Fundamental Compressionist Philosophy.

[Gerry Wolff]

----- Original Message -----
From: "Brendan Macmillan" <bren at mail.csse.monash.edu.au>
To: <gerry at informatics.bangor.ac.uk>
Cc: "Maillist CasC" <casc at sanna.com>
Sent: 30 May 2001 18:28
Subject: Re: Fundamental Compressionist Philosophy.


...
> In effect, *every* point is its own hill.  But in this situation, the
essential
> concept of a hill breaks down, because there is no gradient or slope,
because
> every point is different.
>
> Perhaps a good image is to take the terrain of your caving story, and cut
it up
> into cross-sections of a square foot each (as a practical approximation of
a
> mathematical "point"), and then jumble them all up, so that there is
> "no relationship between the heights of adjacent points".  Now, to find
the
> heighest point now require exhaustive search, as knowing the height of one
> point tells us nothing about the height of an adjacent point.  We truly
are
> lost.

Perhaps we can start a collection of these kinds of problems. Here is a
possible example: we know that the decimal expansion of pi is not random
because it can be generated by a relatively simple formula. But to most
people, the sequence of digits looks completely random. If someone were
given a section of the sequence, (particularly if it were a section
relatively far from the beginning) and asked to find a simple formula that
would generate the section, my guess is that most people, including
professional mathematicians, would find it impossibly difficult.

Would this be a possible example?

With heuristic techniques, it is normally not possible to guarantee that one
has found the best possible answer. If you find yourself on a peak, there is
no way of knowing whether it is the highest peak. All that one can do is
find an answer that is tolerably good. So if you move around amongst a
jumble of points at different heights and accidentally stumble on a
relatively high one, you may decide that that is "good enough". There does
not seem to be much other option.

Regards,

Gerry