This is true. Various compression algorithms will compress something more
than another. All one can say with compressibility is that given 2 sequences
one sequence has more information than the other because it is less
compressible. The maximum known compressibility is what can be used to
relatively quantify information. In other words, one can not use
compressibility as a measure of the absolute quantity of information only
for a comparison. One cannot prove that he has the shortest possible
compression of a sequence. If you could do that, you could prove that a
given sequence is random. But as it is, we can't. Yockey states:
"The considerations in section 2.4.1 might lead us to believe that, having a
definition and a measure of randomness, one could prove a given sequence to be
random. In fact it is impossible to do so. It can easily be shown that a
given sequence is not random. We need only find a program that compresses the
sequence to one substantially shorter than the sequence itself. We need not
prove that the program is the shortest one, just that is is shorter than the
original sequence. On the other hand to prove that a given sequence is indeed
random one must prove that no shorter program exists. This leads immediately
to the paradoxes of sets that have themselves as members. Epimenides, the
Cretan, alleged, 'All Cretans are liars'. Is that statement true or fals?
Consider the Berry paradox: 'Find the smallest positive integer that to be
specified requires more characters than there are in this sentence.' The
sentence supposedly specifies a positive integer of more characters than there
are in the sentence. We are now in the midst of a subtle and fundamental
anomaly in the foundations of mathematics, discovered by Chaitin that is
related to a famous theorem due to Kurt Godel called Godel's incompleteness
theorem. Godel showed that for the axioms defining number theory there are
always theorems that can neither be proven or disproven from the set of
axioms. This theorem caused considerable Angst among mathematicians
comparable to that among physicists caused by quantum theory and relativity
theory."~Hubert Yockey, Information Theory and Molecular Systems, (Cambridge:
Cambridge University Press, 1992), p. 80-81
>And for any particular information, it's possible to make a (custom-made)
>compression engine that will reduce it to 1 bit, right?
Absolutely not. In compressing a sequence you must be able to uncompress it
or you have lost information. Take this post and let us use your patented
1-bit compression algorithm. This note compresses to "1". But the
Encylopedia Brit. also compresses to a "1". Thus you have the contradictory
situation that this post has as much information as the Encyclopedia
Britannica. I know it has as much wisdom. :-)
glenn
Adam, Apes, and Anthropology: Finding the Soul of Fossil Man
and
Foundation, Fall and Flood
http://www.isource.net/~grmorton/dmd.htm