Traits in `std::cmp` and mathematical terminology

[steffahn]

As for what should practically change in the docs IMO

• Ord documentation should mention "total preorder". IMO, it's okay to keep listing "total order", too, since that may help the intuition of people less familiar with more uncommon types of relations. In any case, the most important part is that the axioms - as stated in the docs - are correct. Which they are, except for...
• the axioms for PartialOrd need to be fixed:

  @steffahn

  The closest thing to the axioms given for PartialOrd in the docs is just:

  <= is transitive

  Yup, that's it. You can define the rest:
  a == b iff a <= b && b <= a
a != b iff !(a == b)
a < b iff a <= b && a != b
a > b iff b < a
a >= b iff b <= a

  But it appears there's actually an axiom missing in the documentation of PartialOrd . It doesn't state anywhere that a < b and b == c implies a < c (or similarly for a == b and b < c , giving a < c ). That's most likely an oversight though, I doubt that's intentional. If you add those two implications, the characterization becomes truly equivalent to “ <= is transitive”.

  I.e. the fix is either to add an axiom like "a == b and b < c and c == d implies a < d" Edit: a set of axioms like “a == b and b < c implies a < c. And: a < b and b == c implies a < c”, or to require that <= is transitive. Which is funny because once you require that <= is transitive, a bunch of other stuff (i.e. everything that doesn't just specify how different methods are supposed to be defined in terms of each other) technically becomes really entirely redundant. In any case, I haven't thought about at all yet, what the best presentation here might be. I do like the appeal of keeping things really simple regarding "<= ought to be transitive, and the rest has to be as-if defined in terms of <="; but these properties also have to stay straightforward to check if (which is commonly the case) you do not define all your stuff in terms of <=.

[jbe]

@steffahn

I doubt that's intentional

But the missing axiom isn't required to make PartialOrd (in regard to <) a partial order, right?

@steffahn

I.e. the fix is either to add an axiom like " a == b and b < c and c == d implies a < d ", or to require that <= is transitive.

What exactly would be fixed? It still doesn't make <= act according to a non-strict partial order.

@steffahn

Which is funny because once you require that <= is transitive, a bunch of other stuff (i.e. everything that doesn't just specify how different methods are supposed to be defined in terms of each other) technically becomes really entirely redundant.

Is the opposite also true? That the "bunch of other stuff" assures that <= is transitive?

So if I understand it right (really not sure if I do), we could either demand <= being transitive or keep the (then redudant) other demands?

P.S.: I can also check myself later. This was just a quick response. I should rather be going to sleep

… 5 hours later

[steffahn]

@jbe

@steffahn

I doubt that's intentional

But the missing axiom isn't required to make PartialOrd (in regard to < ) a partial order, right?

Yeah, but the docs already contain extra axioms besides the fact that the < relation of PartialOrd is (technically) a strict partial order. Btw, you cannot make "PartialOrd is a struct partial order" out of this, since e.g. == is a part of PartialOrd, too (via the PartialEq supertrait), and == cannot be defined in terms of < (but everything can be defined in terms of < and == together).

@jbe

@steffahn

I.e. the fix is either to add an axiom like "a == b and b < c and c == d implies a < d" Edit: a set of axioms like “a == b and b < c implies a < c. And: a < b and b == c implies a < c”, or to require that <= is transitive.

What exactly would be fixed?

It would fix that this makes sure that <= is transitive. You cannot seriously tell me the authors of PartialOrd did not want to have <= be transitive. But to be honest, I'm also not familiar with any algorithms / API, that requires just PartialOrd, so I'm not actually entirely sure what these axioms are used for anyways.

@jbe

@steffahn

Which is funny because once you require that <= is transitive, a bunch of other stuff (i.e. everything that doesn't just specify how different methods are supposed to be defined in terms of each other) technically becomes really entirely redundant.

Is the opposite also true? That the "bunch of other stuff" assures that <= is transitive?

Yes, but only as long you add the congruence axiom I've mentioned, i.e. "a == b and b < c and c == d implies a < d", the transitivity of <= is a consequence of this axiom together with the stuff listed in the documentation of PartialOrd. Seems like you didn't do didn't read your "homework":

@steffahn

Now, if you want to, you can verify that having a partial equivalence relation == and a strict partial order < with these two additional axioms is cryptomorphic to having a single, arbitrary, reflexive relation ≤ ,

---

BTW, note that those PartialOrd docs do also list a few truly redundant things like the transitivity of > or ==. (The transitivity of > is even also listed as a corollary, the transitivity of == is part of the conditions of PartialEq.)

[jbe]

@steffahn

Seems like you didn't do didn't read your "homework"

[steffahn]

@steffahn

@jbe

@steffahn

I.e. the fix is either to add an axiom like "a == b and b < c and c == d implies a < d" Edit: a set of axioms like “a == b and b < c implies a < c. And: a < b and b == c implies a < c”, or to require that <= is transitive.

What exactly would be fixed?

It would fix that this makes sure that <= is transitive.

To give a concrete example how <='s transitivity is missing in the currently documented axioms:

#[derive(Copy, Clone)]
pub enum O { A, B, C }
use O::*;

use std::cmp::Ordering::{self, *};

impl PartialOrd for O {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        match (*self, *other) {
            // all values equal themselves, A == B
            (A, A) | (B, B) | (C, C) | (A, B) | (B, A) => Some(Equal),
            // B < C
            (B, C) => Some(Less),
            (C, B) => Some(Greater),
            // A and C not comparable
            (A, C) | (C, A) => None,
        }
    }
}

impl PartialEq for O {
    fn eq(&self, other: &Self) -> bool {
        self.partial_cmp(other) == Some(Equal)
    }
}

impl Eq for O {}

fn main() {
    // violating congruence
    assert_eq!(A == B, true);
    assert_eq!(B < C, true);
    assert_eq!(A < C, false);
    
    // or
    
    // violating `<=`'s transitivity
    assert_eq!(A <= B, true);
    assert_eq!(B <= C, true);
    assert_eq!(A <= C, false);
}

(playground)

this enum fulfills¹ all the axioms listed in the documentation for PartialEq, PartialOrd and even Eq, yet behaves quite surprisingly lMO.

@jbe, looking at this example, turns out that without fixing the documentation of PartialOrd, we don't even get that PartialOrd + Eq is a preorder (or "intuitively a partial order, if we take Eq as equality").

---

¹Going through the axioms: The definition obviously defines an equivalence relation that identifies A and B. (== is reflexive, transitive, symmetric.) The axioms listed as "ensured by the default implementation" on PartialOrd's docs are also fulfilled, and "a == b if and only if `partial_cmp(a, b) == Some(Equal)`` is true by definition.

This leaves:

The comparison must satisfy, for all a , b and c :

• transitivity: a < b and b < c implies a < c . The same must hold for both == and > .
• duality: a < b if and only if b > a .

This transitivity is true. The only instance of ... < ... is B < C, so there's no way to fulfill the precondition a < b and b < c for some a, b, c, in the first place. And duality is also true, since the only instance of ... > ... is C > B, which corresponds to the only instance of ... < ... being B < C

[quinedot]

@jbe

It also helps me to understand that I can use PartialOrd to implement a partial order, but it will confuse me if I'm suddenly surprised by >= and <= not acting accordingly.

The map is not the territory. Here's an implementation consistent with the requirements as I read them.

struct Snowflake(i32);

impl PartialEq<Self> for Snowflake {
    fn eq(&self, _: &Self) -> bool {
        false
    }
}

impl PartialOrd<Self> for Snowflake {
    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
        match self.0.cmp(&other.0) {
            std::cmp::Ordering::Equal => None,
            ne => Some(ne),
        }
    }
}

For Snowflake, <= is a strict partial order (it's the same as <), not a non-strict one.

---

@jbe

And yeah, it is kinda funny that PartialOrd was likely introduced because of IEEE 754. Or were there other reasons?

Yeah, that's why. I think I saw the specific bug in passing when looking it up the stuff below, but didn't note it. There was no split until 2013 or 2014 or such.

I thought maybe the fact that PartialOrd<_> (and PartialEq<_>) can also compare different types might have something to do with it. But no. Turns out that there used to be a different trait (Equiv) for cross-type equality, but they got merged together. Before that, none of the comparison traits were cross-type.

Interestingly, the motivator for Equiv didn't need it anymore (replaced by Borrow I guess), but the generic partial comparison traits got some takers in the standard lib. Also interesting is that Eq and Ord were also briefly made generic, but that got dropped in the stabilization PR. The PR says that the traits match the reform RFC but that's not true (since the type parameter was dropped from Eq and Ord), if anyone wants to take it up .

A related todo is still open and just got pinged by Mara, which is amusing timing.

[tczajka]

What is an example of a practical problem that would be caused if Eq was automatically implemented whenever PartialEq<Self> was implemented? I don't really see any practical reason to separate these.

Reading on the requirements, PartialEq<Self> already requires symmetry and transitivity. Eq adds a requirement for reflexivity. But reflexivity of a relation R can be written as:

a = b => R(a, b)

Which, for equality, is a tautology:
a == b => a == b

As a practical example, I see that HashMap::get requires Eq for keys. But would any problem be caused if you used f64 for hash keys? I don't think so. Every NaN would simply be treated as a separate, unequal key, so every time you insert NaN you get another entry, and every time you look up NaN you don't find anything. So what? Seems fine to me, that's consistent with how equality works for NaN.

There is a problem with implementing Hash for NaN in a good way though, because really it should be different every time. But that's an issue separate from Eq.

[jbe]

@steffahn

To give a concrete example how <= 's transitivity is missing in the currently documented axioms: […]

Thank you very much, it makes things much easier to grasp.

I totally agree that <= not ending up transitive is odd. And I'm not sure if that was intended or not to allow this.

However, I would like to give a counter-argument why it could (maybe) (perhaps) make sense in some cases to demand that < and > describe a partial order while <= and >= aren't transitive.

(I hope I don't make any logic error here.)

Whenever we implement PartialEq without Eq, this means our == operator doesn't mean "equality". It means == is some other weird "partial equivalence relation". I also showed that PartialOrd gives us a strict partial order regarding < and >.

Now let's consider a new floating point type where:

• There are positive and negative finite numbers following comparison rules (regarding ==, < and <=) as usual (e.g. −2.3 < −1.0 <= −1.0 < 6.2).
• There is also −∞ and +∞. Contrary to IEEE 754, we want −∞ and +∞ to be equivalent in regard to == (as it can make sense to treat these equally / as equivalent, as for example done on the Riemann Sphere in mathematics or in the context of negative (Kelvin) temperatures).
• Yet we might want to have a partial order of some kind (using < and >).

I could imagine a complex number type where we want to include −∞ and +∞ as sort-of-limit for some calculations such as 1.0 / 0.0, but treat −∞ == +∞, yet −∞ < 0 < +∞. This doesn't sound more "odd" what IEEE 754 does.

We might still define a partial order on such type, e.g. where < may only return true if the imaginary part is zero.

(Side note: I really dislike that IEEE 754 distinguishes between −∞ and +∞, and even between −0 and +0, despite −0 == +0.)

@steffahn

fn main() {
    // violating congruence
    assert_eq!(A == B, true);
    assert_eq!(B < C, true);
    assert_eq!(A < C, false);
    
    // or
    
    // violating `<=`'s transitivity
    assert_eq!(A <= B, true);
    assert_eq!(B <= C, true);
    assert_eq!(A <= C, false);
}

Let's say

• A represents +∞
• B represents −∞
• C represents zero.

Then we get:

• +∞ == −∞ is true (like on the Riemann Sphere)
• −∞ < 0 is true
• +∞ < 0 is false
• +∞ <= −∞ is true (because +∞ is either smaller than or equivalent to −∞)
• −∞ <= 0 is true (because −∞ is either smaller than or equivalent to 0)
• +∞ <= 0 is false (because +∞ is neither smaller than nor equivalent to 0)

This doesn't seem more crazy (to me) than what IEEE 754 does. What do you think?

I would think of x <= y simply as a short-hand for x < y || x == y, and not more and not less. I don't think we should impose any semantics on <= but keep it strictly formal. Following that, PartialOrd describes a partial order (regarding < and >) and PartialEq describes a partial equivalence relation (regarding == and !=).

P.S.: Of course, PartialOrd and PartialEq are somewhat intertangled, because PartialEq is a supertrait of PartialOrd, but I consider this merely as an implementation detail (it allows to provide optimized implementations for PartialOrd::le, for example).

[kornel]

From a developer perspective: I have arrays of floats, and I need them sorted. I don't care what the proper math says, I have floats, not real numbers. Rust's sound and tight definitions are not helping me when I need to sort floats, and I can't.

The current state simply prevents me from getting my job done. I need to put extra work to do exactly the "bad" thing that Rust is trying to stop me from doing. I use sort_by(partial_cmp().unwrap_or(Equal)) or wrap floats in a newtype that intentionally incorrectly implements Ord just so that I can get floats sorted.

Could we have sort() functions that use a trait like Comparable and work on anything that supports <, while accepting and permitting all of the consequences of this comparison being inconsistent?

[jbe]

@kornel

From a developer perspective: I have arrays of floats, and I need them sorted. I don't care what the proper math says, I have floats, not real numbers. Rust's sound and tight definitions are not helping me when I need to sort floats, and I can't.

The current state simply prevents me from getting my job done. I need to put extra work to do exactly the "bad" thing that Rust is trying to stop me from doing.

I have had the same problem. I always end up with something like:

/// Compare two [`f64`] numbers while treating [NaN](f64::NAN) as greatest.
pub fn cmp_f64_nan_greatest(a: f64, b: f64) -> std::cmp::Ordering {
    a.partial_cmp(&b).unwrap_or_else(|| {
        a.is_nan().cmp(&b.is_nan()).then(std::cmp::Ordering::Equal)
    })
}

And then using that in a closure passed to sort_by.

It's annoying. I agree.

@kornel

I use sort_by(partial_cmp().unwrap_or(Equal)) or wrap floats in a newtype that intentionally incorrectly implements Ord just so that I can get floats sorted.

That I don't do. I'm always worried NaN's could actually appear.

@kornel

Could we have sort() functions that use a trait like Comparable and work on anything that supports < , while accepting and permitting all of the consequences of this comparison being inconsistent?

How about partial_sort, which is happy with PartialOrd? (Maybe I shouldn't have made a new thread, will be more careful next time, as these topics are kinda related.)

P.S.: I just gave an outline of an example in the other thread, as this is related to the original question: "Why sort() need T to be Ord ?"

[tczajka]

@kornel

The current state simply prevents me from getting my job done.

I agree. I have dealt with floating point professionally a lot, and have never seen any good coming out of the weird NaN comparison rules, it's always seen as an unpleasant nuisance.

@jbe

Now let's consider a new floating point type [...]

There are proposals that fix many issues with IEEE-754, such as posits. They have no -0.0, no NaNs at all, better handling of denormals / precision, and only one ∞ (I'm not sure about that last one, I kind of like -∞, +∞ as initializers for minimum / maximum).

[jbe]

@tczajka

I have dealt with floating point professionally a lot, and have never seen any good coming out of the weird NaN comparison rules, it's always seen as an unpleasant nuisance.

I don't agree here, because in C I sometimes rely on NaN not being greater or smaller than any other number, for example.

Like when I write (in C):

if !(a > 0.0 && a < 1.0) error();

I think similar would apply in Rust. But you need to be careful to not "simplify" that to:

if (a <= 0.0 || a >= 1.0) error();

[tczajka]

@jbe

I don't agree here, because in C I sometimes rely on NaN not being greater or smaller than any other number, for example.

Like when I write (in C):

if !(a > 0.0 && a < 1.0) error();

I think similar would apply in Rust. But you need to be careful to not "simplify" that to:

if (a <= 0.0 || a >= 1.0) error();

Both of these would call error if NaN was smaller than all other numbers. Wouldn't that be better than the current behavior, where the first one calls error but the second one doesn't?

[jbe]

@tczajka

Both of these would call error if NaN was smaller than all other numbers. Wouldn't that be better than the current behavior, […]

If NaN is always smaller than any number, then this doesn't work:

if (a < 0.0) { /* negative */ }
else if (a >= 0.0) { /* non-negative */ }
else { /* something bad happened! */ }

If f64::NAN < 0.0, then a < 0.0 would be true (which is somewhat odd).

I have worked with systems where NaN is "sorted" in some way (I don't remember where, maybe it was SQL?), and it didn't feel good.

[steffahn]

Regarding workarounds for floats, there's an obligatory mention of

ordered_float - Rust

missing in this thread.

The crate supports two legitimate approaches with wrapper-types: Either define an order for NaN (in this case arbitrarily, as the greatest element), or disallow NaN entirely (which is checked when constructing the wrapper type).

[tczajka]

@jbe

if (a < 0.0) { /* negative */ }
else if (a >= 0.0) { /* non-negative */ }
else { /* something bad happened! */ }

If f64::NAN < 0.0 , then a < 0.0 would be true (which is somewhat odd).

Odd perhaps, maybe positive NaNs are more intuitive than negative NaNs

The positive/negative example shows why NaNs are a nuisance and it would be better if they didn't exist at all. After all, i32 doesn't have NaNs (even if you divide 0/0), and it works out fine.

But the code you wrote is weird. If you simply forgot about NaNs it would be more natural to write:

if a < 0.0 { /* negative */ }
else { /* positive */ }

The only reason to write something like you wrote is if you remember about NaNs and want to deal with them separately. But in that case it's more natural to write:

if a.is_nan() { /* NaN */ }
else if a < 0.0 { /* negative */ }
else { /* positive */ }

[steffahn]

@kornel

I use sort_by(partial_cmp().unwrap_or(Equal)) or wrap floats in a newtype that intentionally incorrectly implements Ord just so that I can get floats sorted.

I think that's a bad approach; this kind of sort_by call does not meaningfully sort the list in the presence of NaNs. E.g.

use std::cmp::Ordering::Equal;

fn main() {
    let mut list = [
        3.0,
        6.0,
        9.0,
        f64::NAN,
        2.0,
        5.0,
        8.0,
        f64::NAN,
        1.0,
        4.0,
        7.0,
    ];
    list.sort_by(|x, y| x.partial_cmp(y).unwrap_or(Equal));
    dbg!(&list); // order unchanged, definitely not sorted at all, though
}

(playground)

---

If you want to meaningfully sort a list of floats with NaNs, it's better to properly implement Ord for a wrapper type, and instead disregard IEEE. Or to disallow NaNs alltogether. I.e. either of the approaches offered by the ordered_floats crate.

Unless you're saying that you only use this "sort_by(partial_cmp().unwrap_or(Equal))" if you already checked the list for NaNs and there were none, but you don't like wrapper types; in which case you might as well use "sort_by(partial_cmp().unwrap())" though.

[jbe]

But with IEEE 754, I can write:

if (a < 0.0) { /* negative */ }
else if (a >= 0.0) { /* non-negative */ }

And that's something, which I like about them.

I also do like NaN. If we have something like 0.0 / 0.0, what can we do?

• panic?
• give a wrong result such as zero?

Neither of that is nice. Not sure though if NaN != NaN is a good idea. (But it's what we have.)

[tczajka]

@jbe

But with IEEE 754, I can write:

if (a < 0.0) { /* negative */ }
else if (a >= 0.0) { /* non-negative */ }

And that's something, which I like about them.

You wouldn't write such code with i32. The reason you wrote that is because you want to detect NaNs. I don't understand why you prefer the "redundant" comparison with 0.0 to detect NaNs rather than using is_nan() which is a more direct way to say it, and thus nobody would be confused and think the comparison is redundant.

@jbe

If we have something like 0.0 / 0.0 , what can we do?

• panic?
• give a wrong result such as zero?

I would say: return infinity, just as you would when any other number is divided by 0.0. Seems reasonable.

If you're dividing two numbers that have both just rounded to 0.0, the accurate answer could be anything, so it doesn't really matter what you return (except panic, you don't want panic).

[jbe]

@steffahn

@jbe, looking at this example, turns out that without fixing the documentation of PartialOrd , we don't even get that PartialOrd + Eq is a preorder (or "intuitively a partial order, if we take Eq as equality").

This is a stronger argument. If Eq denotes equality (which is a consequence of Ord being a total order), then PartialOrd + Eq should be a partial order in regard to <= and >= (I think).

But I think when a type is !Eq, then <= and >= don't need to be transitive, as explained here:

@jbe

I would think of x <= y simply as a short-hand for x < y || x == y , and not more and not less. […]

When a type is !Eq, then == doesn't denote "equality".