Re: For Sean Pitman: Review of "Meaningful Information"
- From: "Seanpit" <seanpitnospam@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: 12 Mar 2007 10:10:04 -0700
On Mar 12, 12:00 am, "R. Baldwin" <res0k...@xxxxxxxxxxxxxxxxxxxx>
wrote:
Look, you are arguing that you can compress an apparently random
sequence into a one bit representation given the proper reference
computer. I don't see that your pigeon hole method helps you here
because your reference computer must already have the entire sequence
in question associated with your single bit key. That is not what I
would call "compression" and I don't think it is what Chaitin would
call compression either.
It doesn't matter what you would call compression. It matters what the
mathematics says. Chaitin's mathematics do not include the complexity of the
reference computer within the elegant program that represents a string. The
complexity of the reference computer is only accessible if you simulate the
reference computer on a different computer.
Try readinghttp://www.cs.auckland.ac.nz/CDMTCS/chaitin/cup.pdfinstead of
Chaitin's summary articles, and learn the math. You can skip past the LISP
in the first 5 chapters and begin with chapter 6.
True compression, or representation of a longer string by a shorter
one, is based on a shorter program or formula that when carried out
produces the longer string - - such as "list 'A' one million times".
That's very high compression. Your single bit "compression" simply
isn't the same type of compression.
Balderdash. Compression algorithms are optimized for the data they are
intended to compress.
Oh really? What if you are presented with a portion of a sequence
that is previously unknown to you. How would you compress it in a way
that would offer you greater predictive value as to what would come
next in the sequence? - in a way in which you would likely make a lot
of money if you could bet on what character would come next?
Given that you are correct, there would be no way to distinguish the
"orderliness" or "non-randomness" between two different finite strings
since both could be equally non-random depending upon the algorithm
chosen. The whole concept presented by Chaitin that there is an
intuitive as well as mathematical difference between 0101010101010101
and 1001110101100010 would be meaningless.
Yet, this notion is demonstrably mistaken. How so? Because of the
predictive value offered by the first string - which increases as the
pattern continues.
Given an initial sequence 0101010101010101 . . ., even you would
predict that the next digit in the sequence would most likely be 0
based on the previous pattern. If the previous pattern were an
unbroken series of 1 million "01s" you probably be willing to bet a
whole lot of money on the next digit in the series. This is what
algorithmic complexity is all about, or so it seems to me.
You see, if you could find any predictable pattern or formula using
any basis whatsoever, you could make money betting on what the next
digit in the sequence would be. In other words, if you could compress
the digit using any process whatsoever, you'd end up ahead of the game
in your betting on the next digit. Again, this is how KCC relates to
gambling in places like Las Vegas and to pattern detecting in fields
of science like forensics, anthropology, and even SETI.
"Martin-Löf randomness has since been shown to admit many equivalent
characterizations -- in terms of compression, randomness tests, and
gambling -- that bear little outward resemblance to the original
definition, but each of which satisfy our intuitive notion of
properties that random sequences ought to have: random sequences
should be incompressible, they should pass statistical tests for
randomness, and it should be difficult to make money betting on
them. . .
The martingale characterization conveys the intuition that no
effective procedure should be able to make money betting against a
random sequence. A martingale d is a betting strategy. d reads a
finite string w and bets money on the next bit. It bets some fraction
of its money that the next bit will be 0, and then remainder of its
money that the next bit will be 1. d doubles the money it placed on
the bit that actually occurred, and it loses the rest. d(w) is the
amount of money it has after seeing the string w. Since the bet placed
after seeing the string w can be calculated from the values d(w),
d(w0), and d(w1), calculating the amount of money it has is equivalent
to calculating the bet. The martingale characterization says that no
betting strategy implementable by any computer (even in the weak sense
of constructive strategies, which are not necessarily computable) can
make money betting on a random sequence."
http://en.wikipedia.org/wiki/Algorithmically_random_sequence
Sean Pitman
www.DetectingDesign.com
.
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