### Re: [Haskell-cafe] rip in the class-abstraction continuum

Type classes are the approach to constrain type variables, to bound polymorphism and limit the set of types the variables can be instantiated with. If we have two type variables to constrain, multi-parameter type classes are the natural answer then. Let's take this solution and see where it leads to. Here is the original type class class XyConv a where toXy :: a b - [Xy b] and the problematic instance data CircAppr a b = CircAppr a b b -- number of points, rotation angle, radius deriving (Show) instance Integral a = XyConv (CircAppr a) where toXy (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a - am2xy a rad) angles To be more explicit, the type class declaration has the form class XyConv a where toXy :: forall b. a b - [Xy b] with the type variable 'b' universally quantified without any constraints. That means the user of (toXy x) is free to choose any type for 'b' whatsoever. Obviously that can't be true for (toXy (CircAppr x y)) since we can't instantiate pi to any type. It has to be a Floating type. Hence we have to constrain b. As I said, the obvious solution is to make it a parameter of the type class. We get the first solution: class XYConv1 a b where toXy1 :: a b - [Xy b] instance (Floating b, Integral a) = XYConv1 (CircAppr a) b where toXy1 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a - am2xy a rad) angles The type class declaration proclaims that only certain combinations of 'a' and 'b' are admitted to the class XYConv1. In particular, 'a' is (CircAppr a) and 'b' is Floating. This reminds us of collections (with Int keys, for simplicity) class Coll c where empty :: c b insert :: Int - b - c b - c b instance Coll M.IntMap where empty = M.empty insert = M.insert The Coll declaration assumes that a collection is suitable for elements of any type. Later on one notices that if elements are Bools, they can be stuffed quite efficiently into an Integer. If we wish to add ad hoc, efficient collections to the framework, we have to restrict the element type as well: class Coll1 c b where empty1 :: c insert1 :: Int - b - c - c Coll1 is deficient since there is no way to specify the type of elements for the empty collection. When the type checker sees 'empty1', how can it figure out which instance for Coll1 (with the same c but different element types) to choose? We can help the type-checker by declaring (by adding the functional dependency c - b) that for each collection type c, there can be only one instance of Coll1. In other words, the collection type determines the element type. Exactly the same principle works for XYConv. class XYConv2 a b | a - b where toXy2 :: a - [Xy b] instance (Floating b, Integral a) = XYConv2 (CircAppr a b) b where toXy2 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a - am2xy a rad) angles The third step is to move to associated types. At this stage you can consider them just as a different syntax of writing functional dependencies: class XYConv3 a where type XYT a :: * toXy3 :: a - [Xy (XYT a)] instance (Floating b, Integral a) = XYConv3 (CircAppr a b) where type XYT (CircAppr a b) = b toXy3 (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a - am2xy a rad) angles The step from XYConv2 to XYConv3 is mechanical. The class XYConv3 assumes that for each convertible 'a' there is one and only Xy type 'b' to which it can be converted. This was the case for (CircAppr a b). It may not be the case in general. But we can say that for each convertible 'a' there is a _class_ of Xy types 'b' to which they may be converted. This final step brings Tillmann Rendel's solution. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

### Re: [Haskell-cafe] rip in the class-abstraction continuum

Hi, Christopher Howard wrote: class XyConv a where toXy :: a b - [Xy b] [...] I can get a quick fix by adding Floating to the context of the /class/ definition: class XyConv a where toXy :: Floating b = a b - [Xy b] But what I really want is to put Floating in the context of the /instance/ declaration. This is not easily possible. If you could just put the constraint into the instance, there would be a problem when youc all toXy in a polymorphic context, where a is not known. Example: class XyConv a where toXy :: a b - [Xy b] shouldBeFine :: XyConv a = a String - [Xy String] shouldBeFine = toXy This code compiles fine, because the type of shouldBeFine is an instance of the type of toXy. The type checker figures out that here, b needs to be String, and since there is no class constraint visible anywhere that suggests a problem with b = String, the code is accepted. The correctness of this reasoning relies on the fact that whatever instances you add in other parts of your program, they can never constrain b so that it cannot be String anymore. Such an instance would invalidate the above program, but that would be unfair: How would the type checker have known in advance whether or not you'll eventually write this constraining instance. So this is why in Haskell, the type of a method in an instance declaration has to be as general as the declared type of that method in the corresponding class declaration. Now, there is a way out using some of the more recent additions to the language: You can declare, in the class, that each instance can choose its own constraints for b. This is possible by combining constraint kinds and associated type families. {-# LANGUAGE ConstraintKinds, TypeFamilies #-} import GHC.Exts The idea is to add a constraint type to the class declaration: class XyConv a where type C a :: * - Constraint toXy :: C a b = a b - [Xy b] Now it is clear to the type checker that calling toXy requires that b satisfies a constraint that is only known when a is known, so the following is not accepted. noLongerAccepted :: XyConv a = a String - [Xy String] noLongerAccepted = toXy The type checker complains that it cannot deduce an instance of (C a [Char]) from (XyConv a). If you want to write this function, you have to explicitly state that the caller has to provide the (C a String) instance, whatever (C a) will be: haveToWriteThis :: (XyConv a, C a String) = a String - [Xy String] haveToWriteThis = toXy So with associated type families and constraint kinds, the class declaration can explicitly say that instances can require constraints. The type checker will then be aware of it, and require appropriate instances of as-yet-unknown classes to be available. I think this is extremely cool and powerful, but maybe more often than not, we don't actually need this power, and can provide a less generic but much simpler API. Tillmann ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

### Re: [Haskell-cafe] rip in the class-abstraction continuum

On 05/19/2013 10:10 PM, Tillmann Rendel wrote: This is not easily possible. If you could just put the constraint into the instance, there would be a problem when youc all toXy in a polymorphic context, where a is not known. Example: class XyConv a where toXy :: a b - [Xy b] shouldBeFine :: XyConv a = a String - [Xy String] shouldBeFine = toXy This code compiles fine, because the type of shouldBeFine is an instance of the type of toXy. The type checker figures out that here, b needs to be String, and since there is no class constraint visible anywhere that suggests a problem with b = String, the code is accepted. The correctness of this reasoning relies on the fact that whatever instances you add in other parts of your program, they can never constrain b so that it cannot be String anymore. Such an instance would invalidate the above program, but that would be unfair: How would the type checker have known in advance whether or not you'll eventually write this constraining instance. So this is why in Haskell, the type of a method in an instance declaration has to be as general as the declared type of that method in the corresponding class declaration. Now, there is a way out using some of the more recent additions to the language: You can declare, in the class, that each instance can choose its own constraints for b. This is possible by combining constraint kinds and associated type families. {-# LANGUAGE ConstraintKinds, TypeFamilies #-} import GHC.Exts The idea is to add a constraint type to the class declaration: class XyConv a where type C a :: * - Constraint toXy :: C a b = a b - [Xy b] Now it is clear to the type checker that calling toXy requires that b satisfies a constraint that is only known when a is known, so the following is not accepted. noLongerAccepted :: XyConv a = a String - [Xy String] noLongerAccepted = toXy The type checker complains that it cannot deduce an instance of (C a [Char]) from (XyConv a). If you want to write this function, you have to explicitly state that the caller has to provide the (C a String) instance, whatever (C a) will be: haveToWriteThis :: (XyConv a, C a String) = a String - [Xy String] haveToWriteThis = toXy So with associated type families and constraint kinds, the class declaration can explicitly say that instances can require constraints. The type checker will then be aware of it, and require appropriate instances of as-yet-unknown classes to be available. I think this is extremely cool and powerful, but maybe more often than not, we don't actually need this power, and can provide a less generic but much simpler API. Tillmann Thank you for the quick and thorough response. To be honest though, I had some difficulty following your explanation of the constraints problem. I had an even more difficult time when I tried to read up on what Type Families are -- ended up at some wiki page trying to explain Type Families by illustrating them with Generic Finite Maps (a.k.a., Generic Prefix Trees). The rough equivalent of learning German through a Latin-German dictionary. :| Anyway, I played around with my code some more - and it seems like what I am trying to do can be done with multi-parameter type classes: code: {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-} class XyConv a b where toXy :: a b - [Xy b] instance (Integral a, Floating b) = XyConv (CircAppr a) b where toXy (CircAppr divns ang rad) = let dAng = 2 * pi / (fromIntegral divns) in let angles = map ((+ ang) . (* dAng) . fromIntegral) [0..divns] in map (\a - am2xy a rad) angles Seems to work okay: code: h toXy (CircAppr 4 0.0 1.0) [Xy 1.0 0.0,Xy 6.123233995736766e-17 1.0,Xy (-1.0) 1.2246467991473532e-16,Xy (-1.8369701987210297e-16) (-1.0),Xy 1.0 (-2.4492935982947064e-16)] h :t toXy (CircAppr 4 0.0 1.0) toXy (CircAppr 4 0.0 1.0) :: Floating b = [Xy b] Is there anything bad about this approach? -- frigidcode.com signature.asc Description: OpenPGP digital signature ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe