Trouble with injective type families

Wolfgang Jeltsch

5 Jul 2017 5 Jul '17

12:21 a.m.

Hi! Injective type families as supported by GHC 8.0.1 do not behave like I would expect them to behave from my intuitive understanding. Let us consider the following example:

...

{-# LANGUAGE RankNTypes, TypeFamilyDependencies #-}

class C a where

type T a = b | b -> a

instance C Bool where

type T Bool = Int

type X b = forall a . T a ~ b => a

x :: X Int x = False

I would expect this code to be accepted. However, I get the following error message:

...

A.hs:14:5: error: • Could not deduce: a ~ Bool from the context: T a ~ Int bound by the type signature for: x :: T a ~ Int => a at A.hs:13:1-10 ‘a’ is a rigid type variable bound by the type signature for: x :: forall a. T a ~ Int => a at A.hs:11:19 • In the expression: False In an equation for ‘x’: x = False • Relevant bindings include x :: a (bound at A.hs:14:1)

This is strange, since injectivity should exactly make it possible to deduce a ~ Bool from T a ~ Int. Another example is this:

...

{-# LANGUAGE GADTs, TypeFamilyDependencies #-}

class C a where

    type T a = b | b -> a

instance C Bool where

    type T Bool = Int

data G b where

    G :: Eq a => a -> G (T a)

instance Eq (G b) where

    G a1 == G a2 = a1 == a2a

I would also expect this code to be accepted. However, I get the following error message:

...

B.hs:17:26: error: • Could not deduce: a1 ~ a from the context: (b ~ T a, Eq a) bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:5-8 or from: (b ~ T a1, Eq a1) bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:13-16 ‘a1’ is a rigid type variable bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:13 ‘a’ is a rigid type variable bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:5 • In the second argument of ‘(==)’, namely ‘a2’ In the expression: a1 == a2 In an equation for ‘==’: (G a1) == (G a2) = a1 == a2 • Relevant bindings include a2 :: a1 (bound at B.hs:17:15) a1 :: a (bound at B.hs:17:7)

If b ~ T a and b ~ T a1, then T a ~ T a1 and subsequently a ~ a1, because of injectivity. Unfortunately, GHC does not join the two contexts (b ~ T a, Eq a) and (b ~ T a1, Eq a1). Are these behaviors really intended, or are these bugs showing up? All the best, Wolfgang

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Simon Peyton Jones

5 Jul 5 Jul

6:45 a.m.

Functional dependencies and type-family dependencies only induce extra "improvement" constraints, not evidence. For example class C a b | a -> b where foo :: a -> b instance C Bool Int where ... f :: C Bool b => b -> Int f x = x -- Rejected Does the fundep on 'b' allow us to deduce (b ~ Int), GADT-like, in the body of 'f', and hence accept the definition. No, it does not. Think of the translation into System F. We get f = /\b \(d :: C Bool b). \(x::b). x |> ??? What evidence can I used to cast 'x' by to get it from type 'b' to Int? Rather, fundeps resolve ambiguity. Consider g x = foo True + x The call to 'foo True' gives rise to a "wanted" constraint (C Bool beta), where beta is a fresh unification variable. Then by the fundep we get an "improvement" constraint (also "wanted") (beta ~ Int). So we can infer g :: Int -> Int. In your example we have x :: forall a b. (T Int ~ b) => a x = False Think of the System F translation: x = /\a b. \(d :: T Int ~ b). False |> ?? Again, what evidence can we use to cast False to 'a'. In short, fundeps and type family dependencies only add extra unification constraints, which may help to resolve ambiguous types. They don’t provide evidence. That's not to say that they couldn't. But you'd need to extend System FC, GHC's core language, to do so. Simon | -----Original Message----- | From: Glasgow-haskell-users [mailto:glasgow-haskell-users- | bounces@haskell.org] On Behalf Of Wolfgang Jeltsch | Sent: 05 July 2017 01:21 | To: glasgow-haskell-users@haskell.org | Subject: Trouble with injective type families | | Hi! | | Injective type families as supported by GHC 8.0.1 do not behave like I | would expect them to behave from my intuitive understanding. | | Let us consider the following example: | | > {-# LANGUAGE RankNTypes, TypeFamilyDependencies #-} | > | > class C a where | > | > type T a = b | b -> a | > | > instance C Bool where | > | > type T Bool = Int | > | > type X b = forall a . T a ~ b => a | > | > x :: X Int | > x = False | | I would expect this code to be accepted. However, I get the following | error message: | | > A.hs:14:5: error: | > • Could not deduce: a ~ Bool | > from the context: T a ~ Int | > bound by the type signature for: | > x :: T a ~ Int => a | > at A.hs:13:1-10 | > ‘a’ is a rigid type variable bound by | > the type signature for: | > x :: forall a. T a ~ Int => a | > at A.hs:11:19 | > • In the expression: False | > In an equation for ‘x’: x = False | > • Relevant bindings include x :: a (bound at A.hs:14:1) | | This is strange, since injectivity should exactly make it possible to | deduce a ~ Bool from T a ~ Int. | | Another example is this: | | > {-# LANGUAGE GADTs, TypeFamilyDependencies #-} | > | > class C a where | > | > type T a = b | b -> a | > | > instance C Bool where | > | > type T Bool = Int | > | > data G b where | > | > G :: Eq a => a -> G (T a) | > | > instance Eq (G b) where | > | > G a1 == G a2 = a1 == a2a | | I would also expect this code to be accepted. However, I get the | following error message: | | > B.hs:17:26: error: | > • Could not deduce: a1 ~ a | > from the context: (b ~ T a, Eq a) | > bound by a pattern with constructor: | > G :: forall a. Eq a => a -> G (T a), | > in an equation for ‘==’ | > at B.hs:17:5-8 | > or from: (b ~ T a1, Eq a1) | > bound by a pattern with constructor: | > G :: forall a. Eq a => a -> G (T a), | > in an equation for ‘==’ | > at B.hs:17:13-16 | > ‘a1’ is a rigid type variable bound by | > a pattern with constructor: G :: forall a. Eq a => a -> G (T | > a), | > in an equation for ‘==’ | > at B.hs:17:13 | > ‘a’ is a rigid type variable bound by | > a pattern with constructor: G :: forall a. Eq a => a -> G (T | > a), | > in an equation for ‘==’ | > at B.hs:17:5 | > • In the second argument of ‘(==)’, namely ‘a2’ | > In the expression: a1 == a2 | > In an equation for ‘==’: (G a1) == (G a2) = a1 == a2 | > • Relevant bindings include | > a2 :: a1 (bound at B.hs:17:15) | > a1 :: a (bound at B.hs:17:7) | | If b ~ T a and b ~ T a1, then T a ~ T a1 and subsequently a ~ a1, because | of injectivity. Unfortunately, GHC does not join the two contexts (b ~ T | a, Eq a) and (b ~ T a1, Eq a1). | | Are these behaviors really intended, or are these bugs showing up? | | All the best, | Wolfgang | _______________________________________________ | Glasgow-haskell-users mailing list | Glasgow-haskell-users@haskell.org | https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fmail.hask | ell.org%2Fcgi-bin%2Fmailman%2Flistinfo%2Fglasgow-haskell- | users&data=02%7C01%7Csimonpj%40microsoft.com%7Cc333fbaff2f4406c337e08d4c3 | 3bd1bd%7C72f988bf86f141af91ab2d7cd011db47%7C1%7C0%7C636348109082017989&sd | ata=q%2B8ZSqYfNjC7x6%2Bm9vpsgCmCDo%2FIItppqB5Nwzf6Qj0%3D&reserved=0

Wolfgang Jeltsch

1:37 p.m.

Dear Simon, thank you very much for this elaborate explanation. I stumbled on this issue when using functional dependencies years ago. The solution at that time was to use type families. I did not know that injectivity is handled analogously to functional dependencies. Given that it is, the syntax for injectivity makes a lot more sense. All the best, Wolfgang Am Mittwoch, den 05.07.2017, 06:45 +0000 schrieb Simon Peyton Jones:

...

Functional dependencies and type-family dependencies only induce extra "improvement" constraints, not evidence.  For example

class C a b | a -> b where foo :: a -> b instance C Bool Int where ...

f :: C Bool b => b -> Int f x = x -- Rejected

Does the fundep on 'b' allow us to deduce (b ~ Int), GADT-like, in the body of 'f', and hence accept the definition.  No, it does not.  Think of the translation into System F. We get

f = /\b \(d :: C Bool b). \(x::b).  x |> ???

What evidence can I used to cast 'x' by to get it from type 'b' to Int?

Rather, fundeps resolve ambiguity.  Consider

g x = foo True + x

The call to 'foo True' gives rise to a "wanted" constraint (C Bool beta), where beta is a fresh unification variable.  Then by the fundep we get an "improvement" constraint (also "wanted") (beta ~ Int). So we can infer g :: Int -> Int.

In your example we have

   x :: forall a b. (T Int ~ b) => a    x = False

Think of the System F translation:

   x = /\a b. \(d :: T Int ~ b). False |> ??

Again, what evidence can we use to cast False to 'a'.

In short, fundeps and type family dependencies only add extra unification constraints, which may help to resolve ambiguous types.  They don’t provide evidence.  That's not to say that they couldn't.  But you'd need to extend System FC, GHC's core language, to do so.

Simon

...
-----Original Message----- From: Glasgow-haskell-users [mailto:glasgow-haskell-users- bounces@haskell.org] On Behalf Of Wolfgang Jeltsch Sent: 05 July 2017 01:21 To: glasgow-haskell-users@haskell.org Subject: Trouble with injective type families

Hi!

Injective type families as supported by GHC 8.0.1 do not behave like I would expect them to behave from my intuitive understanding.

Let us consider the following example:

...
{-# LANGUAGE RankNTypes, TypeFamilyDependencies #-}

class C a where

    type T a = b | b -> a

instance C Bool where

    type T Bool = Int

type X b = forall a . T a ~ b => a

x :: X Int x = False

I would expect this code to be accepted. However, I get the following error message:

...
A.hs:14:5: error:     • Could not deduce: a ~ Bool       from the context: T a ~ Int         bound by the type signature for:                    x :: T a ~ Int => a         at A.hs:13:1-10       ‘a’ is a rigid type variable bound by         the type signature for:           x :: forall a. T a ~ Int => a         at A.hs:11:19     • In the expression: False       In an equation for ‘x’: x = False     • Relevant bindings include x :: a (bound at A.hs:14:1)

This is strange, since injectivity should exactly make it possible to deduce a ~ Bool from T a ~ Int.

Another example is this:

...
{-# LANGUAGE GADTs, TypeFamilyDependencies #-}

class C a where

    type T a = b | b -> a

instance C Bool where

    type T Bool = Int

data G b where

    G :: Eq a => a -> G (T a)

instance Eq (G b) where

    G a1 == G a2 = a1 == a2a

I would also expect this code to be accepted. However, I get the following error message:

...
B.hs:17:26: error:     • Could not deduce: a1 ~ a       from the context: (b ~ T a, Eq a)         bound by a pattern with constructor:                    G :: forall a. Eq a => a -> G (T a),                  in an equation for ‘==’         at B.hs:17:5-8       or from: (b ~ T a1, Eq a1)         bound by a pattern with constructor:                    G :: forall a. Eq a => a -> G (T a),                  in an equation for ‘==’         at B.hs:17:13-16       ‘a1’ is a rigid type variable bound by         a pattern with constructor: G :: forall a. Eq a => a -> G (T a),         in an equation for ‘==’         at B.hs:17:13       ‘a’ is a rigid type variable bound by         a pattern with constructor: G :: forall a. Eq a => a -> G (T a),         in an equation for ‘==’         at B.hs:17:5     • In the second argument of ‘(==)’, namely ‘a2’       In the expression: a1 == a2       In an equation for ‘==’: (G a1) == (G a2) = a1 == a2     • Relevant bindings include         a2 :: a1 (bound at B.hs:17:15)         a1 :: a (bound at B.hs:17:7)

If b ~ T a and b ~ T a1, then T a ~ T a1 and subsequently a ~ a1, because of injectivity. Unfortunately, GHC does not join the two contexts (b ~ T a, Eq a) and (b ~ T a1, Eq a1).

Are these behaviors really intended, or are these bugs showing up?

All the best, Wolfgang _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fmail .hask ell.org%2Fcgi-bin%2Fmailman%2Flistinfo%2Fglasgow-haskell- users&data=02%7C01%7Csimonpj%40microsoft.com%7Cc333fbaff2f4406c337e0 8d4c3 3bd1bd%7C72f988bf86f141af91ab2d7cd011db47%7C1%7C0%7C6363481090820179 89&sd ata=q%2B8ZSqYfNjC7x6%2Bm9vpsgCmCDo%2FIItppqB5Nwzf6Qj0%3D&reserved=0

Richard Eisenberg

2:16 p.m.

I'd like to add that the reason we never extended System FC with support for injectivity is that the proof of soundness of doing so has remained elusive. A similar unsoundness in type improvement may cause unpredictable behavior in type inference, but it can't ever launch the rockets. So we keep it in type inference but out of FC. Richard

...

On Jul 5, 2017, at 9:37 AM, Wolfgang Jeltsch wrote:

Dear Simon,

thank you very much for this elaborate explanation.

I stumbled on this issue when using functional dependencies years ago. The solution at that time was to use type families.

I did not know that injectivity is handled analogously to functional dependencies. Given that it is, the syntax for injectivity makes a lot more sense.

All the best, Wolfgang

Am Mittwoch, den 05.07.2017, 06:45 +0000 schrieb Simon Peyton Jones:

...
Functional dependencies and type-family dependencies only induce extra "improvement" constraints, not evidence. For example

class C a b | a -> b where foo :: a -> b instance C Bool Int where ...

f :: C Bool b => b -> Int f x = x -- Rejected

Does the fundep on 'b' allow us to deduce (b ~ Int), GADT-like, in the body of 'f', and hence accept the definition. No, it does not. Think of the translation into System F. We get

f = /\b \(d :: C Bool b). \(x::b). x |> ???

What evidence can I used to cast 'x' by to get it from type 'b' to Int?

Rather, fundeps resolve ambiguity. Consider

g x = foo True + x

The call to 'foo True' gives rise to a "wanted" constraint (C Bool beta), where beta is a fresh unification variable. Then by the fundep we get an "improvement" constraint (also "wanted") (beta ~ Int). So we can infer g :: Int -> Int.

In your example we have

x :: forall a b. (T Int ~ b) => a x = False

Think of the System F translation:

x = /\a b. \(d :: T Int ~ b). False |> ??

Again, what evidence can we use to cast False to 'a'.

In short, fundeps and type family dependencies only add extra unification constraints, which may help to resolve ambiguous types. They don’t provide evidence. That's not to say that they couldn't. But you'd need to extend System FC, GHC's core language, to do so.

Simon

...
-----Original Message----- From: Glasgow-haskell-users [mailto:glasgow-haskell-users- bounces@haskell.org] On Behalf Of Wolfgang Jeltsch Sent: 05 July 2017 01:21 To: glasgow-haskell-users@haskell.org Subject: Trouble with injective type families

Hi!

Injective type families as supported by GHC 8.0.1 do not behave like I would expect them to behave from my intuitive understanding.

Let us consider the following example:

...
{-# LANGUAGE RankNTypes, TypeFamilyDependencies #-}

class C a where

type T a = b | b -> a

instance C Bool where

type T Bool = Int

type X b = forall a . T a ~ b => a

x :: X Int x = False

I would expect this code to be accepted. However, I get the following error message:

...
A.hs:14:5: error: • Could not deduce: a ~ Bool from the context: T a ~ Int bound by the type signature for: x :: T a ~ Int => a at A.hs:13:1-10 ‘a’ is a rigid type variable bound by the type signature for: x :: forall a. T a ~ Int => a at A.hs:11:19 • In the expression: False In an equation for ‘x’: x = False • Relevant bindings include x :: a (bound at A.hs:14:1)

This is strange, since injectivity should exactly make it possible to deduce a ~ Bool from T a ~ Int.

Another example is this:

...
{-# LANGUAGE GADTs, TypeFamilyDependencies #-}

class C a where

type T a = b | b -> a

instance C Bool where

type T Bool = Int

data G b where

G :: Eq a => a -> G (T a)

instance Eq (G b) where

G a1 == G a2 = a1 == a2a

I would also expect this code to be accepted. However, I get the following error message:

...
B.hs:17:26: error: • Could not deduce: a1 ~ a from the context: (b ~ T a, Eq a) bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:5-8 or from: (b ~ T a1, Eq a1) bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:13-16 ‘a1’ is a rigid type variable bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:13 ‘a’ is a rigid type variable bound by a pattern with constructor: G :: forall a. Eq a => a -> G (T a), in an equation for ‘==’ at B.hs:17:5 • In the second argument of ‘(==)’, namely ‘a2’ In the expression: a1 == a2 In an equation for ‘==’: (G a1) == (G a2) = a1 == a2 • Relevant bindings include a2 :: a1 (bound at B.hs:17:15) a1 :: a (bound at B.hs:17:7)

If b ~ T a and b ~ T a1, then T a ~ T a1 and subsequently a ~ a1, because of injectivity. Unfortunately, GHC does not join the two contexts (b ~ T a, Eq a) and (b ~ T a1, Eq a1).

Are these behaviors really intended, or are these bugs showing up?

All the best, Wolfgang _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fmail .hask ell.org%2Fcgi-bin%2Fmailman%2Flistinfo%2Fglasgow-haskell- users&data=02%7C01%7Csimonpj%40microsoft.com%7Cc333fbaff2f4406c337e0 8d4c3 3bd1bd%7C72f988bf86f141af91ab2d7cd011db47%7C1%7C0%7C6363481090820179 89&sd ata=q%2B8ZSqYfNjC7x6%2Bm9vpsgCmCDo%2FIItppqB5Nwzf6Qj0%3D&reserved=0

Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://mail.haskell.org/cgi-bin/mailman/listinfo/glasgow-haskell-users

gkaracha

15 Jul 15 Jul

10:48 a.m.

Me and Tom Schrijvers recently got a paper accepted at Haskell '17 concerning the elaboration of functional dependencies in System FC *without extending it*: https://people.cs.kuleuven.be/~george.karachalias/papers/fundeps.pdf https://people.cs.kuleuven.be/~george.karachalias/papers/fundeps.pdf The work is not currently implemented in GHC but we made some extra effort to make our system GHC-compatible so that integration will be possible in the future :-) In addition to the elaboration of FDs, we also noticed that they are unsound as currently implemented (Section 2.2 in the draft), if we treat them as type-level functions. Hence, we had to tighten up the restrictions imposed on them, good thing that until now they didn't affect the generated System FC code! There is still some work to be done to transfer the results to injective type families but I believe we are getting there :) Cheers, George -- View this message in context: http://haskell.1045720.n5.nabble.com/Trouble-with-injective-type-families-tp... Sent from the Haskell - Glasgow-haskell-users mailing list archive at Nabble.com.

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gkaracha
Richard Eisenberg
Simon Peyton Jones
Wolfgang Jeltsch