Hi
I'm trying to show that a system of rules for manipulating Haskell expressions is terminating. The rules can be applied in any order, to any subexpression - and there is a problem if there is any possible infinite sequence. The rule that is giving me particular problems is: (\v -> x) y => x[v/y] (I realise this rule may duplicate the computation of y, but that is not relevant for this purpose.) In particular, given the expression (\x -> x x) (\x -> x x) we can keep applying this rule infinitely. However, I don't believe this expression is type safe in Haskell. Are any such expressions that would cause this to non-terminate not type safe? What are the necessary conditions for this to be safe? Does GHC perform lambda reduction, probably by introducing a let binding? Does the combination of reducing lambdas and creating let bindings give similar problems, particularly if bindings are inlined? I'm wondering if this rule is genuinely unsafe in Haskell. If it is, why isn't it an issue in real simplifiers. If it isn't, what makes it safe. Thanks for any help, Neil _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On 3/12/08, Neil Mitchell <[hidden email]> wrote:
> However, I don't believe this expression is type safe in Haskell. Using higher-order polymorphism: f (x :: forall a. a -> a) = x x -- Taral <[hidden email]> "Please let me know if there's any further trouble I can give you." -- Unknown _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Neil Mitchell
On Wed, Mar 12, 2008 at 09:05:03PM +0000, Neil Mitchell wrote:
> Hi > > I'm trying to show that a system of rules for manipulating Haskell > expressions is terminating. The rules can be applied in any order, to > any subexpression - and there is a problem if there is any possible > infinite sequence. > > The rule that is giving me particular problems is: > > (\v -> x) y => x[v/y] > > (I realise this rule may duplicate the computation of y, but that is > not relevant for this purpose.) > > In particular, given the expression > > (\x -> x x) (\x -> x x) > > we can keep applying this rule infinitely. > > However, I don't believe this expression is type safe in Haskell. Are > any such expressions that would cause this to non-terminate not type > safe? topic here, for me at least.) Haskell's core type system (let, lambda, and application) can be mapped into the simply-typed lambda calculus by macro-expanding lets, and (by the construction of the Hindley-Milner type system) this mapping is typability-preserving. The simply-typed lambda calculus is not only terminating, it is strongly normalizing - ALL possible reduction sequences stop eventually. (There is a very elegant proof of this). The addition of data types in full generality breaks this: newtype Curry o = Curry { unCurry :: Curry o -> o } loop :: o loop = (\c -> unCurry c c) (Curry (\c -> unCurry c c)) Systems like Coq, which need to preserve strong normalization for logical soundness, adopt a variety of syntactic restrictions. Coq specifically forbids contravariant recursion in types. > What are the necessary conditions for this to be safe? Does GHC > perform lambda reduction, probably by introducing a let binding? Does > the combination of reducing lambdas and creating let bindings give > similar problems, particularly if bindings are inlined? Yes. If you feed that example into GHC, GHC will crash (even with the optimizer off). Works fine in Hugs/Yhc. http://www.haskell.org/ghc/docs/latest/html/users_guide/bugs.html#bugs-ghc > I'm wondering if this rule is genuinely unsafe in Haskell. If it is, > why isn't it an issue in real simplifiers. If it isn't, what makes it > safe. Nobody but us logic fans actually writes encodings of Curry's Paradox. Stefan _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe signature.asc (196 bytes) Download Attachment |
In reply to this post by Taral
On Wed, Mar 12, 2008 at 02:30:41PM -0700, Taral wrote:
> On 3/12/08, Neil Mitchell <[hidden email]> wrote: > > However, I don't believe this expression is type safe in Haskell. > > Using higher-order polymorphism: > > f (x :: forall a. a -> a) = x x Interestingly, this doesn't work - f is a self-application function, but it does not have a type that can be made to look like forall a. a -> a. Indeed, higher-order polymorphism as implemented in GHC can be implemented in System F-omega, a strongly normalizing calculus. (The usual datatype caveats apply). Stefan _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe signature.asc (196 bytes) Download Attachment |
In reply to this post by Neil Mitchell
As Stefan says, System Fw is strongly normalising. This is a remarkable result because (as you observe) it's very non-obvious how to prove it.
However GHC goes beyond Fw by adding data types letrec This blows strong normalisation out of the water. (Assuming you have reasonable rules for case and letrec.) But perhaps if you restrict data types a bit, and place draconian restrictions on letrec (e.g. never inline one) you could retain strong normalisation. It depends how much you want your rules to do. GHC's simplifier deals with the letrec problem by cutting each recursive loop -- see the paper "Secrets of the GHC inliner". GHC doesn't solve the data type problem at all -- you can make the Simplifier loop if you try. Simon | -----Original Message----- | From: [hidden email] [mailto:[hidden email]] On Behalf Of Neil Mitchell | Sent: 12 March 2008 21:05 | To: Haskell Cafe | Subject: [Haskell-cafe] Termination of substitution | | Hi | | I'm trying to show that a system of rules for manipulating Haskell | expressions is terminating. The rules can be applied in any order, to | any subexpression - and there is a problem if there is any possible | infinite sequence. | | The rule that is giving me particular problems is: | | (\v -> x) y => x[v/y] | | (I realise this rule may duplicate the computation of y, but that is | not relevant for this purpose.) | | In particular, given the expression | | (\x -> x x) (\x -> x x) | | we can keep applying this rule infinitely. | | However, I don't believe this expression is type safe in Haskell. Are | any such expressions that would cause this to non-terminate not type | safe? What are the necessary conditions for this to be safe? Does GHC | perform lambda reduction, probably by introducing a let binding? Does | the combination of reducing lambdas and creating let bindings give | similar problems, particularly if bindings are inlined? | | I'm wondering if this rule is genuinely unsafe in Haskell. If it is, | why isn't it an issue in real simplifiers. If it isn't, what makes it | safe. | | Thanks for any help, | | Neil | _______________________________________________ | Haskell-Cafe mailing list | [hidden email] | http://www.haskell.org/mailman/listinfo/haskell-cafe _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Hi
Thanks for the interesting comments. Its looks like learning some of the basics about System Fw is probably the way forward. > However GHC goes beyond Fw by adding > data types > letrec > This blows strong normalisation out of the water. (Assuming you have reasonable rules for case and letrec.) But perhaps if you restrict data types a bit, and place draconian restrictions on letrec (e.g. never inline one) you could retain strong normalisation. It depends how much you want your rules to do. I have restricted letrec appropriately. I haven't looked at the data type problem, but given that GHC manages to ignore it, I think its probably fair for me to ignore it for the time being. Thanks very much for the helpful comments, Neil _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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