On 29 Dec 2013, at 22:51, meekerdb wrote:
On 12/29/2013 1:28 PM, Jason Resch wrote:
On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <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.
It works up to a constant related to the choice of the universal base
to do the compression. "11111111111111" is probably random in the SK
combinator language. But for strings which are greater than the
description of the universal bases used, the same strings will be
random or not.
Bruno
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.
http://iridia.ulb.ac.be/~marchal/
--
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.
