = maxLit' e1 >> maxLit' e2 maxLit' (Lit i) = get >>= \k -> when (i > k) (put i) -- Output toStr :: Integer -> Expr -> State Integer String toStr k e = toStr' e where toStr' (Lit i) | i >= k = return $ 'x':show i -- variable | otherwise = return $ show i -- literal toStr' (App e1 e2) = do s1 <- toStr' e1 s2 <- toStr' e2 return $ "(" ++ s1 ++ " " ++ s2 ++ ")" toStr' (Lam f) = do i <- get modify (+ 1) s <- toStr' (f (Lit i)) return $ "\\x" ++ show i ++ " " ++ s instance Show Expr where show e = evalState (toStr m e) m where m = 2 + maxLit e -- Examples ex2, ex3, ex4 :: Expr ex2 = Lam(\a -> Lam(\b -> App a (App b (Lit 3)))) ex3 = Lam(\a -> Lam(\b -> Lam(\c -> App a (App b c)))) ex4 = Lam(\a -> Lam(\b -> Lam(\c -> Lam(\d -> App (App a b) (App c d))))) check :: Expr -> IO () check e = putStrLn(show e ++ " ===> \n" ++ show (rot e) ++ "\n") main = check ex2 >> check ex3 >> check ex4 </code></pre> with the following result: <pre><code>\x5 \x6 (x5 (x6 3)) ===> \x5 \x6 (x6 (x5 3)) \x2 \x3 \x4 (x2 (x3 x4)) ===> \x2 \x3 \x4 (x4 (x2 x3)) \x2 \x3 \x4 \x5 ((x2 x3) (x4 x5)) ===> \x2 \x3 \x4 \x5 ((x5 x2) (x3 x4)) </code></pre> (Don't be fooled by the similar-looking variable names. This is the rotation you seek, modulo alpha-conversion.)
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