Type families vs. functional dependencies -- how to express something
Hello all, for the past few hours I've been struggling to express a certain idea using type families and I really can't get it to typecheck. It all works fine using functional dependencies, but it could be more readable with TFs. From those papers about TFs I got this feeling that they should be as expressive as FDs, and we should use them (some people even occasionally mentioning that FDs may be deprecated in favor of them). Can someone help me make the following work, please? The working code with functional dependencies:
{-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FunctionalDependencies #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE ScopedTypeVariables #-}
__ = undefined
data HNil = HNil data HCons a b = HCons a b
data A a = A a data B a = B a
class X a l p | a -> l p where ll :: a -> l pp :: a -> p
instance X (B l) HNil (B l) where ll _ = HNil pp p = p
instance (X x l p) => X (A t -> x) (HCons t l) p where ll _ = HCons __ (ll (__ :: x)) pp p = pp (p (A (__ :: t)))
-- (inferred) -- testX :: (X a l (B l)) => a -> B l testX x = pp x `asTypeOf` B (ll x)
The motivation here is that A a represents a variable of type a, B b represents a computation with state b, and functions of type A a -> B b represent a computations with state b that use the first parameter as an accessor for a variable (or, more precisely, state component) of type a. Now, I want testX to take such function (with n parameters) and provide it with those accessors, fixing the type b to contain components of the corresponding accessors only. (The example uses dummy types and undefineds for simplicity.) This pretty much works: testX (B __) :: B HNil testX (\(A _) -> B __) :: B (HCons t HNil) testX (\(A True) -> B __) :: B (HCons Bool HNil) testX (\(A _) (A _) -> B __) :: B (HCons t (HCons t1 HNil)) testX (\(A _) (A _) (A _) -> B __) :: B (HCons t (HCons t1 (HCons t2 HNil))) testX (\(A _) -> B HNil) :: error This is my attempt to rephrase it using type families:
class X' a where type L a type P a ll' :: a -> L a pp' :: a -> P a
instance X' (B l) where type L (B l) = HNil type P (B l) = B l ll' _ = HNil pp' p = p
instance (X' x) => X' (A t -> x) where type L (A t -> x) = HCons t (L x) type P (A t -> x) = P x ll' _ = HCons __ (ll' (__ :: x)) pp' p = pp' (p (A (__ :: t)))
-- (inferred) -- testX' :: (B (L a) ~ P a, X' a) => a -> P a testX' x = pp' x `asTypeOf` B (ll' x)
Only the X' (B l) instance works, the other produces a strange error: testX' (B __) :: P (B HNil) :: B HNil testX' (\(A _) -> B __) :: error Couldn't match expected type `a' against inferred type `Main.R:L(->) t (B a)' Expected type: P (A t -> B a) Inferred type: B (L (A t -> B a)) Unifying those two types by hand, I get: P (A t -> B a) ~> P (B a) ~> B a B (L (A t -> B a)) ~> B (HCons t (L (B a))) ~> B (HCons t HNil) Hence, a = HCons t HNil. But GHC doesn't get that. I'm using GHC 6.12.1. Thanks for any hints. Best regards, -- Tomáš Janoušek, a.k.a. Liskni_si, http://work.lisk.in/
Unifying those two types by hand, I get:
P (A t -> B a) ~> P (B a)
Maybe the problem is that type families (and associated types, their class cousins) are not injective: P x ~ P y does not imply that x ~ y. Maybe you need a data type (with appropriate wrapping and unwrapping) to ensure injectivity. Cf: http://www.haskell.org/haskellwiki/GHC/Type_families#Injectivity.2C_type_inf... http://www.mail-archive.com/haskell-cafe@haskell.org/msg63359.html Dan Tomáš Janoušek wrote:
Hello all,
for the past few hours I've been struggling to express a certain idea using type families and I really can't get it to typecheck. It all works fine using functional dependencies, but it could be more readable with TFs. From those papers about TFs I got this feeling that they should be as expressive as FDs, and we should use them (some people even occasionally mentioning that FDs may be deprecated in favor of them). Can someone help me make the following work, please?
The working code with functional dependencies:
{-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FunctionalDependencies #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE ScopedTypeVariables #-}
__ = undefined
data HNil = HNil data HCons a b = HCons a b
data A a = A a data B a = B a
class X a l p | a -> l p where ll :: a -> l pp :: a -> p
instance X (B l) HNil (B l) where ll _ = HNil pp p = p
instance (X x l p) => X (A t -> x) (HCons t l) p where ll _ = HCons __ (ll (__ :: x)) pp p = pp (p (A (__ :: t)))
-- (inferred) -- testX :: (X a l (B l)) => a -> B l testX x = pp x `asTypeOf` B (ll x)
The motivation here is that A a represents a variable of type a, B b represents a computation with state b, and functions of type A a -> B b represent a computations with state b that use the first parameter as an accessor for a variable (or, more precisely, state component) of type a. Now, I want testX to take such function (with n parameters) and provide it with those accessors, fixing the type b to contain components of the corresponding accessors only. (The example uses dummy types and undefineds for simplicity.)
This pretty much works:
testX (B __) :: B HNil testX (\(A _) -> B __) :: B (HCons t HNil) testX (\(A True) -> B __) :: B (HCons Bool HNil) testX (\(A _) (A _) -> B __) :: B (HCons t (HCons t1 HNil)) testX (\(A _) (A _) (A _) -> B __) :: B (HCons t (HCons t1 (HCons t2 HNil))) testX (\(A _) -> B HNil) :: error
This is my attempt to rephrase it using type families:
class X' a where type L a type P a ll' :: a -> L a pp' :: a -> P a
instance X' (B l) where type L (B l) = HNil type P (B l) = B l ll' _ = HNil pp' p = p
instance (X' x) => X' (A t -> x) where type L (A t -> x) = HCons t (L x) type P (A t -> x) = P x ll' _ = HCons __ (ll' (__ :: x)) pp' p = pp' (p (A (__ :: t)))
-- (inferred) -- testX' :: (B (L a) ~ P a, X' a) => a -> P a testX' x = pp' x `asTypeOf` B (ll' x)
Only the X' (B l) instance works, the other produces a strange error:
testX' (B __) :: P (B HNil) :: B HNil testX' (\(A _) -> B __) :: error
Couldn't match expected type `a' against inferred type `Main.R:L(->) t (B a)' Expected type: P (A t -> B a) Inferred type: B (L (A t -> B a))
Unifying those two types by hand, I get:
P (A t -> B a) ~> P (B a) ~> B a
B (L (A t -> B a)) ~> B (HCons t (L (B a))) ~> B (HCons t HNil)
Hence, a = HCons t HNil. But GHC doesn't get that.
I'm using GHC 6.12.1.
Thanks for any hints. Best regards,
Hello, On Tue, May 18, 2010 at 04:47:50PM -0700, Dan Weston wrote:
Unifying those two types by hand, I get:
P (A t -> B a) ~> P (B a)
Maybe the problem is that type families (and associated types, their class cousins) are not injective: P x ~ P y does not imply that x ~ y. Maybe you need a data type (with appropriate wrapping and unwrapping) to ensure injectivity. Cf:
http://www.haskell.org/haskellwiki/GHC/Type_families#Injectivity.2C_type_inf... http://www.mail-archive.com/haskell-cafe@haskell.org/msg63359.html
That's probably it, thanks. I think I'll just stay with functional dependencies this time, a data type with wrapping and unwrapping can't possibly make the code more readable :-). Regards, -- Tomáš Janoušek, a.k.a. Liskni_si, http://work.lisk.in/
I'm pretty sure this is a bug. My code works if I put a type
signature on your second example.
*TFInjectivity> :t testX'
testX' :: (P a ~ B (L a), X' a) => a -> B (L a)
*TFInjectivity> :t testX' (\(A _) -> B __) :: B (HCons a HNil)
testX' (\(A _) -> B __) :: B (HCons a HNil) :: B (HCons a HNil)
*TFInjectivity> :t testX' (\(A _) -> B __)
Top level:
Couldn't match expected type `t TFInjectivity.:R1L B a'
against inferred type `a'
Expected type: B (L (A t -> B a))
Inferred type: P (A t -> B a)
Here's my back-of-the-envelope thoughts on how typechecking should go:
*TFInjectivity> :t (\(A _) -> B __)
(\(A _) -> B __) :: A t -> B a
*TFInjectivity> :t testX'
testX' :: (P a ~ B (L a), X' a) => a -> B (L a)
testX' (\(A _) -> B__)
introduce variables for (\(A _) -> B__)
(\(A _) -> B __) :: A t1 -> B t2
introduce variables and desired context for testX'
testX' :: t3 -> B (L t3)
needed:
1) P t3 ~ B (L t3),
2) X' t3
have:
3) t3 ~ (A t1 -> B t2)
apply (3) to (1) and (2)
needed:
4) P (A t1 -> B t2) ~ B (L (A t1 -> B t2))
5) X' (A t1 -> B t2)
apply X' instances to discharge (5)
apply P from X' instance (A t -> x) to (4)
6) P (B t2) ~ B (L (A t1 -> B t2))
apply P from X' instance (B t) to (6)
7) B t2 ~ B (L (A t1 -> B t2))
injectivity of data constructor
8) t2 ~ L (A t1 -> B t2)
(this is a very interesting unification to have to make, and might be
the heart of the problem? it looks like an infinite type, but it's
not!)
apply L from X' instance (A t -> x) to (8)
9) t2 ~ HCons t1 (L (B t2))
apply L from X' instance (B t2) to (9)
10) t2 ~ HCons t1 HNil
unify t2 and HCons t1 HNil, discharge
testX' (\(A _) -> B __) :: B (L (A t1 -> B t2))
or, simplifying,
testX' (\(A _) -> B __) :: B (HCons t1 HNil)
(Code follows)
-- ryan
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE ScopedTypeVariables #-}
module TFInjectivity where
__ = undefined
data HNil = HNil
data HCons a b = HCons a b
data A a = A a
data B a = B a
class X' a where
type L a
type P a
ll' :: a -> L a
pp' :: a -> P a
instance X' (B l) where
type L (B l) = HNil
type P (B l) = B l
ll' _ = HNil
pp' p = p
instance (X' x) => X' (A t -> x) where
type L (A t -> x) = HCons t (L x)
type P (A t -> x) = P x
ll' _ = HCons __ (ll' (__ :: x))
pp' p = pp' (p (A (__ :: t)))
testX' :: (L a ~ l, P a ~ B l, X' a) => a -> B l
testX' x = pp' x
2010/5/18 Tomáš Janoušek
Hello all,
for the past few hours I've been struggling to express a certain idea using type families and I really can't get it to typecheck. It all works fine using functional dependencies, but it could be more readable with TFs. From those papers about TFs I got this feeling that they should be as expressive as FDs, and we should use them (some people even occasionally mentioning that FDs may be deprecated in favor of them). Can someone help me make the following work, please?
The working code with functional dependencies:
{-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FunctionalDependencies #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE ScopedTypeVariables #-}
__ = undefined
data HNil = HNil data HCons a b = HCons a b
data A a = A a data B a = B a
class X a l p | a -> l p where ll :: a -> l pp :: a -> p
instance X (B l) HNil (B l) where ll _ = HNil pp p = p
instance (X x l p) => X (A t -> x) (HCons t l) p where ll _ = HCons __ (ll (__ :: x)) pp p = pp (p (A (__ :: t)))
-- (inferred) -- testX :: (X a l (B l)) => a -> B l testX x = pp x `asTypeOf` B (ll x)
The motivation here is that A a represents a variable of type a, B b represents a computation with state b, and functions of type A a -> B b represent a computations with state b that use the first parameter as an accessor for a variable (or, more precisely, state component) of type a. Now, I want testX to take such function (with n parameters) and provide it with those accessors, fixing the type b to contain components of the corresponding accessors only. (The example uses dummy types and undefineds for simplicity.)
This pretty much works:
testX (B __) :: B HNil testX (\(A _) -> B __) :: B (HCons t HNil) testX (\(A True) -> B __) :: B (HCons Bool HNil) testX (\(A _) (A _) -> B __) :: B (HCons t (HCons t1 HNil)) testX (\(A _) (A _) (A _) -> B __) :: B (HCons t (HCons t1 (HCons t2 HNil))) testX (\(A _) -> B HNil) :: error
This is my attempt to rephrase it using type families:
class X' a where type L a type P a ll' :: a -> L a pp' :: a -> P a
instance X' (B l) where type L (B l) = HNil type P (B l) = B l ll' _ = HNil pp' p = p
instance (X' x) => X' (A t -> x) where type L (A t -> x) = HCons t (L x) type P (A t -> x) = P x ll' _ = HCons __ (ll' (__ :: x)) pp' p = pp' (p (A (__ :: t)))
-- (inferred) -- testX' :: (B (L a) ~ P a, X' a) => a -> P a testX' x = pp' x `asTypeOf` B (ll' x)
Only the X' (B l) instance works, the other produces a strange error:
testX' (B __) :: P (B HNil) :: B HNil testX' (\(A _) -> B __) :: error
Couldn't match expected type `a' against inferred type `Main.R:L(->) t (B a)' Expected type: P (A t -> B a) Inferred type: B (L (A t -> B a))
Unifying those two types by hand, I get:
P (A t -> B a) ~> P (B a) ~> B a
B (L (A t -> B a)) ~> B (HCons t (L (B a))) ~> B (HCons t HNil)
Hence, a = HCons t HNil. But GHC doesn't get that.
I'm using GHC 6.12.1.
Thanks for any hints. Best regards, -- Tomáš Janoušek, a.k.a. Liskni_si, http://work.lisk.in/ _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
participants (3)
-
Dan Weston -
Ryan Ingram -
Tomáš Janoušek