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 generatecomputationally a random number, and that is right, if we wantgenerate only that numbers. but a simple counting algorithmgenerating all numbers, 0, 1, 2, .... 6999500235148668, ...generates all random finite incompressible strings,How can a finite string be incompressible? 6999500235148668 in base6999500235148669 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. Each particular language will have`

`some exception on the incompressibility issue. That should be part of`

`the role of the variable constant in the general universal definition.`

Bruno 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.