Break `abs` into two aspects

[Zemyla]

There is a homomorphism from the Naturals to any Semiring, which obeys:

fromNatural 0 = zero
fromNatural 1 = one
fromNatural (m + n) = fromNatural m `plus` fromNatural n
fromNatural (m * n) = fromNatural m `times` fromNatural n

The simplest implementation is this, but it's nowhere near the most
efficient:

fromNatural :: Semiring a => Natural -> a
fromNatural 0 = zero
fromNatural n = one `plus` fromNatural (n - 1)

One which takes O(log n) time instead of O(n) would go like this:

fromNatural :: Semiring a => Natural -> a
fromNatural = go 0 zero one
  go i s m n | i `seq` s `seq` m `seq` n `seq` False = undefined
  go _ s _ 0 =  s
  go i s m n
    | testBit n i = go (i + 1) (plus s m) (plus m m) (clearBit n i)
    | otherwise = go (i + 1) s (plus m m) n

On Tue, Feb 4, 2020, 02:21 Andreas Abel <andreas.abel at ifi.lmu.de> wrote:

>  >         class Semiring a where
>  >            zero  :: a
>  >            plus  :: a -> a -> a
>  >            one   :: a
>  >            times :: a -> a -> a
>  >            fromNatural :: Natural -> a
>
> I think `fromNatural` should not be part of the `Semiring` class, but we
> could have an extension (NaturalSemiring) that adds this method.
>
> In the Agda code base, we have, for lack of a standard, rolled our own
> semiring class,
>
>
> https://github.com/agda/agda/blob/master/src/full/Agda/Utils/SemiRing.hs
>
> and we use it for several finite semirings, e.g.,
>
>
>
> https://github.com/agda/agda/blob/64c0c2e813a84f91b3accd7c56efaa53712bc3f5/src/full/Agda/TypeChecking/Positivity/Occurrence.hs#L127-L155
>
> Cheers,
> Andreas
>
> On 2020-02-03 22:34, Carter Schonwald wrote:
> > Andrew: could you explain the algebra notation you were using for short
> > hand?  I think I followed, but for people the libraries list might be
> > their first exposure to advanced / graduate abstract algebra (which
> > winds up being simpler than most folks expect ;) )
> >
> > On Fri, Jan 31, 2020 at 4:36 PM Carter Schonwald
> > <carter.schonwald at gmail.com <mailto:carter.schonwald at gmail.com>> wrote:
> >
> >     that actually sounds pretty sane. I think!
> >
> >     On Fri, Jan 31, 2020 at 3:38 PM Andrew Lelechenko
> >     <andrew.lelechenko at gmail.com <mailto:andrew.lelechenko at gmail.com>>
> >     wrote:
> >
> >         On Tue, 28 Jan 2020, Dannyu NDos wrote:
> >
> >          > Second, I suggest to move `abs` and `signum` from `Num` to
> >         `Floating`
> >
> >         I can fully relate your frustration with `abs` and `signum` (and
> >         numeric type classes in Haskell altogether). But IMO breaking
> >         both in `Num` and in `Floating` at once is not a promising way
> >         to make things proper.
> >
> >         I would rather follow the beaten track of Applicative Monad and
> >         Semigroup Monoid proposals and - as a first step - introduce a
> >         superclass (probably, borrowing the design from `semirings`
> >         package):
> >
> >         class Semiring a where
> >            zero  :: a
> >            plus  :: a -> a -> a
> >            one   :: a
> >            times :: a -> a -> a
> >            fromNatural :: Natural -> a
> >         class Semiring a => Num a where ...
> >
> >         Tangible benefits in `base` include:
> >         a) instance Semiring Bool,
> >         b) a total instance Semiring Natural (in contrast to a partial
> >         instance Num Natural),
> >         c) instance Num a => Semiring (Complex a) (in contrast to
> >         instance RealFloat a => Num (Complex a)),
> >         d) newtypes Sum and Product would require only Semiring
> >         constraint instead of Num.
> >
> >         Best regards,
> >         Andrew
> >
> >
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> >
> >
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