On 12/29/2013 1:28 PM, Jason Resch wrote:
On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 12/29/2013 5:56 AM, Bruno Marchal wrote:
On 28 Dec 2013, at 22:23, meekerdb wrote:
On 12/28/2013 4:09 AM, Bruno Marchal wrote:
For a long time I got opponent saying that we cannot generate
computationally a
random number, and that is right, if we want generate only that numbers.
but a
simple counting algorithm generating all numbers, 0, 1, 2, ....
6999500235148668,
... generates all random finite incompressible strings,
How can a finite string be incompressible? 6999500235148668 in base
6999500235148669 is just 10.
You can define a finite string as incompressible when the shorter
combinators to
generate it is as lengthy as the string itself.
This definition is not universal for a finite amount of short sequences
which
indeed will depend of the language used (here combinators).
Then you can show that such a definition can be made universal by adding
some
constant, which will depend of the universal language.
It can be shown that most (finite!) numbers, written in any base, are
random in
that sense.
Of course, 10 is a sort of compression of any string X in some base, but if
you
allow change of base, you will need to send the base with the number in the
message. If you fix the base, then indeed 10 will be a compression of that
particular number base, for that language, and it is part of
incompressibility
theory that no definition exist working for all (small) numbers.
Since all finite numbers are small, I think this means the theory only
holds in the
limit.
Brent
Brent,
It is easy to see with the pigeon hole principal. There are more 2 digit numbers than 1
digit numbers, and more 3 digit numbers than 2 digit numbers, and so on. For any string
you can represent using a shorter string, another "shorter string" must necessarily be
displaced. You can't keep replacing things with shorter strings because there aren't
enough of them, so as a side-effect, every compression strategy must represent some
strings by larger ones. In fact, the average size of all possible compressed messages
(with some upper-bound length n) can never be smaller than the average size of all
uncompressed messages.
The only reason compression algorithms are useful is because they are tailored to
represent some class of messages with shorter strings, while making (the vast majority
of) other messages slightly larger.
A good explanation. But just because you cannot compress all numbers of a given size
doesn't imply that any particular number is incompressible. So isn't it the case that
every finite number string is compressible in some algorithm? So there's no sense to
saying 6999500235148668 is random, but 11111111111111 is not, except relative to some
given compression algorithm.
Brent
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
