# Break `abs` into two aspects

Andreas Abel andreas.abel at ifi.lmu.de
Wed Feb 5 18:34:09 UTC 2020

```Well, I see your arguments, but cannot help the feeling that you are
reasoning from a specific instance family of semirings, namely numerical
ones (N, Z, Q, ...).

For idempotent semirings (e.g. the example I gave), repetitively adding
one gets you nowhere.  (Cf. also lattices, many of which are semirings.)

I'd be convinced if Natural was something like the free semiring, but
this is certainly not the case.

Semirings are really diverse, I don't think the Semiring class should be
hijacked for a particular flavor of semirings.  We do not have any such
pretext for Semigroup or Monoid either.

Enjoy the diversity at https://en.wikipedia.org/wiki/Semiring

On 2020-02-04 17:32, Zemyla wrote:
> It really doesn't matter if it's not "interesting" or not surjective for
> some Semirings. It should be included, because:
>
> (a) Even for semirings where it is "interesting", it's not surjective
> (for instance, Rational or Double)
> (b) It's a method with a default definition, so you don't have to expend
> any mental effort on it
> (c) A lot of instances have uninteresting methods: for instance, (*>)
> and (<*) for Applicative ((->) e) are const id and const respectively.
> sometimes useful/interesting/faster, rather than when they're always
> interesting.
> (d) It's useful for Semiring-generic methods and instances.
> (e) It can achieve an asymptotic speedup on some instances. Like, if you
> have Semiring a => Semiring (f a) for some type f, then you can have
> fromNatural n = pure (fromNatural n) instead of doing the whole O(log n)
> song and dance with the default definition. Also, your example admits a
> simple definition:
>    fromNatural n = if n == 0 then S.empty else S.singleton True
> (f) "zero" and "one" can be defined in terms of fromNatural, for
> programmers who only need to define that:
>    zero = fromNatural 0
>    one = fromNatural 1
> This leads to the MINIMAL pragma on Semiring being {-# MINIMAL plus,
> times, (zero, one | fromNatural) #-}
> (g) If it's not included in the class, but in some subclass
> (NaturalSemiring, you proposed), but it's possible from the class, then
> people will just define and use the O(log n) version instead of
> requiring the subclass, leading to wasted effort and duplicated code.
>
> On Tue, Feb 4, 2020, 09:20 Andreas Abel <andreas.abel at ifi.lmu.de
> <mailto:andreas.abel at ifi.lmu.de>> wrote:
>
>       > 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>
>      > <mailto: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
>     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>>
>      >     <mailto: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>>
>      >     <mailto: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
>      >      >         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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