Re: [boost] Interval Library and comparison operators

From: Guillaume Melquiond (gmelquio_at_[hidden])
Date: 2002-09-06 11:48:56

On Fri, 6 Sep 2002, Joel Young wrote:

> Hi Doug,
> 
> From: Douglas Gregor <gregod_at_[hidden]>
> > where a <= x <= c, b <= y <= d
> is there a typo in this?  Should it be:
>   where a <= x <= b, c <= y <= d
> 
> assuming there is a typo,
> 
> Just to understand this I quickly came up with the following
> three-valued logic partionings of the interval-interval space for:
> 
> [a,b] RelOp [c,d] iff 
>   Conjunction of x RelOp y for all (x, y) 
>     where a <= x <= b, c <= y <= d
> 
> Where Conjunction is three-valued conjunction.
> 
>        true              false           indeterminate
> ------------------------------------------------------
> <  ==  {<}               {>,mi}          A - {<,>,mi}            
> <= ==  {<,m}             {>}             A - {<,>,m} 
> =  ==  {}                {<,>}           A - {<,>} 
> != ==  {<,>}             {}              A - {<,>} 
> >= ==  {>,mi}            {<}             A - {<,>,mi}
> >  ==  {>}               {<,m}           A - {<,>,m}
> 
> where A = {<,>,m, mi, o, oi, s, si, f, fi, d, di, =}
> 
> Except for (IMHO verboten) point-intervals, true(=) is empty?

Yes, you get it right. But why should point-intervals be forbidden?

> Are the existential versions of interest?: 
> 
> [a,b] RelOp [c,d] iff 
>   Conjunction of 
>     (for all x in [a,b] there exists a y in [c,d] such that x RelOp y) 
> 
> or
> 
> [a,b] RelOp [c,d] iff 
>   Conjunction of 
>     (for all y in [c,d] there exists a x in [a,b] such that x RelOp y) 
> 
> or
> 
> [a,b] RelOp [c,d] iff 
>   Conjunction of 
>     (there exists a x in [a,b] and a y in [c,d]) such that x RelOp y) 

The last one is the one I use. And I use it as often as the "forall"
version.

Regards,

Guillaume