Re: Break `abs` into two aspects

4 Feb 2020

      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  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/s...
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
mailto:carter.schonwald@gmail.com> wrote:
that actually sounds pretty sane. I think!
On Fri, Jan 31, 2020 at 3:38 PM Andrew Lelechenko
    mailto:andrew.lelechenko@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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Re: Break `abs` into two aspects

Zemyla