# Context alias

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## Revision as of 12:34, 16 April 2009

Context aliases, also known as class aliases, are a long-requested feature of Haskell. This feature would allow class hierarchies to be restructured without breaking compatibility to a certain degree. Also, it would make fine-grained class hierarchies usable.

## Contents |

## 1 The proposal

### 1.1 The original class alias proposal

The original proposal can be found on a page on John Meachem’s website.

#### 1.1.1 Class aliases with new methods

I would like to emphasize an important point from the original proposal that was not emphasized enough:

Lets look at one of the examples from the original proposal:

class SemiLatticeJoin a where join :: a -> a -> a

class BoundedBelow a where bottom :: a

class BoundedBelowJoinable a = (BoundedBelow a, SemiLatticeJoin a) where joins :: [a] -> a joins xs = foldl join bottom xs

Notice that *BoundedBelowJoinable* doesn't have the *alias* keyword. Is this a syntax error or is it allowed? It is allowed because *BoundedBelowJoinable* is not just an alias for *(BoundedBelow a, SemiLatticeJoin a)*. It also declares a new method called *joins*.

So why is this usefull?

Users can declare instances for *BoundedBelow* and *SemiLatticeJoin* and get *joins* for free or they can declare an instance for *BoundedBelowJoinable* and define an optimized *joins* for their type.

Lets look at another example why this ability, to give a class alias new methods, is useful. Again I take an example from the original proposal but I slightly change it:

The current *Num* class in the Prelude is (more or less) this

class Num a where (+) :: a -> a -> a (*) :: a -> a -> a (-) :: a -> a -> a negate :: a -> a fromInteger :: Integer -> a

Ideally we would want to split it up using classes from the monoids package:

class Monoid a where mempty :: a mappend :: a -> a -> a class Monoid a => Group a where gnegate :: a -> a minus :: a -> a -> a gsubtract :: a -> a -> a

gnegate = minus mempty a `minus` b = a `mappend` gnegate b a `gsubtract` b = gnegate a `mappend` b

class Multiplicative a where one :: a times :: a -> a -> a

class FromInteger a where fromInteger :: Integer -> a

But this creates some problems as mentioned in the proposal:

- People using the new prelude have to write the ungainly (Monoid a, Group a, Multiplicative a, FromInteger a) and declare separate instances for all of them.
- If at some point a HasZero class is separated out then everyone needs to modify their instance declarations.
- Num still must be declared if you want it to work with old prelude functions, containing completely redundant information.
- All the problems mentioned in the second section of the proposal about alternate preludes in general.

We can solve all of them by creating a class alias:

class alias Num a = (Monoid a, Group a, Multiplicative a, FromInteger a)

Or can we? Unfortunately this *Num* is different than the original *Num*. Because instead of the methods *(+)*, *(-)*, *(*)* and *negate* we have *mappend*, *minus*, *times* and *gnegate*.

Fortunately we can add the original names as new methods to the class alias and give them default definitions in terms of the new names:

class Num a = (Monoid a, Group a, Multiplicative a, FromInteger a) where -- Default implementations of existing methods: mempty = 0 mappend = (+)

one = 1 times = (*)

minus = (-) gnegate = negate

-- New methods with default implementations: (+) :: a -> a -> a (+) = mappend

(*) :: a -> a -> a (*) = times

(-) :: a -> a -> a (-) = minus negate :: a -> a negate = gnegate

The question is: how is the above translated?

The new methods from *Num* should be placed in a new "internal" class: *Num_NEW_METHODS*:

class Num_NEW_METHODS a where (+) :: a -> a -> a (*) :: a -> a -> a (-) :: a -> a -> a negate :: a -> a

What happens when a user defines an instance for *Num*? Lets look at an example:

Say a user defines the natural numbers and makes them an instance of the *Num* class alias:

data N = Z | S N instance Num N where Z + y = y S x + y = S (x + y)

Z * _ = Z S Z * y = y S x * y = y + x * y

x - Z = x S x - S y = x - y

fromInteger 0 = Z fromInteger (n+1) = S n -- You gotta love n+k patterns!

Note that the other methods of *Num* like *mempty*, *mappend*, *one* and *times* have default implementations in terms of the above.

First of all an instance for *Num_NEW_METHODS* will be defined:

instance Num_NEW_METHODS N where Z + y = y S x + y = S (x + y)

Z * _ = Z S Z * y = y S x * y = y + x * y

x - Z = x S x - S y = x - y

negate = gnegate

Then the other instances are defined using methods from *Num_NEW_METHODS*:

instance Monoid N where mempty = 0 mappend = (+)

instance Group N where minus = (-)

instance Multiplicative N where one = 1 times = (*)

instance FromInteger N where fromInteger 0 = Z fromInteger (n+1) = S n -- You gotta love n+k patterns!

In conclusion, a class alias is a name for a context plus optionally a new class. The question is how useful this ability is.

The *BoundedBelowJoinable* could also be defined as a normal class with the necessary superclasses:

class (BoundedBelow a, SemiLatticeJoin a) => BoundedBelowJoinable a where joins :: [a] -> a joins xs = foldl join bottom xs

However, user now don't get a *BoundedBelowJoinable* for free when they have defined instances for *BoundedBelow* and *SemiLatticeJoin*.

The *Num* class alias could also be rewritten to a simple alias together with some functions that translate the original names to the new names:

class alias Num a = (Monoid a, Group a, Multiplicative a, FromInteger a)

(+) = mappend (-) = minus (*) = times negate = gnegate

### 1.2 Improvements

#### 1.2.1 “Context alias” instead of “class alias”

A “class alias” actually doesn’t stand for a class but for a context (or a part of a context). So it might be better to choose a slightly different syntax:

context Foobar a = (Foo a, Bar a)

However if we allow class "aliases" to be extended with new methods then a class "alias" is not just a name for a context. (It is actually a context with a new class)

Maybe we should keep the syntax really light like:

class Foobar a = (Foo a, Bar a)

#### 1.2.2 Superclass constraints

John Meacham proposes the following syntax for class aliases (context aliases) with superclass constraints:

class alias Num a = Eq a => (Additive a, Multiplicative a)

This is not consistent with the superclass syntax of class declarations. I think, we should use this syntax:

class alias Eq a => Num a = (Additive a, Multiplicative a)

Or better:

context Eq a => Num a = (Additive a, Multiplicative a)

#### 1.2.3 Functional dependencies

Does the following make sense?

class alias A a b = (B a b, C a b) | a -> b where ...

#### 1.2.4 Associated data types and type synonyms

When `{-# LANGUAGE TypeFamilies #-}` is enabled, classes may declare associated data types or associated type synonyms.

If we allow class aliases to be extended with new methods, I think it make sense to also allow them to be extended with associated data types and type synonyms:

class A a = (B a, C a) where type T a data D a

#### 1.2.5 Equality constraints

When `{-# LANGUAGE TypeFamilies #-}` is enabled, type contexts can include equality constraints (`t1 ~ t2`).

It makes sense to also allow them in class aliases (context aliases)

#### 1.2.6 Things to have in mind

In order to get the context alias extension well, we should have an eye on problems we might want to solve with the help of context aliases. Here are some:

- should just be a combination ofMonadPlusandAlternative(actually,Monadshould just be a combination ofAlternative fandApplicative f)forall a. Monoid (f a)

- should be a superclass ofApplicativeMonad

## 2 Implementation

Starting an implementation of context aliases is planned for the 5th Haskell Hackathon.