Lambda.hs

{-# LANGUAGE TemplateHaskell, TypeOperators, MultiParamTypeClasses,
  FlexibleInstances, FlexibleContexts, UndecidableInstances,
  OverlappingInstances, Rank2Types, GADTs, KindSignatures,
  ScopedTypeVariables, TypeFamilies #-}
--------------------------------------------------------------------------------
-- |
-- Module      :  Examples.MultiParam.Lambda
-- Copyright   :  (c) 2011 Patrick Bahr, Tom Hvitved
-- License     :  BSD3
-- Maintainer  :  Tom Hvitved <hvitved@diku.dk>
-- Stability   :  experimental
-- Portability :  non-portable (GHC Extensions)
--
-- Tagless (monadic) interpretation of extended lambda calculus
--
--------------------------------------------------------------------------------

module Examples.MultiParam.Lambda where

import Data.Comp.MultiParam
import Data.Comp.MultiParam.Show ()
import Data.Comp.MultiParam.Equality ()
import Data.Comp.MultiParam.Derive
import Control.Monad (liftM2)
import Control.Monad.Error (MonadError, throwError)

data Lam :: (* -> *) -> (* -> *) -> * -> * where
  Lam :: (a i -> b j) -> Lam a b (i -> j)
data App :: (* -> *) -> (* -> *) -> * -> * where
  App :: b (i -> j) -> b i -> App a b j
data Const :: (* -> *) -> (* -> *) -> * -> * where
  Const :: Int -> Const a b Int
data Plus :: (* -> *) -> (* -> *) -> * -> * where
  Plus :: b Int -> b Int -> Plus a b Int
data Err :: (* -> *) -> (* -> *) -> * -> * where
  Err :: Err a b i
type Sig = Lam :+: App :+: Const :+: Plus :+: Err

$(derive [smartConstructors, makeHDifunctor, makeShowHD, makeEqHD]
         [''Lam, ''App, ''Const, ''Plus, ''Err])

-- * Tagless interpretation
class Eval f where
  evalAlg :: f I I i -> i -- I . evalAlg :: Alg f I is the actual algebra

$(derive [liftSum] [''Eval])

eval :: (HDifunctor f, Eval f) => Term f i -> i
eval = unI . cata (I . evalAlg)

instance Eval Lam where
  evalAlg (Lam f) = unI . f . I

instance Eval App where
  evalAlg (App (I f) (I x)) = f x

instance Eval Const where
  evalAlg (Const n) = n

instance Eval Plus where
  evalAlg (Plus (I x) (I y)) = x + y

instance Eval Err where
  evalAlg Err = error "error"

-- * Tagless monadic interpretation
type family Sem (m :: * -> *) i
type instance Sem m (i -> j) = Sem m i -> m (Sem m j)
type instance Sem m Int = Int

newtype M m i = M {unM :: m (Sem m i)}

class Monad m => EvalM m f where
  evalMAlg :: f (M m) (M m) i -> m (Sem m i) -- M . evalMAlg :: Alg f (M m)

$(derive [liftSum] [''EvalM])

evalM :: (Monad m, HDifunctor f, EvalM m f) => Term f i -> m (Sem m i)
evalM = unM . cata (M . evalMAlg)

instance Monad m => EvalM m Lam where
  evalMAlg (Lam f) = return $ unM . f . M . return

instance Monad m => EvalM m App where
  evalMAlg (App (M mf) (M mx)) = do f <- mf; f =<< mx
  
instance Monad m => EvalM m Const where
  evalMAlg (Const n) = return n

instance Monad m => EvalM m Plus where
  evalMAlg (Plus (M mx) (M my)) = liftM2 (+) mx my

instance MonadError String m => EvalM m Err where
  evalMAlg Err = throwError "error" -- 'throwError' rather than 'error'

e :: Term Sig Int
e = Term ((iLam $ \x -> (iLam (\y -> y `iPlus` x) `iApp` iConst 3)) `iApp` iConst 2)

v :: Either String Int
v = evalM e

e' :: Term Sig (Int -> Int)
e' = Term iErr --(iLam id)

v' :: Either String (Int -> Either String Int)
v' = evalM e'