On 30 Dec 2013, at 05:54, meekerdb wrote:

On 12/29/2013 7:45 PM, Jason Resch wrote:



On Sun, Dec 29, 2013 at 6:58 PM, meekerdb <meeke...@verizon.net> wrote:
On 12/29/2013 3:49 PM, Jason Resch wrote:



On Sun, Dec 29, 2013 at 5:42 PM, meekerdb <meeke...@verizon.net> wrote:
On 12/29/2013 2:08 PM, Jason Resch wrote:



On Sun, Dec 29, 2013 at 4:51 PM, meekerdb <meeke...@verizon.net> 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.

Thanks.

But just because you cannot compress all numbers of a given size doesn't imply that any particular number is incompressible.

That is true if you consider the size of the compression program to be of no relevance. In such a case, you can of course have a number of very small strings map directly to very large ones.

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.

Right, but this leads to the concept of Kolmogorov complexity. If you consider the size of the minimum string and algorithm together, necessary to represent some number, you will find there are some patterns of data that are more compressible than others. In your previous example with base 6999500235148668, you would need to include both that base, and the string "10" in order to encode 6999500235148669.

But that seems to make the randomness of a number dependent on the base used to write it down? Did I have to write down "And this is in base 10" to show that 6999500235148668 is random? There seems to be an equivocation here on "computing a number" and "computing a representation of a number".



A number containing regular patterns in some base, will also contain regular patterns in some other base (even if they are not obvious to us), compression algorithms are good at recognizing them.

The text of this sentence may not seem very redundant, but english text can generally be compressed somewhere between 20% - 30% of its original size. If you convert a number like "55555555555" to base 2, its patterns should be more evident in the pattern of bits.

For the majority of numbers, you will find the Kolmogorov complexity of the number to almost always be on the order of the number of digits in that number. The exceptions like 1111111111 are few and far between.

111111111 looks a lot messier in base 9.



base 10: 1111111111111111111
base 9: 7355531854711617707
base 2: 11110110101101110101101010110010101111000100011100

In base 9, there is a high proportion of 7's compared to other digits. In base 2, the sequence '110' seems more common than statistics would suggest. In any case, the number is far from incompressible. It takes only 9 characters to represent that 19 digit number in Kolmogorov complexity "1r19inb10" = "1 repeated 19 times in base 10", in my encoding language.

So you are agreeing with me that to cite a specific number and say "That number is random." is meaningless.

I agree with that in the sense of "random as unpredictable", but I disagree in the sense of "random as uncompressible". Some numbers are objectively not compressible, just like some shufflings of a deck of cards are uncompressible, because the shortest possible description of the ordering of the cards requires more information to describe than merely giving the list itself. So it is with a number and its digits.

I think we're talking past each other. What you're calling a number is what I'd call a string of digits. I can understand a string of digits being incompressible...but the number it names has many representations. To say a number is incompressible?? There's an old joke proof in mathematics that every number is interesting, otherwise there would be a smallest uninteresting number - which would peforce be interesting. It seems that interesting means something like "has a short description".

This shows that "interesting" needs some caution to be defined, and that the notion will not be algorithmically constructive (like most notion/objects in classical math). But for incompressibility, it means that the notion will also not be algorithmic:constructive, and that it will be based on some constant depending on the choice of the universal base. This makes the notion clear for all numbers except finite number of exceptions.

Bruno




Brent

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