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

Dave Abrahams dabrahams at apple.com
Fri Jul 22 21:28:40 CDT 2016

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

> On Fri, Jul 22, 2016 at 9:13 PM, Matthew Johnson <matthew at anandabits.com>
> wrote:
>
>>
>> On Jul 22, 2016, at 9:10 PM, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>>
>> On Fri, Jul 22, 2016 at 9:08 PM, Matthew Johnson <matthew at anandabits.com>
>> wrote:
>>
>>>
>>> On Jul 22, 2016, at 9:04 PM, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>>>
>>> On Fri, Jul 22, 2016 at 8:57 PM, Matthew Johnson <matthew at anandabits.com>
>>>  wrote:
>>>
>>>>
>>>> On Jul 22, 2016, at 8:54 PM, Xiaodi Wu via swift-evolution <
>>>> swift-evolution at swift.org> wrote:
>>>>
>>>> On Fri, Jul 22, 2016 at 8:52 PM, Jaden Geller via swift-evolution <
>>>> swift-evolution at swift.org> wrote:
>>>>
>>>>> "The totalOrder predicate will order these cases, and it also
>>>>> distinguishes between different representations of NaNs and between the
>>>>> same decimal floating point number encoded in different ways."
>>>>> - [Wikipedia](
>>>>> https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate
>>>>> )
>>>>>
>>>>> Sounds like `===` should not return `true` for zeros of different
>>>>> signs, then.
>>>>>
>>>>
>>>> Fair enough; the result of that will be, as Pyry noted above, that:
>>>>
>>>> ```
>>>> [-0.0, 1.0, .nan, 0.0].firstIndex(of: 0.0) //=> 3, not 0
>>>> ```
>>>>
>>>>
>>>> Maybe we need floating point specific implementations of some algorithms
>>>> to resolve this problem?
>>>>
>>>> It doesn’t seem like there is a way to provide the semantics required by
>>>> generic algorithms and still provide the expected behavior for floating
>>>> point values.
>>>>
>>>
>>> Well, what I'm trying to say is that generic algorithms such as
>>> `index(of:)` require only an equivalence relation. For floating point
>>> types, there are three ways to slice it:
>>>
>>> 1. NaN != NaN and +0 == -0 [what the traditional comparison operators are
>>> constrained to do]
>>> 2. NaN == NaN, +0 == -0, and the same number encoded different ways
>>> compare equal
>>> 3. NaN == NaN, +0 != -0, and the same number encoded different ways
>>> compare not equal
>>>
>>> Both #2 and #3 can fall out of valid equivalence relations; if `===`
>>> behaved like #2 for FloatingPoint types, then generic algorithms work just
>>> fine. If we insist on using a total ordering defined by `<=>` all the time,
>>> then we've got problems.
>>>
>>>
>>> And if we don’t then we’re back to 3 different concepts of equality.
>>> There is definitely a tradeoff no matter what we choose.
>>>
>>
>> If some types have three concepts of equality, each with their particular
>> use, why must we eliminate one of them?
>>
>>
>> This isn’t about eliminating concepts of equality for a type.  They can
>> have 42 if they want.
>>
>> This is about the right way to define the semantics of specific
>> protocols.  It says nothing about additional notions of equality a type may
>> have available.
>>
>> The difficulty is finding a design for the protocols that makes sense with
>> floating point types because we want them to be able to conform to the
>> protocols.
>>
>
> Agreed. My argument is that if a Comparable can define its own `===`, still
> supplying a valid equivalence relation but not being constrained by a
> contract that `(a <=> b) == .same` iff `a === b`, then we are good to go
> with floating point types.

How would that work?  Can you spell out the implications, show how <=>
and === would be implemented, and describe what it would mean for
algorithms?

>
> On Jul 22, 2016, at 6:48 PM, Dave Abrahams via swift-evolution <
>>>>> swift-evolution at swift.org> wrote:
>>>>>
>>>>>
>>>>> on Fri Jul 22 2016, Jaden Geller <swift-evolution at swift.org> wrote:
>>>>>
>>>>> 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)".
>>>>>
>>>>>
>>>>> I don't see why both `(+0) === (-0)` and `(+0) <=> (-0)` can't return
>>>>> `true` and `.same`, respectively. This doesn't break the total
>>>>> ordering of values. `===` doesn't do raw memory comparison. They're
>>>>> "identical", so it ought to return `true`.
>>>>>
>>>>>
>>>>> It ought to do whatever IEEE-754 specifies that its total ordering test
>>>>> does.  That is, IEEE-754 gets to decide whether the difference between
>>>>> +0 and -0 is “essential” to IEEE-754 floating point types, or not.
>>>>>
>>>>>
>>>>> On Jul 22, 2016, at 6:37 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)".
>>>>>
>>>>>  /// - Note: conforming types will gain `<`, `<=`, `>`, and `>=`
>>>>>  ///   operators defined in terms of `<=>`.
>>>>>  /// - Note: Types that require domain-specific `<`, etc.
>>>>>  ///   implementations with different semantics (e.g. floating
>>>>>  ///   point) should define more-specific overloads of those
>>>>>  ///   operators, which will be used in contexts where the
>>>>>  ///   static type is known to the compiler.
>>>>>  /// - Note: Generic code can freely use `<=>` or the traditional
>>>>>  ///   comparison operators to compare conforming instances;
>>>>>  ///   the result will always be supplied by `<=>`
>>>>>  ///   and will be unaffected by more specific overloads of
>>>>>  ///   the other operators.
>>>>>  protocol Comparable : Identifiable {
>>>>>    func <=> (lhs: Self, rhs: Self) -> Ordering
>>>>>  }
>>>>>
>>>>>  /// Default implementations of `<`, `<=`, `>`, and `>=`.
>>>>>  extension Comparable {
>>>>>    static func <(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (lhs <=> rhs) == .ascending
>>>>>    }
>>>>>    static func <=(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (rhs <=> lhs) != .ascending
>>>>>    }
>>>>>    static func >(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (lhs <=> rhs) == .descending
>>>>>    }
>>>>>    static func >=(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (rhs <=> lhs) != .descending
>>>>>    }
>>>>>  }
>>>>>
>>>>> I like this idea. If we keep === as a separate thing, now users have 3
>>>>> “opportunities” to define
>>>>> equality. The must be few, if any, use cases for this.
>>>>>
>>>>> Would love to see if anyone on the list can give us an example.
>>>>> Otherwise we should make
>>>>> areSame === again™!
>>>>>
>>>>>
>>>>> Daniel Duan
>>>>> Sent from my iPhone
>>>>>
>>>>> On Jul 21, 2016, at 6:32 PM, Robert Widmann via swift-evolution
>>>>> <swift-evolution at swift.org
>>>>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>
>>>>> wrote:
>>>>>
>>>>>
>>>>> On Jul 21, 2016, at 6:19 PM, Xiaodi Wu
>>>>> <xiaodi.wu at gmail.com
>>>>> <mailto:xiaodi.wu at gmail.com <xiaodi.wu at gmail.com>>>
>>>>> wrote:
>>>>>
>>>>> This is nice. Is `areSame()` being proposed because static `==` is
>>>>> the status quo and you're trying to make the point that `==` in the
>>>>> future need not guarantee the same semantics?
>>>>>
>>>>>
>>>>> Yep!  Equivalence and equality are strictly very different things.
>>>>>
>>>>>
>>>>> Nit: I think the more common term in stdlib would be
>>>>> `areEquivalent()`. Do you think `same` in that context (independent
>>>>> of the word "ordering") might erroneously suggest identity?
>>>>>
>>>>>
>>>>> There is room for improvement here.  Keep ‘em coming.
>>>>>
>>>>>
>>>>>
>>>>> On Thu, Jul 21, 2016 at 8:11 PM, Robert Widmann via
>>>>> swift-evolution
>>>>> <swift-evolution at swift.org
>>>>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>>
>>>>> wrote:
>>>>> Hello Swift Community,
>>>>>
>>>>> Harlan Haskins, Jaden Geller, and I have been working on a
>>>>> proposal to clean up the semantics of ordering relations in the
>>>>> standard library.  We have a draft that you can get as a gist.
>>>>>
>>>>> Cheers,
>>>>>
>>>>> ~Robert Widmann
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> _______________________________________________
>>>>> swift-evolution mailing list
>>>>> swift-evolution at swift.org
>>>>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>
>>>>>
>>>>>
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>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>
>>>>> _______________________________________________
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>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>
>>>>>
>>>>> --
>>>>> Dave
>>>>>
>>>>> _______________________________________________
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>>>>> <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>>>>
>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>>>
>>>>>
>>>>> _______________________________________________
>>>>> swift-evolution mailing list
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>>>>> <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>>>
>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>
>>>>>
>>>>> --
>>>>> Dave
>>>>>
>>>>> _______________________________________________
>>>>> swift-evolution mailing list
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>>>>> <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>>>
>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>
>>>>>
>>>>> --
>>>>> Dave
>>>>> _______________________________________________
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>>>>> <mailto:swift-evolution at swift.org <swift-evolution at swift.org>>
>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution
>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>
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>>>>>
>>>>> _______________________________________________
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>>>>>
>>>>>
>>>>> --
>>>>> Dave
>>>>>
>>>>> _______________________________________________
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>>>>>
>>>>>
>>>>>
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>>>>>
>>>> _______________________________________________
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>>>>
>>>>
>>>
>>
>>

--
Dave
```