Rob, (01)
RF> I agree there may be some systems usually expressed in terms
> of reals which seem to have the property I want. Chaos and
> catastrophe theory would be two. But I'm not sure that "reals"
> sums them up. (02)
I agree that just using real numbers (or floating point on computers)
is a weak constraint that does not narrow down the options very much.
It's a necessary, but not sufficient condition. But it does widen
the range of mathematical structures and algorithms to consider. (03)
RF> It struck me that the power of computation might explain certain
> odd properties of these non-linear systems (their "random"
> character, to be frank :-) Perhaps the dialogue can be productive
> the other way round as well. But let us be specific what it is
> about either which is important. (04)
As Blum et al. discussed, many NP-complete problems can be solved
to a very high degree of approximation by using some kind of model
based on real numbers. The traveling salesman problem is a good
example. Unfortunately, they noted that the various examples they
have studied use widely different kinds of math for each problem.
Their goal is to find some underlying principles or systematic
approach to finding solutions for all or at least a large number
of NP-complete problems. (05)
RF> For instance, all the "real" systems which interest me seem
> to be "non-linear". Would that narrow it down a bit? (06)
Yes, but there are infinitely many nonlinear equations, almost
all of which are extremely difficult to solve. Furthermore,
some NP-complete problems can be solved without using nonlinear
equations. Also, the Fourier transforms, which are mathematical
analogs of holograms, use linear equations. (07)
One approach that I recommend is to look at sources that have
not been exhaustively analyzed. Those could be very new studies
that nobody else has yet read or very old studies that have been
forgotten. There are also many mathematical structures and
algorithms used in science and engineering that are seldom or
never used by the mainstream of AI and comp. sci. Holograms,
for example, fall in that category, but there are many others
that are less well known. (08)
Summary: If you have a problem that people know how to solve, look
at mainstream methods. But if the mainstream guys can't solve it,
look in the nooks and crannies and dusty old tomes. (09)
John (010)
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