# Re: Bruno's mathematical reality

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On 29 Dec 2013, at 22:51, meekerdb wrote:```
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```On 12/29/2013 1:28 PM, Jason Resch wrote:
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On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <meeke...@verizon.net> wrote:
```On 12/29/2013 5:56 AM, Bruno Marchal wrote:
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On 28 Dec 2013, at 22:23, meekerdb wrote:

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```On 12/28/2013 4:09 AM, Bruno Marchal wrote:
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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,
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How can a finite string be incompressible? 6999500235148668 in base 6999500235148669 is just 10.
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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).
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Then you can show that such a definition can be made universal by adding some constant, which will depend of the universal language.
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It can be shown that most (finite!) numbers, written in any base, are random in that sense.
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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.
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Since all finite numbers are small, I think this means the theory only holds in the limit.
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Brent

Brent,

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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.
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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.
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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.
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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.
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Bruno

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Brent

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http://iridia.ulb.ac.be/~marchal/

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