On Mon, Jan 07, 2002 at 11:47:29AM -0500, Mathieu Bouchard wrote:
> On Fri, 4 Jan 2002, Tomasz Wegrzanowski wrote:
> > Let P be the problem space, that is set of all problems that can be
> > solved by any language. Requirements like speed are part of promlem.
> > Let difficulty of solving a problem be defined with positive real
> > number. No let's f_L(p) be function that gives difficulty of solving
> > problem p \in P in language L. This function changes to some extend
> > from one programmer to another. Now let's D(p)->R_{0+}, p \in P be
> > interest density function, that is, function that shows what problems
> > you are most interested in.
>
> 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.

I can't find a prove that it is or is not continuous now.

> > Our task is to find L that minimizes expresion \int D(p)*f_L(p) dp
>
> I think there are infinitely many "interesting" (non-zero interest)
> solvable problems that are arbitrarily difficult to solve in whatever
> language... which would mean that this integral diverges... which is not a
> good think if you're trying to optimise its value. Ok, I don't have much
> to back my claims, and I'm can't convince myself terribly.

Well, interest density is function of what you are likely to actually do,
not what you would like to do.

This function diverges only if you have too ambitious plans,
and it has good meaning in that case - you aren't going to finish them.