Compression theory reference?

Victor Duchovni Victor.Duchovni at MorganStanley.com
Tue Aug 31 18:31:49 EDT 2004


On Tue, Aug 31, 2004 at 04:56:25PM -0400, John Denker wrote:

>  4) Don't forget the _recursion_ argument.  Take their favorite
> algorithm (call it XX).  If their claims are correct, XX should
> be able to compress _anything_.   That is, the output of XX
> should _always_ be at least one bit shorter than the input.
> Then the compound operation XX(XX(...)) should produce something
> two bits shorter than the original input.  If you start with a
> N-bit message and apply the XX function N-1 times, you should be
> able to compress each and every message down to a single bit.
> 

This proof as written (the theorem is still true of course) relies on
the algorithm always compressing, rather than never expanding. It is
much simpler as a result, is there an equally simple argument to prove
that all non-expanding codes never compress?

Note that it is possible to turn any compressor into one whose expansion
is at most one 1 extra bit:

If F(x) is shorter than x by at least one bit output 0|F(x) if F(x)
is the same length as x or longer output 1|x. So we can lose 1 bit of
efficiency in compressed strings to gain at most 1 bit of overhead in
uncompressed strings.

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