Re: [boost] Please don't define a fake"operator<"just for ordered containers

From: Christoph Ludwig (ludwig_at_[hidden])
Date: 2006-07-12 06:27:42

On Tue, Jul 11, 2006 at 10:44:50AM -0700, Sean Parent wrote:
> What are the semantics of operator <?
> 
> I'd state the following:
[...]
> 4. In the absence of established convention (such as from the domain  
> of mathematics), lexicographical ordering by comparing the parts  
> should be used.
> 	Examples from the standard: std::string (which does not sort by  
> established linguistic rules), std::pair, std::vector, std::set,  
> tr1::tuple.
> 	Counter examples (which should be fixed): std::complex (real,  
> imaginary).
 
Do I understand correctly that you ask for 

  namespace std {
    template<class T>
    bool operator<(const complex<T>& lhs, const complex<T>& rhs);
  }

in the standard? That makes me certainly cringe! 

complex<T> models the _arithmetic_ type of complex numbers. 
If you order an arithmetic type A, then the ordering better respects the
arithmetic. That is (among other conditions), if a,b,c in A, a < b, and 0 < c,
then a * c < b * c.  That implies, though, there cannot be an ordering
that respects the complex arithmetic. [Let a = 0, b = 1i, c = 1i and assume
you have an ordering with 0 < 1i. Then a < b, 0 = a * c > b * c = -1, i.e.,
your ordering does not respect the arithmetic. (If 0 > 1i, then
consider b = c = -1i.)]

Sure, you can "forget" the arithmetic structure of the complex numbers and
restrict them to their structure as a two dimensional real vector space; I
agree this is often useful and you can order finite dimensional vector
spaces. Bu then either implement the ordering in a named function or define
an appropriate type that models (2D) real vector spaces. 

Defining operator< for complex<T> for convenience's sake seems to me an abuse
of operator overloading.

> Mathematicians do not have memory or complexity to deal with  
> (mathematicians don't sort) - so arguing that the lack of a rule in  
> mathematics implies that one shouldn't exist in computer science is  
> vacuous.

You forgot the theory of Groebner bases. Mathematicians most certainly sort,
sometimes even with non-standard orders. :-) 

Regards

Christoph
-- 
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