Haskell Interpreter for System T Combinator Language

Question

In a previous question SystemT Compiler and dealing with Infinite Types in Haskell I asked about how to parse a SystemT Lambda Calculus to SystemT Combinators. I decided to use plain algebraic data types for encoding both the SystemT Lambda calculus and SystemT Combinator calculus (based on the comment: SystemT Compiler and dealing with Infinite Types in Haskell).

SystemTCombinator.hs:

module SystemTCombinator where

data THom = Id
          | Unit
          | Zero
          | Succ
          | Compose THom THom
          | Pair THom THom
          | Fst
          | Snd
          | Curry THom
          | Eval
          | Iter THom THom
          deriving (Show)

SystemTLambda.hs:

{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE FlexibleInstances         #-}
{-# LANGUAGE MultiParamTypeClasses     #-}
{-# LANGUAGE PartialTypeSignatures     #-}
{-# LANGUAGE TypeSynonymInstances      #-}

module SystemTLambda where

import           Control.Monad.Catch
import           Data.Either (fromRight)
import qualified SystemTCombinator    as SystemTC

type TVar = String

data TType = One | Prod TType TType | Arrow TType TType | Nat deriving (Eq)

instance Show TType where
  show ttype = case ttype of
    One -> "'Unit"
    Nat -> "'Nat"
    Prod ttype1 ttype2 ->
      "(" ++ show ttype1 ++ " * " ++ show ttype2 ++ ")"
    Arrow ttype1@(Arrow _ _) ttype2 ->
      "(" ++ show ttype1 ++ ") -> " ++ show ttype2
    Arrow ttype1 ttype2 -> show ttype1 ++ " -> " ++ show ttype2

data TTerm = Var TVar
           | Let TVar TTerm TTerm
           | Lam TVar TTerm
           | App TTerm TTerm
           | Unit
           | Pair TTerm TTerm
           | Fst TTerm
           | Snd TTerm
           | Zero
           | Succ TTerm
           | Iter TTerm TTerm TVar TTerm
           | Annot TTerm TType
           deriving (Show)

-- a context is a list of hypotheses/judgements
type TContext = [(TVar, TType)]

-- our exceptions for SystemT
data TException = TypeCheckException String
                | BindingException String
  deriving (Show)

instance Exception TException

newtype Parser a = Parser { run :: TContext -> Either SomeException a }

instance Functor Parser where
  fmap f xs = Parser $ \ctx ->
    either Left (\v -> Right $ f v) $ run xs ctx

instance Applicative Parser where
  pure a = Parser $ \ctx -> Right a
  fs <*> xs = Parser $ \ctx ->
    either Left (\f -> fmap f $ run xs ctx) (run fs ctx)

instance Monad Parser where
  xs >>= f = Parser $ \ctx ->
    either Left (\v -> run (f v) ctx) $ run xs ctx

instance MonadThrow Parser where
  throwM e = Parser (const $ Left $ toException e)

instance MonadCatch Parser where
  catch p f = Parser $ \ctx ->
    either
      (\e -> case fromException e of
        Just e' -> run (f e') ctx -- this handles the exception
        Nothing -> Left e) -- this propagates it upwards
      Right
      $ run p ctx

withHypothesis :: (TVar, TType) -> Parser a -> Parser a
withHypothesis hyp cmd = Parser $ \ctx -> run cmd (hyp : ctx)

tvarToHom :: TVar -> Parser (SystemTC.THom, TType)
tvarToHom var = Parser $ \ctx ->
  case foldr transform Nothing ctx of
    Just x -> Right x
    Nothing -> throwM $ BindingException ("unbound variable " ++ show var)
  where
    transform (var', varType) homAndType =
      if var == var'
      then Just (SystemTC.Snd, varType)
      else homAndType >>= (\(varHom, varType) -> Just (SystemTC.Compose SystemTC.Fst varHom, varType))

check :: TTerm -> TType -> Parser SystemTC.THom
-- check a lambda
check (Lam var bodyTerm) (Arrow varType bodyType) =
  withHypothesis (var, varType) $
  check bodyTerm bodyType >>= (\bodyHom -> return $ SystemTC.Curry bodyHom)
check (Lam _    _    ) ttype                 = throwM
  $ TypeCheckException ("expected function type, got '" ++ show ttype ++ "'")
-- check unit
check Unit One = return SystemTC.Unit
check Unit ttype =
  throwM $ TypeCheckException ("expected unit type, got '" ++ show ttype ++ "'")
-- check products
check (Pair term1 term2) (Prod ttype1 ttype2) = do
  hom1 <- check term1 ttype1
  hom2 <- check term2 ttype2
  return $ SystemTC.Pair hom1 hom2
check (Pair _      _     ) ttype                = throwM
  $ TypeCheckException ("expected product type, got '" ++ show ttype ++ "'")
-- check primitive recursion
check (Iter baseTerm inductTerm tvar numTerm) ttype = do
  baseHom   <- check baseTerm ttype
  inductHom <- withHypothesis (tvar, ttype) (check inductTerm ttype)
  numHom    <- check numTerm Nat
  return $ SystemTC.Compose (SystemTC.Pair SystemTC.Id numHom)
                            (SystemTC.Iter baseHom inductHom)
-- check let bindings
check (Let var valueTerm exprTerm) exprType = do
  (valueHom, valueType) <- synth valueTerm
  exprHom <- withHypothesis (var, valueType) (check exprTerm exprType)
  return $ SystemTC.Compose (SystemTC.Pair SystemTC.Id valueHom) exprHom
check tterm ttype = do
  (thom, ttype') <- synth tterm
  if ttype == ttype'
    then return thom
    else throwM $ TypeCheckException
      (  "expected type '"
      ++ show ttype
      ++ "', inferred type '"
      ++ show ttype'
      ++ "'"
      )

synth :: TTerm -> Parser (SystemTC.THom, TType)
synth (Var tvar) = tvarToHom tvar
synth (Let var valueTerm exprTerm) = do
  (valueHom, valueType) <- synth valueTerm
  (exprHom, exprType) <- withHypothesis (var, valueType) (synth exprTerm)
  return (SystemTC.Compose (SystemTC.Pair SystemTC.Id valueHom) exprHom, exprType)
synth (App functionTerm valueTerm) = do
  (functionHom, functionType) <- synth functionTerm
  case functionType of
    Arrow headType bodyType -> do
      valueHom <- check valueTerm headType
      return (SystemTC.Compose (SystemTC.Pair functionHom valueHom) SystemTC.Eval, bodyType)
    _ -> throwM $ TypeCheckException ("expected function, got '" ++ show functionType ++ "'")
synth (Fst pairTerm) = do
  (pairHom, pairType) <- synth pairTerm
  case pairType of
    Prod fstType sndType -> return (SystemTC.Compose pairHom SystemTC.Fst, fstType)
    _ -> throwM $ TypeCheckException ("expected product, got '" ++ show pairType ++ "'")
synth (Snd pairTerm) = do
  (pairHom, pairType) <- synth pairTerm
  case pairType of
    Prod fstType sndType -> return (SystemTC.Compose pairHom SystemTC.Snd, sndType)
    _ -> throwM $ TypeCheckException ("expected product, got '" ++ show pairType ++ "'")
synth Zero = return (SystemTC.Compose SystemTC.Unit SystemTC.Zero, Nat)
synth (Succ numTerm) = do
  numHom <- check numTerm Nat
  return (SystemTC.Compose numHom SystemTC.Succ, Nat)
synth (Annot term ttype) = do
  hom <- check term ttype
  return (hom, ttype)
synth _ = throwM $ TypeCheckException "unknown synthesis"

I use the above bidirectional type checker to parse SystemTLambda.TTerm into SystemTCombinator.THom.

systemTLambda :: TTerm
systemTLambda =
  Let "sum"
    (Annot
      (Lam "x" $ Lam "y" $
       Iter (Var "y") (Succ $ Var "n") "n" (Var "x"))
      (Arrow Nat $ Arrow Nat Nat))
    (App
      (App
        (Var "sum")
        (Succ $ Succ Zero))
      (Succ $ Succ $ Succ Zero))

systemTCombinator :: Either SomeException (SystemTC.THom, SystemTC.TType)
systemTCombinator = fromRight Unit $ run (synth result) []

The combinator expression is:

Compose (Pair Id (Curry (Curry (Compose (Pair Id (Compose Fst Snd)) (Iter Snd (Compose Snd Succ)))))) (Compose (Pair (Compose (Pair Snd (Compose (Compose (Compose Unit Zero) Succ) Succ)) Eval) (Compose (Compose (Compose (Compose Unit Zero) Succ) Succ) Succ)) Eval)

The problem I have now is how to interpret this combinator expression. I know that all combinator expressions are meant to be interpreted as a function. The problem is that I don't know the input and output types of this function, and I expect that the "interpreter" function will be partial, in that if it tries to interpret something incorrectly it should result in a RuntimeException because the combinator expression is untyped, it is possible to have bad combinator expressions. However my type checker should ensure that once reaching the interpreter the combinators should be well typed already.

According to the original blog post, http://semantic-domain.blogspot.com/2012/12/total-functional-programming-in-partial.html the combinators have all functional equivalents. Something like:

evaluate Id = id
evaluate Unit = const ()
evaluate Zero = \() -> Z
evaluate (Succ n) = S n
evaluate (Compose f g) = (evaluate g) . (evaluate f)
evaluate (Pair l r) = (evaluate l, evaluate r)
evaluate Fst = fst
evaluate Snd = snd
evaluate (Curry h) = curry (evaluate h)
evaluate Eval = \(f, v) -> f v
evaluate (Iter base recurse) = \(a, n) ->
  case n of
    Z   -> evaluate base a
    S n -> evaluate recurse (a, evaluate (Iter base recurse) (a, n))

But obviously that doesn't work. It seems that there must be some way of interpreting the THom tree, such that I get some sort of function back, that can be executed in partial manner.

Answer 1

To interpret THom in a guaranteed well-typed way, you will need to "explain" its types to the Haskell type checker. I see you have already considered the GADT version of THom, which would be the conventional way to do this explanation, and that's still the approach I would go with. If I understand correctly, the trouble you were facing is that the logic of synth was too complex to prove that it would always produce a well-typed THom, and this comes as no surprise.

I think you can keep your synth (rougly) as it is, if you add a simple pass that type checks the resulting untyped THom into a typed GADT, say StrongTHom a b. Returning existentials seems risky, it would be preferable to provide it an incoming context:

checkTHom :: THom -> TType a -> TType b -> Maybe (StrongTHom a b)

(where TType is the singleton form in the previous answer). It just requires you to know at the top level what your input and output types are going to be. This is usually fine, because in order to actually use the result you will eventually have to know the types at which it is instantiated anyway. (You might have to punt this expected type information outward a few levels until a concrete type is known)

If you absolutely must be able to infer the input and output types, then I suppose there is no choice but to return an existential. It just means that your type checker will include a lot more type equality checks (cf. typeEq below), and the untyped THom may also need to include more type information.

In either case THom will definitely have to include any types that it erases. For example, in Compose :: THom a b -> THom b c -> THom a c, b is erased and checkTHom will have to reconstruct it. So Compose needs to include enough information so that this is possible. At this point an existential (SomeType from the previous answer) will probably be fine, because the only way you have to use it is by unwrapping it and passing it on recursively.

To write this checker I have a feeling you will need a strong equality check:

typeEq :: TType a -> TType b -> Maybe (a :~: b)

(where :~: is standard type equality) which is easy to write; I'm just making sure you're aware of the technique.

Once you have this, then eval :: StrongTHom a b -> a -> b should go through like warm butter. Good luck!

Answer 2

Alternatively, it is pretty easy to do runtime type checking by declaring a type of all possible values.

data Value
    = VUnit                          -- of type One
    | VPair Value Value              -- of type Pair
    | VFunc (Value -> Interp Value)  -- of type Func
    | VNat Integer                   -- of type Nat

Then you can very directly use your untyped THom, for an appropriate interpreter monad Interp (maybe just an Except monad):

eval :: THom -> Value -> Interp Value
eval Id v  = v
eval Unit _ = VUnit
eval Zero VUnit = VNat Zero
eval Succ (VNat n) = VNat (n + 1)
...
eval _ _ = throwE "type error"

Notice also that VFunc above has the same type as the codomain of eval, since embedded functions may also fail.