On 24.10.2010 03:38, wren ng thornton wrote:
On 10/23/10 4:53 PM, Alexey Khudyakov wrote:
On 23.10.2010 05:11, wren ng thornton wrote:
I'd rather see,
class Additive v where -- or AdditiveMonoid, if preferred
zeroV :: v
(^+^) :: v -> v -> v
class Additive v => AdditiveGroup v where
negateV :: v -> v
Seems good for me. One more instance declaration to write and no changes
in usage.
However when written this way it becomes obvious that
`zeroV' == `mempty' and ^+^ = mappend. Is Additive really needed then?
It depends on the usage, since we don't have a nice way of having
multiple Monoid instances in scope with different identifiers for their
respective mzero/mappend. For example, in Edward Kmett's monoids[1]
library he reuses Monoid for additive monoids and adds a new
Multiplicative class for multiplicative monoids; that way you can use
operators for a semiring without needing newtype wrappers everywhere in
order to distinguish the two structures on the same type.
When dealing with modules and vector spaces we have three or four
different monoids in play: the additive and multiplicative monoids of
the underlying semiring/ring/field, and the additive and multiplicative
monoids of the module/vectorspace. Lacking the aforementioned feature,
that means there are good reasons to have duplicate classes (i.e.,
they're all monoids) so long as they are documented as capturing
different notions (e.g., distinguishing "scalar" and "vectorial" uses).
I don't care much about the name of the class, I'd just like support for
monoids, semirings,... when they lack a group, ring,... structure.
Then what about following type class hierarchy? I think it supports
those structures. Only restriction is that it forces one to have both
left and right modules. It's possible to split them but I think it will
be to painful for vector spaces over R and C.
class AdditiveMonoid v where
(^+^) :: v → v → v
zeroV :: v
class AdditiveMonoid ⇒ AdditiveGroup v where
negateV :: v → v
-- For performance sake
(^-^) :: v → v → v
v ^-^ u = v ^+^ negateV u
class Module v where
type Scalar v :: *
(*^) :: Scalar v → v → v
(^*) :: v → Scalar v → v
(^*) = flip (*^)
class (AdditiveGroup v, Module v) ⇒ VectorSpace v
class VectorSpace v ⇒ InnerSpace v where
(<.>) :: v → v → Scalar v
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