Re: Pi as the Mother Number
- From: <dougwedel@xxxxxxxxxxxxx>
- Date: Sun, 12 Feb 2006 15:50:56 GMT
"Jim Black" <tramspap@xxxxxxxxx> wrote
Let's say you have a string of up to 100 digits, chosen at random.
There are 10^100+10^99+10^98+... = 1.111....*10^100 possible strings.
Now suppose you run them through some sort of compression algorithm.
The compression algorithm outputs a string of digits which you could
uncompress if you wanted the original string of digits back. There
must therefore be at least 1.111...*10^100 possible output strings, or
the compression would not be reversible. Since we want to make the
output strings as short as possible, we wouldn't use a 101-digit string
as a possible output if there was an string of 100 digits or fewer that
wasn't being used to represent something else. Thus, we will end up
using all 1.111...*10^100 possible 100-digit-or-fewer strings as
outputs. So the average length of a string after compression will be
the same as the average length of a string before compression.
The reason that compression works, despite this argument, is that not
all inputs are equally probable. For example, for a program designed
to compress text, the input "Have a nice day" is much more probable
than the input "dkjrc4jg5ghd3bv" is. But if the input is random, i.e.,
all inputs are equally likely, then compression is impossible.
I totally agree with your analysis. I never imagined that it would
be possible to compress _most_ random numbers by finding them in pi
and then if necessary compressing the index into pi -- my only point is
that it would be possible to compress a SUBSET of what would otherwise
appear to be random incompressible numbers by these means. I think this is
quite consistent with your formulation. You talk about the "average
length" of a string after compression. I am not so much interested in
the average as I am interested in finding out the size of the total universe
of
finite random numbers that could be compressed by this method (even if this
is only a tiny subset of all random numbers).
.
- References:
- Pi as the Mother Number
- From: dougwedel
- Re: Pi as the Mother Number
- From: Jim Black
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