On Fri, Jan 11, 2002 at 08:30:07PM -0500, Mathieu Bouchard wrote:
> 
> On Tue, 8 Jan 2002, Tomasz Wegrzanowski wrote:
> > On Mon, Jan 07, 2002 at 11:47:29AM -0500, Mathieu Bouchard wrote:
> > > On Fri, 4 Jan 2002, Tomasz Wegrzanowski wrote:
> > > I think you're better off defining a measure... well I don't know, but I
> > > wouldn't think of a problem space as continuous enough to warrant defining
> > > a density. The question would be "what is your total interest in this set
> > > of problems", that is, a power(Problems) -> R_{0+} function, such that it
> > > is homomorphic under addition (set addition is union presuming mutually
> > > exclusive sets), and M(x)+M(not x) = M(everything)...
> > Problem space is at least dense, as you need dense set to
> > represent some of requirements that are part of your problem,
> > like all performance-related issues.
> 
> Well, if a problem, to be solvable, has to be of finite length in a finite
> alphabet, then the problem space is discrete, even though it's
> theoretically infinite. If you think about real numbers that can be
> incorporated in the problem definition: the subset of the real numbers
> that can actually be written down (in whatever form) is countable. It's
> "bigger" than the Rational set just like the latter is "bigger" than the
> Integer set, but it's still countable.

Solution set is countable, problem set doesn't have to.
Problems are represented in human mind usually, without finite symbol alphabet.
Infinite number of problems can map into the same solution.

This discusion is quite esoteric :)