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From: David Bergman (davidb_at_[hidden])
Date: 20020905 13:41:37
Joel,
No matter what ordering chosen, the equivalence and '<=' syntactic sugar
(yes, it should be regarded as syntactic sugar for '<  ==') still
should be as proposed by you.
So, back to the ordering. Which one is the most natural choice? Hmm,
there are obviously three orderings that popup:
1. The lexicographic (yes, that is a properly chosen name) ordering,
where
' [a1, a2] < [b1, b2] ' iff ' a1 < b1  a1 == b1 && a2 < b2 '
2. The (strict; some people are strict about strictness ;) subset
ordering, where
' A < B ' iff A is a subset of B; when all end points reside in the
same chain, this is reduced to ' b1 <= a1 && a2 <= b2 && !(A == B) '
3. The "complete position" ordering (have no proper name for it...),
where
' [a1, a2] < [b1, b2] ' iff ' a2 < b1 && a1 < b1 '
the extra conjunct ' a1 < b1 ' is needed to ensure that all end
points are in the same chain, since we else can get A < B && B < A, if
the points lie in a circle.
Note that all these orderings are nontotal if the underlying "number"
system is not totally ordered.
In the case the underlying "number" system is totally ordered, ordering
(1) is the only one being total. We have obviously discussed ordering
(3) the most in this thread.
Partial orderings always have, and should have, these "undefined
regions". What is the problem with that? Forcing those regions to be
defined by either (1) introducing inconsistencies, such as defining '<='
to be '!>', or (2) using obscure tribools will not help.
We all have to realize that at least two of the orderings above are
partial, there is not much we can do about it.
No matter what ordering chosen, '<=' should be regarded as syntactic
sugar for '<  ==' in the case there exists an equivalence, such as
here...
I must be extremely stupid/ignorant not to see the problem by these
fairly straightforward definitions, so I hope someone will enlighten
me, so I also can join the "tribool and complex ordering" choire ;)
David
Original Message
From: boostbounces_at_[hidden]
[mailto:boostbounces_at_[hidden]] On Behalf Of Joel Young
Sent: Thursday, September 05, 2002 2:13 PM
To: boost_at_[hidden]
Cc: jdy_at_[hidden]
Subject: Re: [boost] Interval Library and comparison operators
From: "David Bergman" <davidb_at_[hidden]>
> Guillaume,
>
> What is the rationale behind *not* defining
>
> [a, b] == [c, d] as a == c && b == d
>
> And
>
> A <= B A < B  A == B
The key question is how are you going to define less than and still keep
it symmetric with greater than and not have an undefined region?
When you project the rich intervalinterval relation space into only
three relations, it is nontrivial to have a nice projection that
maintains a nice definition of equality.
Joel
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