Break `abs` into two aspects
Carter Schonwald
carter.schonwald at gmail.com
Wed Feb 5 19:24:19 UTC 2020
ok, so would you make the case that
"fromNatural" always can have a definition thats sound/satisfactory, even
fro those ones that Andreas Abel is saying we're ignoring?
On Wed, Feb 5, 2020 at 2:00 PM Zemyla <zemyla at gmail.com> wrote:
> If your semiring is idempotent, then you can simply have
>
> fromNatural = idempotentFromNatural
>
> where
>
> idempotentFromNatural :: Semiring a => Natural -> a
> idempotentFromNatural n = if n == 0 then zero else one
>
> It's like stimes in Semigroup. The default implementation is almost always
> sensible, and sometimes it can have more meaning (like how the Sum monoid
> allows negative values as the repeat argument in stimes).
>
> And again, it's an operation that can be defined by default on all
> Semirings, and can be vastly faster than the default on some. That, in my
> opinion, justifies its inclusion. If you don't feel it's meaningful for
> your Semiring, just let it be defined by default. But it's definitely
> useful for numeric and derived ones.
>
> On Wed, Feb 5, 2020, 12:51 Carter Schonwald <carter.schonwald at gmail.com>
> wrote:
>
>> ooo, thats a good point about lattices/partial orders! (we like those
>> here too, but sometimes forget :) )
>>
>> On Wed, Feb 5, 2020 at 1:34 PM Andreas Abel <andreas.abel at ifi.lmu.de>
>> wrote:
>>
>>> 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.
>>> > Haskell adds methods to classes when they're always possible and
>>> > 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
>>> > > > 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>>
>>> > > <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
>>> > > 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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