Break `abs` into two aspects

[Andreas Abel]

 > There is a homomorphism from the Naturals to any Semiring

Sure, but there are many finite semirings where I would not care about 
such a homomorphism, thus, why force me to define it?

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

This might not be surjective, and also not very interesting.  For 
instance consider the semiring

   Set Bool
   zero  = Set.empty
   one   = Set.singleton True
   plus  = Set.union
   times s t = { x == y | x <- s, y <- t }

This semiring models variances (covariant = {True}, contravariant = 
{False}, constant = {}, dontknow = {True,False}).  times is for function 
composition and plus combination of information.

The fromNatural targets only the zero/one-fragment since plus is 
idempotent.  I conjecture there is not a single surjective semiring-hom 
from Nat to Set Bool.  Thus, a function fromNatural is totally 
uninteresting for the general case of semirings.

On 2020-02-04 13:42, Zemyla wrote:
> 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 
> <mailto: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>
>     <mailto: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>
>     <mailto: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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