Re: [Haskell-cafe] Type Family Relations
Thomas DuBuisson: Cafe, I am wondering if there is a way to enforce compile time checking of an axiom relating two separate type families. Mandatory contrived example: type family AddressOf h type family HeaderOf a -- I'm looking for something to the effect of: type axiom HeaderOf (AddressOf x) ~ x -- Valid: type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header -- Invalid type instance AddressOf AppHeader = AppAddress type instance HeaderOf AppAddress = [AppHeader] So this is a universally enforced type equivalence. The stipulation could be arbitrarily complex, checked at compile time, and must hold for all instances of either type family. Am I making this too hard? Is there already a solution I'm missing? There are no type-level invariants, like your type axiom, at the moment, although there is active work in this area http://www.cs.kuleuven.be/~toms/Research/papers/plpv2009_draft.pdf -- Type Invariants for Haskell, T. Schrijvers, L.-J. Guillemette, S. Monnier. Accepted at PLPV 2009. However, there is a simple solution to your problem. To enforce a side condition on the type instances of two separate families, you need to bundle the families as associated types into a class. Then, you can impose side conditions by way of super class constraints. In your example, that *should* work as follows -- GHC currently doesn't accept this, due to superclass equalities not being fully implemented, but we'll solve this in a second step: class (HeaderOf a ~ h, AddressOf h ~ a) = Protocol h a where type AddressOf h type HeaderOf a -- Valid: instance Protocol IPv4Header IPv4 where type AddressOf IPv4Header = IPv4 type HeaderOf IPv4 = IPv4Header -- Invalid instance Protocol AppHeader AppAddress where type AddressOf AppHeader = AppAddress type HeaderOf AppAddress = [AppHeader] Superclass equalities are currently only partially implemented and GHC rejects them for this reason. However, your application doesn't require the full power of superclass equalities and can be realised with normal type classes: class EQ a b instance EQ a a class (EQ (HeaderOf a) h, EQ (AddressOf h) a) = Protocol h a where type AddressOf h type HeaderOf a -- Valid: instance Protocol IPv4Header IPv4 where type AddressOf IPv4Header = IPv4 type HeaderOf IPv4 = IPv4Header -- Invalid instance Protocol AppHeader AppAddress where type AddressOf AppHeader = AppAddress type HeaderOf AppAddress = [AppHeader] With this definition, the invalid definition is rejected with the message /Users/chak/Code/haskell/main.hs:34:9: No instance for (EQ [AppHeader] AppHeader) arising from the superclasses of an instance declaration at /Users/chak/Code/haskell/main.hs:34:9-37 Possible fix: add an instance declaration for (EQ [AppHeader] AppHeader) In the instance declaration for `Protocol AppHeader AppAddress' Manuel ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Type Family Relations
Generalizing the previous post, with:
-
{-# LANGUAGE GADTs #-}
module Equ where
data a:==:b where
Equ :: (a ~ b) = a:==:b
symm :: (a:==:a)
symm = Equ
refl :: (a:==:b) - (b:==:a)
refl Equ = Equ
trans :: (a:==:b) - (b:==:c) - (a:==:c)
trans Equ Equ = Equ
cast :: (a:==:b) - (a - b)
cast Equ = id
-
We can do (e.g.):
data IPv4Header = C1
data IPv4 = C2
type instance HeaderOf IPv4 = IPv4Header
type instance AddressOf IPv4Header = IPv4
t0 :: IPv4 :==: AddressOf IPv4Header
t0 = Equ
t1 :: IPv4Header :==: HeaderOf IPv4
t1 = Equ
t2 :: IPv4 :==: AddressOf (HeaderOf IPv4)
t2 = Equ
t3 :: IPv4Header :==: HeaderOf (AddressOf IPv4Header)
t3 = Equ
-- And interestingly `t6' shows that the type family option
-- in the previous case is slightly stronger that the funcdeps
-- option, ie the type fams one corresponds to the funcdeps
-- addr - hdr, hdr - addr (instead of the weaker addr - hdr).
-- If this isn't desired I'd bet there's a way to modify the type
-- instances to get the desired behavior.
t5 :: AddrHdrPair a b
- a :==: AddressOf (HeaderOf a)
t5 AddrHdrPair = Equ
t6 :: AddrHdrPair a b
- b :==: HeaderOf (AddressOf b)
t6 AddrHdrPair = Equ
Matt
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Re: [Haskell-cafe] Type Family Relations
Hi,
I like collecting examples of different type system related issues,
and I am curious in what way is the solution that I posted limited. Do
you happen to have an example?
Thanks,
Iavor
On Sat, Jan 3, 2009 at 8:35 PM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Thank you all for the responses. I find the solution that omits type
families [Diatchki] to be too limiting while the solution 'class (Dual
(Dual s) ~ s) =' [Ingram] isn't globally enforced. I've yet to
closely study your first solution, Ryan, but it appears to be what I
was looking for - I'll give it a try in the coming week.
Tom
On Sat, Jan 3, 2009 at 8:18 PM, Iavor Diatchki iavor.diatc...@gmail.com
wrote:
Hello,
Usually, you can program such things by using super-classes. Here is
how you could encode your example:
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies,
FlexibleInstances #-}
class HeaderOf addr hdr | addr - hdr
class HeaderOf addr hdr = AddressOf hdr addr | addr - hdr
data IPv4Header = C1
data IPv4 = C2
data AppAddress = C3
data AppHeader = C4
instance AddressOf IPv4Header IPv4
instance HeaderOf IPv4 IPv4Header
{- results in error:
instance AddressOf AppHeader AppAddress
instance HeaderOf AppAddress [AppHeader]
-}
Hope that this helps,
Iavor
On Sat, Jan 3, 2009 at 7:22 AM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Cafe,
I am wondering if there is a way to enforce compile time checking of
an axiom relating two separate type families.
Mandatory contrived example:
type family AddressOf h
type family HeaderOf a
-- I'm looking for something to the effect of:
type axiom HeaderOf (AddressOf x) ~ x
-- Valid:
type instance AddressOf IPv4Header = IPv4
type instance HeaderOf IPv4 = IPv4Header
-- Invalid
type instance AddressOf AppHeader = AppAddress
type instance HeaderOf AppAddress = [AppHeader]
So this is a universally enforced type equivalence. The stipulation
could be arbitrarily complex, checked at compile time, and must hold
for all instances of either type family.
Am I making this too hard? Is there already a solution I'm missing?
Cheers,
Tom
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Re: [Haskell-cafe] Type Family Relations
Hi Thomas, The specific problem you describe has a simple solution using multiple parameter classes with syntactic sugar for functional notation instead of type families [1]: class AddressOf h a | h - a, a - h -- bijections class HeaderOf a h | a - h, h - a instance HeaderOf (AddressOf h) h -- your axiom By the way, without the syntactic sugar, the instance declaration shown here is just: instance AddressOf h a = HeaderOf a h -- also your axiom Individual (address,header) pairs are documented with a single instance, as in: instance AddressOf IPv4Header IPv4 And now the type system can derive HeaderOf IPv4 IPv4Header automatically using your axiom. But you won't be able to add the invalid AppHeader example because it will conflict (either with overlapping instances or with covering, depending on how you approach it). I like this approach because we only have to give a single instance instead of writing a pair of declarations that have to be checked for mutual consistency. I conclude that perhaps your example doesn't illustrate the kind of arbitrarily complex constraints you had in mind. Maybe you could elaborate? All the best, Mark [1] http://web.cecs.pdx.edu/~mpj/pubs/fundeps-design.html Thomas DuBuisson wrote: Cafe, I am wondering if there is a way to enforce compile time checking of an axiom relating two separate type families. Mandatory contrived example: type family AddressOf h type family HeaderOf a -- I'm looking for something to the effect of: type axiom HeaderOf (AddressOf x) ~ x -- Valid: type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header -- Invalid type instance AddressOf AppHeader = AppAddress type instance HeaderOf AppAddress = [AppHeader] So this is a universally enforced type equivalence. The stipulation could be arbitrarily complex, checked at compile time, and must hold for all instances of either type family. Am I making this too hard? Is there already a solution I'm missing? Cheers, Tom ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Type Family Relations
Hi,
I like collecting examples of different type system related issues,
and I am curious in what way is the solution that I posted limited. Do
you happen to have an example?
Hi Iavor,
I think that
class HeaderOf addr hdr | addr - hdr
does not enforce that there must be a corresponding instance
AddressOf hdr addr. Hence, the type checker cannot use that information
either. Do you have a way to remedy that?
Cheers,
Tom
On Sat, Jan 3, 2009 at 8:35 PM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Thank you all for the responses. I find the solution that omits type
families [Diatchki] to be too limiting while the solution 'class (Dual
(Dual s) ~ s) =' [Ingram] isn't globally enforced. I've yet to
closely study your first solution, Ryan, but it appears to be what I
was looking for - I'll give it a try in the coming week.
Tom
On Sat, Jan 3, 2009 at 8:18 PM, Iavor Diatchki iavor.diatc...@gmail.com wrote:
Hello,
Usually, you can program such things by using super-classes. Here is
how you could encode your example:
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies,
FlexibleInstances #-}
class HeaderOf addr hdr | addr - hdr
class HeaderOf addr hdr = AddressOf hdr addr | addr - hdr
data IPv4Header = C1
data IPv4 = C2
data AppAddress = C3
data AppHeader = C4
instance AddressOf IPv4Header IPv4
instance HeaderOf IPv4 IPv4Header
{- results in error:
instance AddressOf AppHeader AppAddress
instance HeaderOf AppAddress [AppHeader]
-}
Hope that this helps,
Iavor
On Sat, Jan 3, 2009 at 7:22 AM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Cafe,
I am wondering if there is a way to enforce compile time checking of
an axiom relating two separate type families.
Mandatory contrived example:
type family AddressOf h
type family HeaderOf a
-- I'm looking for something to the effect of:
type axiom HeaderOf (AddressOf x) ~ x
-- Valid:
type instance AddressOf IPv4Header = IPv4
type instance HeaderOf IPv4 = IPv4Header
-- Invalid
type instance AddressOf AppHeader = AppAddress
type instance HeaderOf AppAddress = [AppHeader]
So this is a universally enforced type equivalence. The stipulation
could be arbitrarily complex, checked at compile time, and must hold
for all instances of either type family.
Am I making this too hard? Is there already a solution I'm missing?
Cheers,
Tom
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--
Tom Schrijvers
Department of Computer Science
K.U. Leuven
Celestijnenlaan 200A
B-3001 Heverlee
Belgium
tel: +32 16 327544
e-mail: tom.schrijv...@cs.kuleuven.be
url: http://www.cs.kuleuven.be/~toms/
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[Haskell-cafe] Type Family Relations
Cafe, I am wondering if there is a way to enforce compile time checking of an axiom relating two separate type families. Mandatory contrived example: type family AddressOf h type family HeaderOf a -- I'm looking for something to the effect of: type axiom HeaderOf (AddressOf x) ~ x -- Valid: type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header -- Invalid type instance AddressOf AppHeader = AppAddress type instance HeaderOf AppAddress = [AppHeader] So this is a universally enforced type equivalence. The stipulation could be arbitrarily complex, checked at compile time, and must hold for all instances of either type family. Am I making this too hard? Is there already a solution I'm missing? Cheers, Tom ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Type Family Relations
Excerpts from Thomas M. DuBuisson's message of Sat Jan 03 09:22:47 -0600 2009: Mandatory contrived example: type family AddressOf h type family HeaderOf a -- I'm looking for something to the effect of: type axiom HeaderOf (AddressOf x) ~ x -- Valid: type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header -- Invalid type instance AddressOf AppHeader = AppAddress type instance HeaderOf AppAddress = [AppHeader] So this is a universally enforced type equivalence. The stipulation could be arbitrarily complex, checked at compile time, and must hold for all instances of either type family. Am I making this too hard? Is there already a solution I'm missing? The problem is that type classes are open - anybody can extend this family i.e. type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header type instance AddressOf IPv6Header = IPv4 ipv4func :: (AddressOf x ~ IPv4) = x - ... And it will perfectly accept arguments with a type of IPv6Header. There is a proposal for extending GHC to support type invariants of this nature, but it is not implemented: http://tomschrijvers.blogspot.com/2008/11/type-invariants-for-haskell.html Austin ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Type Family Relations
I've been fighting this same problem for a while. The solution I've come up with is to encode the axioms into a typeclass which gives you a proof of the axioms. Here's an excerpt from some code I've been playing around with; HaskTy and Lift are type families. -- Theorem: for all t instance of Lift, (forall env. HaskTy (Lift t) env == t) data HaskTy_Lift_Id t env = (t ~ HaskTy (Lift t) env) = HaskTy_Lift_Id class Thm_HaskTy_Lift_Id t where thm_haskty_lift_id :: forall env. HaskTy_Lift_Id t env instance Thm_HaskTy_Lift_Id Int where thm_haskty_lift_id = HaskTy_Lift_Id instance Thm_HaskTy_Lift_Id Bool where thm_haskty_lift_id = HaskTy_Lift_Id lemma_haskty_lift_id_app :: HaskTy_Lift_Id a env - HaskTy_Lift_Id b env - HaskTy_Lift_Id (a - b) env lemma_haskty_lift_id_app HaskTy_Lift_Id HaskTy_Lift_Id = HaskTy_Lift_Id instance (Thm_HaskTy_Lift_Id a, Thm_HaskTy_Lift_Id b) = Thm_HaskTy_Lift_Id (a - b) where thm_haskty_lift_id = lemma_haskty_lift_id_app thm_haskty_lift_id thm_haskty_lift_id As you can see, I encode a witness of the type equality into a concrete data type. You then direct the typechecker as to how to prove the type equality using the typeclass mechanism; the class instances closely mirror the type family instances. You then add this typeclass as a context on functions that require the equality. Control.Coroutine[1] uses a similar restriction: class Connect s where connect :: (s ~ Dual c, c ~ Dual s) = InSession s a - InSession c b - (a,b) A cleaner solution, that sadly doesn't work in GHC6.10 [2], would be something like: class (s ~ Dual (Dual s)) = Connect s where connect :: InSession s a - InSession (Dual s) b - (a,b) The difference is mainly one of efficiency; even though there is only one constructor for Thm_HaskTy_Lift_Id t env, at runtime the code still has to prove that evaluation terminates (to avoid _|_ giving an unsound proof of type equality). This means that deeply nested instances of the (a - b) instance require calling dictionary constructors to match the tree, until eventually we see that each leaf is a valid constructor. If the type equality could be brought into scope by just bringing the typeclass into scope, the dictionaries themselves could remain unevaluated at runtime. -- ryan [1] http://hackage.haskell.org/cgi-bin/hackage-scripts/package/Coroutine [2] http://hackage.haskell.org/trac/ghc/ticket/2102 On Sat, Jan 3, 2009 at 7:22 AM, Thomas DuBuisson thomas.dubuis...@gmail.com wrote: Cafe, I am wondering if there is a way to enforce compile time checking of an axiom relating two separate type families. Mandatory contrived example: type family AddressOf h type family HeaderOf a -- I'm looking for something to the effect of: type axiom HeaderOf (AddressOf x) ~ x -- Valid: type instance AddressOf IPv4Header = IPv4 type instance HeaderOf IPv4 = IPv4Header -- Invalid type instance AddressOf AppHeader = AppAddress type instance HeaderOf AppAddress = [AppHeader] So this is a universally enforced type equivalence. The stipulation could be arbitrarily complex, checked at compile time, and must hold for all instances of either type family. Am I making this too hard? Is there already a solution I'm missing? Cheers, Tom ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Type Family Relations
Hello,
Usually, you can program such things by using super-classes. Here is
how you could encode your example:
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies,
FlexibleInstances #-}
class HeaderOf addr hdr | addr - hdr
class HeaderOf addr hdr = AddressOf hdr addr | addr - hdr
data IPv4Header = C1
data IPv4 = C2
data AppAddress = C3
data AppHeader = C4
instance AddressOf IPv4Header IPv4
instance HeaderOf IPv4 IPv4Header
{- results in error:
instance AddressOf AppHeader AppAddress
instance HeaderOf AppAddress [AppHeader]
-}
Hope that this helps,
Iavor
On Sat, Jan 3, 2009 at 7:22 AM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Cafe,
I am wondering if there is a way to enforce compile time checking of
an axiom relating two separate type families.
Mandatory contrived example:
type family AddressOf h
type family HeaderOf a
-- I'm looking for something to the effect of:
type axiom HeaderOf (AddressOf x) ~ x
-- Valid:
type instance AddressOf IPv4Header = IPv4
type instance HeaderOf IPv4 = IPv4Header
-- Invalid
type instance AddressOf AppHeader = AppAddress
type instance HeaderOf AppAddress = [AppHeader]
So this is a universally enforced type equivalence. The stipulation
could be arbitrarily complex, checked at compile time, and must hold
for all instances of either type family.
Am I making this too hard? Is there already a solution I'm missing?
Cheers,
Tom
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Re: [Haskell-cafe] Type Family Relations
Thank you all for the responses. I find the solution that omits type
families [Diatchki] to be too limiting while the solution 'class (Dual
(Dual s) ~ s) =' [Ingram] isn't globally enforced. I've yet to
closely study your first solution, Ryan, but it appears to be what I
was looking for - I'll give it a try in the coming week.
Tom
On Sat, Jan 3, 2009 at 8:18 PM, Iavor Diatchki iavor.diatc...@gmail.com wrote:
Hello,
Usually, you can program such things by using super-classes. Here is
how you could encode your example:
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies,
FlexibleInstances #-}
class HeaderOf addr hdr | addr - hdr
class HeaderOf addr hdr = AddressOf hdr addr | addr - hdr
data IPv4Header = C1
data IPv4 = C2
data AppAddress = C3
data AppHeader = C4
instance AddressOf IPv4Header IPv4
instance HeaderOf IPv4 IPv4Header
{- results in error:
instance AddressOf AppHeader AppAddress
instance HeaderOf AppAddress [AppHeader]
-}
Hope that this helps,
Iavor
On Sat, Jan 3, 2009 at 7:22 AM, Thomas DuBuisson
thomas.dubuis...@gmail.com wrote:
Cafe,
I am wondering if there is a way to enforce compile time checking of
an axiom relating two separate type families.
Mandatory contrived example:
type family AddressOf h
type family HeaderOf a
-- I'm looking for something to the effect of:
type axiom HeaderOf (AddressOf x) ~ x
-- Valid:
type instance AddressOf IPv4Header = IPv4
type instance HeaderOf IPv4 = IPv4Header
-- Invalid
type instance AddressOf AppHeader = AppAddress
type instance HeaderOf AppAddress = [AppHeader]
So this is a universally enforced type equivalence. The stipulation
could be arbitrarily complex, checked at compile time, and must hold
for all instances of either type family.
Am I making this too hard? Is there already a solution I'm missing?
Cheers,
Tom
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