On 12/29/2013 5:56 AM, Bruno Marchal wrote:

## Advertising

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 randomnumber, and that is right, if we want generate only that numbers. but a simplecounting 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 isjust 10.You can define a finite string as incompressible when the shorter combinators togenerate it is as lengthy as the string itself.This definition is not universal for a finite amount of short sequences which indeedwill 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 allowchange of base, you will need to send the base with the number in the message. If youfix the base, then indeed 10 will be a compression of that particular number base, forthat language, and it is part of incompressibility theory that no definition existworking for all (small) numbers.

Since all finite numbers are small, I think this means the theory only holds in the limit. Brent

Each particular language will have some exception on the incompressibility issue. Thatshould be part of the role of the variable constant in the general universal definition.Bruno http://iridia.ulb.ac.be/~marchal/ <http://iridia.ulb.ac.be/%7Emarchal/> --You received this message because you are subscribed to the Google Groups "EverythingList" group.To unsubscribe from this group and stop receiving emails from it, send an email toeverything-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.

-- 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.