Numerics & implementing different instances of the same class
Is there a good way of doing this? My running example is Monoid:
class Monoid a where operation :: a -> a -> a identity :: a
With the obvious examples on Num:
instance (Num a) => Monoid a where operation = (+) identity = 1
instance (Num a) => Monoid a where operation = (*) identity = 0
Of course, this won't work. I could introduce a newtype wrapper:
newtype (Num a) => MulNum a = MulNum a newtype (Num a) => AddNum a = AddNum a
instance (Num a) => Monoid (MulNum a) where operation (MulNum x) (MulNum y) = MulNum (x * y) identity = MulNum 1
instance (Num a) => Monoid (AddNum a) where ... -- etc
However, when it comes to defining (e.g.) a Field class you have two Abelian groups over the same type, which won't work straight off:
class Field a where ... instance (AbelianGroup a, AbelianGroup a) => Field a where ...
Could try using the newtypes again:
instance (AbelianGroup (x a), AbelianGroup (y a) => Field a where ...
... but this requires undecidable instances. I'm not even sure if it will do what I want. (For one thing it would also require an indication of which group distributes over the other, and this may restore decidability.) I'm beginning to think that the best way to do things would be to drop the newtype wrappers and include instead an additional parameter of a type-level Nat to allow multiple definitions per type. Is this a good way to do things? Has anyone else done something similar? I've taken a look at the Numeric Prelude but it seems to be doing things a bit differently. (e.g. there aren't constraints on Ring that require Monoid, etc) - George
What about something like data AddMult a b = AddMult a b class Monoid a where operation :: a -> a -> a identity :: a instance (Monoid a, Monoid b) => Monoid (AddMult a b) where operation (AddMult a1 m1) (AddMult a2 m2) = AddMult (operation a1 a2) (operation m1 m2) identity = AddMult identity identity class Commutative a where -- Nothing, this is a programmer proof obligation class Monoid a => Group a where inverse :: a -> a class (Commutative a, Group a) => AbelianGroup a where class (AbelianGroup a, AbelianGroup b) => Field a b where instance AbelianGroup a => Field a a where George Pollard wrote:
Is there a good way of doing this? My running example is Monoid:
class Monoid a where operation :: a -> a -> a identity :: a
With the obvious examples on Num:
instance (Num a) => Monoid a where operation = (+) identity = 1
instance (Num a) => Monoid a where operation = (*) identity = 0
Of course, this won't work. I could introduce a newtype wrapper:
newtype (Num a) => MulNum a = MulNum a newtype (Num a) => AddNum a = AddNum a
instance (Num a) => Monoid (MulNum a) where operation (MulNum x) (MulNum y) = MulNum (x * y) identity = MulNum 1
instance (Num a) => Monoid (AddNum a) where ... -- etc
However, when it comes to defining (e.g.) a Field class you have two Abelian groups over the same type, which won't work straight off:
class Field a where ... instance (AbelianGroup a, AbelianGroup a) => Field a where ...
Could try using the newtypes again:
instance (AbelianGroup (x a), AbelianGroup (y a) => Field a where ...
... but this requires undecidable instances. I'm not even sure if it will do what I want. (For one thing it would also require an indication of which group distributes over the other, and this may restore decidability.)
I'm beginning to think that the best way to do things would be to drop the newtype wrappers and include instead an additional parameter of a type-level Nat to allow multiple definitions per type. Is this a good way to do things?
Has anyone else done something similar? I've taken a look at the Numeric Prelude but it seems to be doing things a bit differently. (e.g. there aren't constraints on Ring that require Monoid, etc)
- George
2008/12/12 George Pollard
However, when it comes to defining (e.g.) a Field class you have two Abelian groups over the same type, which won't work straight off:
Especially since you generally can't take the multiplicative inverse of the additive identity.
I'm beginning to think that the best way to do things would be to drop the newtype wrappers and include instead an additional parameter of a type-level Nat to allow multiple definitions per type. Is this a good way to do things?
That depends on what you're trying to do. I don't think this is an area where there is a single best solution. I've occasionally toyed with labeled monoid classes, like this one: class Monoid label a where unit :: label -> a mult :: label -> a -> a -> a data Plus instance (Num a) => Monoid Plus a where unit _ = 0 mult _ = (+) ... and so forth. Even here, there are several design possibilities. For example, here the label and the carrier jointly determine the operation, but you can also have the label determine the operation and the carrier. Moving on, you can then have: class (Monoid label a) => Group label a where inverse :: label -> a -> a class (Group labP a, Monoid labM a) => Ring labP labM a Of course, you now need to provide labels for all your operations. I suspect the overhead isn't worth it.
Has anyone else done something similar? I've taken a look at the Numeric Prelude but it seems to be doing things a bit differently. (e.g. there aren't constraints on Ring that require Monoid, etc)
A couple of years ago, I suggested breaking Num into Monoid, Semiring,
Group, Ring, and something else for abs and signum.
http://www.haskell.org/pipermail/haskell-cafe/2006-September/018118.html
Thus,
class Monoid a where
zero :: a
(+) :: a -> a -> a
class (Monoid a) => Semiring a where
one :: a
(*) :: a -> a -> a
Semiring has laws which require one and (*) to form a monoid, so:
newtype Product a = Product a
instance (Semiring a) => Monoid (Product a) where
zero = Product one
Product x + Product y = Product (x * y)
Note that the Monoid instance is now a consequence of the Semiring
instance, rather than a requirement.
--
Dave Menendez
participants (3)
-
Dan Weston -
David Menendez -
George Pollard