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Re: Comparisons and decorations, part 2

Arnold Neumaier wrote:

Nate Hayes wrote:

John Pryce wrote:

What I get from this thread is that

- We have had at least four valid formulas presented for the topological
"A interiorTo B" when A and B are nonempty, all of which, in principle,
seem to allay Nate's worst fears of performance hit. My nextDown/nextUp
method is clearly bad if these are done in software, but they are such
trivial operations given the 754 number encoding, and so useful, that I
would hope a bit of chip area would be devoted to hard-wiring them in the 
near future.

- I am very sceptical that anyone could predict which of the 4 will work
best on a given commonly used architecture, and that probably depends
strongly on how vectorised the code is.

- An "overall best way" to represent Empty in an inf-sup interval type
depends on many things and I am equally sceptical that we can predict it. 
So, reluctantly, I believe P1788 should NOT standardise a representation
of Empty this time around. It can do so at the next revision of 1788, if
existing implementations show a "best way" has emerged.




If as mentioned recently [1,Overflow] is a family of compact intervals,


According to Motion 3, intervals are definite sets of real numbers,
not families of intervals. Thus [1,Overflow] makes no sense.


I know what Motion 3 says.
This doesn't mean it doesn't make sense, only that P1788 hasn't passed a motion on such a thing. 

This is why I said _if_.








then
by reasoning mentioned in Arnold's 9/24 e-mail (appended below as a
postscript)
   [1,Overflow] \interior [0,Overflow]
should be false.
No, it would have to be undecided, since the unknown value of Overflow on the left and the unknowen value of Overflow on the right can have 
any of the relations < = >.


Undecided implies something like a bool_set result {true,false}.

I don't have any problem if comparision relations return bool_set results.

I also don't have any problem with what you said previously:



Therefore, one needs to ensure that a test for disjointness,
containment, or interiorness, is never false positive.
This rule is enough to decide all ambiguities for these
three comparisons.
Clearly, returning "false" instead of "{true,false}" avoids any potential false positive. 

Nate