So Terminating contains all the terminating terms in the untyped lambda calculus and none of the non-terminating ones. And the type checker checks this. So it sounds to me like the (terminating) type checker solves the halting problem. Can you please explain which part of this I have misunderstood? -- Lennart On Jan 7, 2007, at 17:31 , Jim Apple wrote:
To show how expressive GADTs are, the datatype Terminating can hold any term in the untyped lambda calculus that terminates, and none that don't. I don't think that an encoding of this is too surprising, but I thought it might be a good demonstration of the power that GADTs bring.
{-# OPTIONS -fglasgow-exts #-}
{- Using GADTs to encode normalizable and non-normalizable terms in the lambda calculus. For definitions of normalizable and de Bruin indices, I used:
Christian Urban and Stefan Berghofer - A Head-to-Head Comparison of de Bruijn Indices and Names. In Proceedings of the International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP 2006). Seattle, USA. ENTCS. Pages 46-59
http://www4.in.tum.de/~urbanc/Publications/lfmtp-06.ps
@incollection{ pierce97foundational, author = "Benjamin Pierce", title = "Foundational Calculi for Programming Languages", booktitle = "The Computer Science and Engineering Handbook", publisher = "CRC Press", address = "Boca Raton, FL", editor = "Allen B. Tucker", year = "1997", url = "citeseer.ist.psu.edu/pierce95foundational.html" }
-}
module Terminating where
-- Terms in the untyped lambda-calculus with the de Bruijn representation
data Term t where Var :: Nat n -> Term (Var n) Lambda :: Term t -> Term (Lambda t) Apply :: Term t1 -> Term t2 -> Term (Apply t1 t2)
-- Natural numbers
data Nat n where Zero :: Nat Z Succ :: Nat n -> Nat (S n)
data Z data S n
data Var t data Lambda t data Apply t1 t2
data Less n m where LessZero :: Less Z (S n) LessSucc :: Less n m -> Less (S n) (S m)
data Equal a b where Equal :: Equal a a
data Plus a b c where PlusZero :: Plus Z b b PlusSucc :: Plus a b c -> Plus (S a) b (S c)
{- We can reduce a term by function application, reduction under the lambda, or reduction of either side of an application. We don't need this full power, since we could get by with normal-order evaluation, but that required a more complicated datatype for Reduce. -} data Reduce t1 t2 where ReduceSimple :: Replace Z t1 t2 t3 -> Reduce (Apply (Lambda t1) t2) t3 ReduceLambda :: Reduce t1 t2 -> Reduce (Lambda t1) (Lambda t2) ReduceApplyLeft :: Reduce t1 t2 -> Reduce (Apply t1 t3) (Apply t2 t3) ReduceApplyRight :: Reduce t1 t2 -> Reduce (Apply t3 t1) (Apply t3 t2)
{- Lift and Replace use the de Bruijn operations as expressed in the Urban and Berghofer paper. -} data Lift n k t1 t2 where LiftVarLess :: Less i k -> Lift n k (Var i) (Var i) LiftVarGTE :: Either (Equal i k) (Less k i) -> Plus i n r -> Lift n k (Var i) (Var r) LiftApply :: Lift n k t1 t1' -> Lift n k t2 t2' -> Lift n k (Apply t1 t2) (Apply t1' t2') LiftLambda :: Lift n (S k) t1 t2 -> Lift n k (Lambda t1) (Lambda t2)
data Replace k t n r where ReplaceVarLess :: Less i k -> Replace k (Var i) n (Var i) ReplaceVarEq :: Equal i k -> Lift k Z n r -> Replace k (Var i) n r ReplaceVarMore :: Less k (S i) -> Replace k (Var (S i)) n (Var i) ReplaceApply :: Replace k t1 n r1 -> Replace k t2 n r2 -> Replace k (Apply t1 t2) n (Apply r1 r2) ReplaceLambda :: Replace (S k) t n r -> Replace k (Lambda t) n (Lambda r)
{- Reflexive transitive closure of the reduction relation. -} data ReduceEventually t1 t2 where ReduceZero :: ReduceEventually t1 t1 ReduceSucc :: Reduce t1 t2 -> ReduceEventually t2 t3 -> ReduceEventually t1 t3
-- Definition of normal form: nothing with a lambda term applied to anything. data Normal t where NormalVar :: Normal (Var n) NormalLambda :: Normal t -> Normal (Lambda t) NormalApplyVar :: Normal t -> Normal (Apply (Var i) t) NormalApplyApply :: Normal (Apply t1 t2) -> Normal t3 -> Normal (Apply (Apply t1 t2) t3)
-- Something is terminating when it reduces to something normal data Terminating where Terminating :: Term t -> ReduceEventually t t' -> Normal t' -> Terminating
{- We can encode terms that are non-terminating, even though this set is only co-recursively enumerable, so we can't actually prove all of the non-normalizable terms of the lambda calculus are non-normalizable. -}
data Reducible t1 where Reducible :: Reduce t1 t2 -> Reducible t1 -- A term is non-normalizable if, no matter how many reductions you have applied, -- you can still apply one more. type Infinite t1 = forall t2 . ReduceEventually t1 t2 -> Reducible t2
data NonTerminating where NonTerminating :: Term t -> Infinite t -> NonTerminating
-- x test1 :: Terminating test1 = Terminating (Var Zero) ReduceZero NormalVar
-- (\x . x)@y test2 :: Terminating test2 = Terminating (Apply (Lambda (Var Zero))(Var Zero)) (ReduceSucc (ReduceSimple (ReplaceVarEq Equal (LiftVarGTE (Left Equal) PlusZero))) ReduceZero) NormalVar
-- omega = \x.x@x type Omega = Lambda (Apply (Var Z) (Var Z)) omega = Lambda (Apply (Var Zero) (Var Zero))
-- (\x . \y . y)@(\z.z@z) test3 :: Terminating test3 = Terminating (Apply (Lambda (Lambda (Var Zero))) omega) (ReduceSucc (ReduceSimple (ReplaceLambda (ReplaceVarLess LessZero))) ReduceZero) (NormalLambda NormalVar)
-- (\x.x@x)(\x.x@x) test4 :: NonTerminating test4 = NonTerminating (Apply omega omega) help3
help1 :: Reducible (Apply Omega Omega) help1 = Reducible (ReduceSimple (ReplaceApply (ReplaceVarEq Equal (LiftLambda (LiftApply (LiftVarLess LessZero) (LiftVarLess LessZero)))) (ReplaceVarEq Equal (LiftLambda (LiftApply (LiftVarLess LessZero) (LiftVarLess LessZero))))))
help2 :: ReduceEventually (Apply Omega Omega) t -> Equal (Apply Omega Omega) t help2 ReduceZero = Equal help2 (ReduceSucc (ReduceSimple (ReplaceApply (ReplaceVarEq _ (LiftLambda (LiftApply (LiftVarLess _) (LiftVarLess _)))) (ReplaceVarEq _ (LiftLambda (LiftApply (LiftVarLess _) (LiftVarLess _)))))) y) = case help2 y of Equal -> Equal
help3 :: Infinite (Apply Omega Omega) help3 x = case help2 x of Equal -> help1 _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe