[swift-evolution] [Draft][Proposal] Formalized Ordering

[Dave Abrahams]

on Fri Jul 22 2016, Xiaodi Wu <xiaodi.wu-AT-gmail.com> wrote:

> On Fri, Jul 22, 2016 at 9:46 PM, Dave Abrahams <dabrahams at apple.com> wrote:
>
>>
>> on Fri Jul 22 2016, Xiaodi Wu <xiaodi.wu-AT-gmail.com> wrote:
>>
>> > On Fri, Jul 22, 2016 at 9:23 PM, Matthew Johnson <matthew at anandabits.com
>> >
>> > wrote:
>> >
>> >>
>> >> On Jul 22, 2016, at 9:17 PM, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>> >>
>> >> On Fri, Jul 22, 2016 at 9:15 PM, Matthew Johnson via swift-evolution <
>> >> swift-evolution at swift.org> wrote:
>> >>
>> >>>
>> >>> On Jul 22, 2016, at 9:04 PM, Dave Abrahams via swift-evolution <
>> >>> swift-evolution at swift.org> wrote:
>> >>>
>> >>>
>> >>> on Fri Jul 22 2016, Matthew Johnson <swift-evolution at swift.org> wrote:
>> >>>
>> >>> On Jul 22, 2016, at 8:37 PM, Xiaodi Wu via swift-evolution
>> >>> <swift-evolution at swift.org> wrote:
>> >>>
>> >>> On Fri, Jul 22, 2016 at 8:20 PM, Dave Abrahams via swift-evolution
>> >>> <swift-evolution at swift.org
>> >>>
>> >>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>
>> >>> wrote:
>> >>>
>> >>> on Fri Jul 22 2016, Daniel Duan <daniel-AT-duan.org
>> >>> <http://daniel-at-duan.org/>> wrote:
>> >>>
>> >>> On Jul 22, 2016, at 3:00 PM, Dave Abrahams via swift-evolution
>> >>> <swift-evolution at swift.org
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>
>> >>> wrote:
>> >>>
>> >>>
>> >>> on Fri Jul 22 2016, Daniel Duan
>> >>> <swift-evolution at swift.org
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>>
>> >>> wrote:
>> >>>
>> >>>
>> >>> On Jul 22, 2016, at 11:05 AM, Dave Abrahams via swift-evolution
>> >>> <swift-evolution at swift.org
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>>
>> >>> wrote:
>> >>>
>> >>>
>> >>> on Thu Jul 21 2016, Duan
>> >>>
>> >>>
>> >>> <swift-evolution at swift.org
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>
>> >>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>>>
>> >>> wrote:
>> >>>
>> >>> Great proposal. I want to second that areSame may mislead user to
>> >>> think this is about identity.
>> >>>
>> >>> I like areEquivalent() but there may be better names.
>> >>>
>> >>>
>> >>> It really *is* about identity as I posted in a previous message.  But
>> >>> that doesn't change the fact that areEquivalent might be a better name.
>> >>> It's one of the things we considered; it just seemed long for no real
>> >>> benefit.
>> >>>
>> >>>
>> >>> If the addresses of the arguments aren’t being used, then we don’t
>> >>> consider
>> >>> them part of their *identity*. I can follow this logic. My fear is most
>> >>> users
>> >>> won’t make this leap on their own and get the same initial impression
>> as
>> >>> I did.
>> >>> It's entirely possible this fear is unfounded. Some educated
>> bikesheding
>> >>> wouldn't hurt here IMO :)
>> >>>
>> >>>
>> >>> Well, it's still a very real question whether we ought to have the
>> >>> additional API surface implied by areSame, or wether we should collapse
>> >>> it with ===.
>> >>>
>> >>>
>> >>> To spell this out (because I had to think about it for a second): ===
>> >>> will be derived from
>> >>> <=>,
>> >>> but also becomes default implementation for ==, which remains open for
>> >>> customization.
>> >>>
>> >>>
>> >>> I was imagining roughly this (untested):
>> >>>
>> >>>  /// Two references are identical if they refer to the same
>> >>>  /// instance.
>> >>>  ///
>> >>>  /// - Note: Classes with a more-refined notion of “identical”
>> >>>  ///   should conform to `Identifiable` and implement `===`.
>> >>>  func ===(lhs: AnyObject, rhs: AnyObject) -> Bool {
>> >>>    ObjectIdentifier(lhs) == ObjectIdentifier(rhs)
>> >>>  }
>> >>>
>> >>>  /// Supports testing that two values of `Self` are identical
>> >>>  ///
>> >>>  /// If `a` and `b` are of type `Self`, `a === b` means that
>> >>>  /// `a` and `b` are interchangeable in most code.  A conforming
>> >>>  /// type can document that specific observable characteristics
>> >>>  /// (such as the `capacity` of an `Array`) are inessential and
>> >>>  /// thus not to be considered as part of the interchangeability
>> >>>  /// guarantee.
>> >>>  ///
>> >>>  /// - Requires: `===` induces an equivalence relation over
>> >>>  ///   instances.
>> >>>  /// - Note: conforming types will gain an `==` operator that
>> >>>  ///   forwards to `===`.
>> >>>  /// - Note: Types that require domain-specific `==`
>> >>>  ///   implementations with different semantics (e.g. floating
>> >>>  ///   point) should define a more-specific overload of `==`,
>> >>>  ///   which will be used in contexts where the static type is
>> >>>  ///   known to the compiler.
>> >>>  /// - Note: Generic code should usually use `==` to compare
>> >>>  ///   conforming instances; that will always dispatch to `===`
>> >>>  ///   and will be unaffected by more specific overloads of
>> >>>  ///   `==`.
>> >>>  protocol Identifiable { // née Equatable name is negotiable
>> >>>    func ===(_: Self, _: aSelf) -> Bool
>> >>>  }
>> >>>
>> >>>  /// Default definition of `==` for Identifiable types.
>> >>>  func ==<T: Identifiable>(lhs: T, rhs: T) -> Bool {
>> >>>    return lhs === rhs
>> >>>  }
>> >>>
>> >>>  /// Conforming types have a default total ordering.
>> >>>  ///
>> >>>  /// If `a` and `b` are of type `Self`, `a <=> b` means that
>> >>>  /// `a` and `b` are interchangeable in most code.  A conforming
>> >>>  /// type can document that specific observable characteristics
>> >>>  /// (such as the `capacity` of an `Array`) are inessential and
>> >>>  /// thus not to be considered as part of the interchangeability
>> >>>  /// guarantee.
>> >>>  ///
>> >>>  /// - Requires: `<=>` induces a total ordering over
>> >>>  ///   instances.
>> >>>  /// - Requires: the semantics of `<=>` are  consistent with
>> >>>  ///   those of `===`.  That is, `(a <=> b) == .equivalent`
>> >>>  ///   iff `a === b`.
>> >>>
>> >>> For floating point, I'd hope that `a === b` if `(a <=> b) == .same`
>> *but
>> >>> not iff*. This is to satisfy IEEE 754: "Comparisons shall ignore the
>> sign
>> >>> of zero (so +0 = −0)”.
>> >>>
>> >>>
>> >>> The point of this design is that `===` means identity and that `.same `
>> >>> also means identity.
>> >>>
>> >>> Since this is new territory I suppose we get to decide what identity
>> >>> means for floating point.  Should +0 and -0 have the same identity or
>> >>> not?  I’ll leave the answer to folks more knowledgable about numerics
>> >>> than I.
>> >>>
>> >>>
>> >>> It's settled law
>> >>>
>> https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate
>> >>> :-)
>> >>>
>> >>>
>> >>> Yes, assuming we want to define identity in terms of the IEEE
>> definition
>> >>> of total ordering.
>> >>>
>> >>
>> >> I see what you're saying here. That could work. Comparable `===` and
>> >> Equatable `<=>` could do its own thing, and FloatingPoint
>> >> `isTotallyOrdered(below:)` can preserve the IEEE definition of total
>> >> ordering
>> >>
>> >>
>> >> Actually, I was hinting at your argument that `===` true iff `<=>` same
>> >> shouldn’t be a semantic requirement of the protocols.
>> >>
>> >> This is another option, but I don’t think it’s going to fly.  It seems
>> >> reasonable to assume that `<=>` will have IEEE semantics.  We will trip
>> a
>> >> lot of people up if it doesn’t.  That’s a big reason we can’t consider
>> >> changing floating point `==` to define an equivalence relation.
>> >>
>> >
>> > Actually, here I doubt it. The total ordering isn't exposed as part of
>> any
>> > comparison operator defined in the IEEE spec. In fact, the total ordering
>> > wasn't introduced until a (fairly) recent IEEE revision, IIUC. Breaking
>> > `==` would definitely cause people to jump, but `<=>` needn't be the IEEE
>> > totalOrder predicate IMO.
>>
>> Wait, I thought we were saying that `<=>` could be IEEE totalOrder, and
>> `===` could be like `==` but with well-behaved NaNs, so it's still an
>> equivalence relation, thus declaring the signedness of 0 to be
>> inessential.
>>
>
> I was (that was the "=== if but not iff <=>" business above), then I
> thought Matthew was saying something different and agreed with him.
>
> What I thought that Matthew thought was actually very insightful. He didn't
> actually think this, apparently, but: IEEE totalOrder does exactly what it
> says on the tin. But, it is not useful for any generic comparisons or (as
> far as I'm aware) any generic sorting algorithms. I cannot conceive of a
> numeric algorithm or a generic algorithm that relies on two equal floating
> point values being ordered based on their binary representation. We should
> have some way of exposing totalOrder to a user of a BinaryFloatingPoint
> type, but I don't know that it should be the basis for floating point
> *identity* with respect to protocol conformance. It's explicitly *not* what
> IEEE recommends for comparison anyway.

That makes sense.  Perhaps IEEE hasn't actually made a principled
decision about which aspects of floating point numbers are essential,
and we have to do it for them.

-- 
Dave