interpret Parigot's lambda-mu calculus in Haskell - haskell

Question

One can interpret the lambda calculus in Haskell:
data Expr = Var String | Lam String Expr | App Expr Expr
data Value a = V a | F (Value a -> Value a)
interpret :: [(String, Value a)] -> Expr -> Value a
interpret env (Var x) = case lookup x env of
Nothing -> error "undefined variable"
Just v -> v
interpret env (Lam x e) = F (\v -> interpret ((x, v):env) e)
interpret env (App e1 e2) = case interpret env e1 of
V _ -> error "not a function"
F f -> f (interpret env e2)
How could the above interpreter be extended to the lambda-mu calculus? My guess is that it should use continuations for interpreting the additional constructs in this calculus. (15) and (16) from the Bernardi&Moortgat paper are the kind of translations I expect.
It is possible since Haskell is Turing-complete, but how?
Hint: See the comment on page 197 on this research paper for the intuitive meaning of the mu binder.

Answer 1

Here's a mindless transliteration of the reduction rules from the paper, using #user2407038's representation (as you'll see, when I say mindless, I really do mean mindless):
{-# LANGUAGE DataKinds, KindSignatures, GADTs #-}
{-# LANGUAGE StandaloneDeriving #-}
import Control.Monad.Writer
import Control.Applicative
import Data.Monoid
data TermType = Named | Unnamed
type Var = String
type MuVar = String
data Expr (n :: TermType) where
Var :: Var -> Expr Unnamed
Lam :: Var -> Expr Unnamed -> Expr Unnamed
App :: Expr Unnamed -> Expr Unnamed -> Expr Unnamed
Freeze :: MuVar -> Expr Unnamed -> Expr Named
Mu :: MuVar -> Expr Named -> Expr Unnamed
deriving instance Show (Expr n)
substU :: Var -> Expr Unnamed -> Expr n -> Expr n
substU x e = go
where
go :: Expr n -> Expr n
go (Var y) = if y == x then e else Var y
go (Lam y e) = Lam y $ if y == x then e else go e
go (App f e) = App (go f) (go e)
go (Freeze alpha e) = Freeze alpha (go e)
go (Mu alpha u) = Mu alpha (go u)
renameN :: MuVar -> MuVar -> Expr n -> Expr n
renameN beta alpha = go
where
go :: Expr n -> Expr n
go (Var x) = Var x
go (Lam x e) = Lam x (go e)
go (App f e) = App (go f) (go e)
go (Freeze gamma e) = Freeze (if gamma == beta then alpha else gamma) (go e)
go (Mu gamma u) = Mu gamma $ if gamma == beta then u else go u
appN :: MuVar -> Expr Unnamed -> Expr n -> Expr n
appN beta v = go
where
go :: Expr n -> Expr n
go (Var x) = Var x
go (Lam x e) = Lam x (go e)
go (App f e) = App (go f) (go e)
go (Freeze alpha w) = Freeze alpha $ if alpha == beta then App (go w) v else go w
go (Mu alpha u) = Mu alpha $ if alpha /= beta then go u else u
reduceTo :: a -> Writer Any a
reduceTo x = tell (Any True) >> return x
reduce0 :: Expr n -> Writer Any (Expr n)
reduce0 (App (Lam x u) v) = reduceTo $ substU x v u
reduce0 (App (Mu beta u) v) = reduceTo $ Mu beta $ appN beta v u
reduce0 (Freeze alpha (Mu beta u)) = reduceTo $ renameN beta alpha u
reduce0 e = return e
reduce1 :: Expr n -> Writer Any (Expr n)
reduce1 (Var x) = return $ Var x
reduce1 (Lam x e) = reduce0 =<< (Lam x <$> reduce1 e)
reduce1 (App f e) = reduce0 =<< (App <$> reduce1 f <*> reduce1 e)
reduce1 (Freeze alpha e) = reduce0 =<< (Freeze alpha <$> reduce1 e)
reduce1 (Mu alpha u) = reduce0 =<< (Mu alpha <$> reduce1 u)
reduce :: Expr n -> Expr n
reduce e = case runWriter (reduce1 e) of
(e', Any changed) -> if changed then reduce e' else e
It "works" for the example from the paper: with
example 0 = App (App t (Var "x")) (Var "y")
where
t = Lam "x" $ Lam "y" $ Mu "delta" $ Freeze "phi" $ App (Var "x") (Var "y")
example n = App (example (n-1)) (Var ("z_" ++ show n))
I can reduce example n to the expected result:
*Main> reduce (example 10)
Mu "delta" (Freeze "phi" (App (Var "x") (Var "y")))
The reason I put scare quotes around "works" above is that I have no intuition about the λμ calculus so I don't know what it should do.

Answer 2

Note: this is only a partial answer since I'm not sure how to extend the interpreter.
This seems like a good use case for DataKinds. The Expr datatype is indexed on a type which is named or unnamed. The regular lambda constructs produce named terms only.
{-# LANGUAGE GADTs, DataKinds, KindSignatures #-}
data TermType = Named | Unnamed
type Var = String
type MuVar = String
data Expr (n :: TermType) where
Var :: Var -> Expr Unnamed
Lam :: Var -> Expr Unnamed -> Expr Unnamed
App :: Expr Unnamed -> Expr Unnamed -> Expr Unnamed
and the additional Mu and Name constructs can manipulate the TermType.
...
Name :: MuVar -> Expr Unnamed -> Expr Named
Mu :: MuVar -> Expr Named -> Expr Unnamed

Answer 3

How about something like the below. I don't have a good idea on how to traverse Value a, but at least I can see it evaluates example n into MuV.
import Data.Maybe
type Var = String
type MuVar = String
data Expr = Var Var
| Lam Var Expr
| App Expr Expr
| Mu MuVar MuVar Expr
deriving Show
data Value a = ConV a
| LamV (Value a -> Value a)
| MuV (Value a -> Value a)
type Env a = [(Var, Value a)]
type MuEnv a = [(MuVar, Value a -> Value a)]
varScopeErr :: Var -> Value a
varScopeErr v = error $ unwords ["Out of scope λ variable:", show v]
appErr :: Value a
appErr = error "Trying to apply a non-lambda"
muVarScopeErr :: MuVar -> (Value a -> Value a)
muVarScopeErr alpha = id
app :: Value a -> Value a -> Value a
app (LamV f) x = f x
app (MuV f) x = MuV $ \y -> f x `app` y
app _ _ = appErr
eval :: Env a -> MuEnv a -> Expr -> Value a
eval env menv (Var v) = fromMaybe (varScopeErr v) $ lookup v env
eval env menv (Lam v e) = LamV $ \x -> eval ((v, x):env) menv e
eval env menv (Mu alpha beta e) = MuV $ \u ->
let menv' = (alpha, (`app` u)):menv
wrap = fromMaybe (muVarScopeErr beta) $ lookup beta menv'
in wrap (eval env menv' e)
eval env menv (App f e) = eval env menv f `app` eval env menv e
example 0 = App (App t (Var "v")) (Var "w")
where
t = Lam "x" $ Lam "y" $ Mu "delta" "phi" $ App (Var "x") (Var "y")
example n = App (example (n-1)) (Var ("z_" ++ show n))