Type Family Relations

Thomas DuBuisson

3 Jan 2009 3 Jan '09

10:22 a.m.

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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Austin Seipp

3 Jan 3 Jan

12:45 p.m.

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

Ryan Ingram

4:58 p.m.

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 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

Iavor Diatchki

8:18 p.m.

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 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

Thomas DuBuisson

11:35 p.m.

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 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 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

Iavor Diatchki

4 Jan 4 Jan

4:45 a.m.

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 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 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 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

Tom Schrijvers

7:58 a.m.

...

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 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 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 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

_______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

-- Tom Schrijvers Department of Computer Science K.U. Leuven Celestijnenlaan 200A B-3001 Heverlee Belgium tel: +32 16 327544 e-mail: tom.schrijvers@cs.kuleuven.be url: http://www.cs.kuleuven.be/~toms/

Mark P. Jones

12:40 p.m.

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

Manuel M T Chakravarty

16 Jan 16 Jan

1:21 a.m.

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'

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participants (7)

Austin Seipp
Iavor Diatchki
Manuel M T Chakravarty
Mark P. Jones
Ryan Ingram
Thomas DuBuisson
Tom Schrijvers