On 29 Dec 2013, at 23:42, meekerdb wrote:
On 12/29/2013 2:08 PM, Jason Resch wrote:
On Sun, Dec 29, 2013 at 4:51 PM, meekerdb <meeke...@verizon.net>
On 12/29/2013 1:28 PM, Jason Resch wrote:
On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <meeke...@verizon.net>
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
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.
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
A good explanation.
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
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".
Only for the numbers or strings with size similar to the size of the
universal number use for the compression. This means it works for
almost all numbers (= all except a finite number of exception).
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.
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