`On Fri, Aug 15, 2003 at 10:06:42AM +0200, M?ns Rullg?rd wrote:> Andries Brouwer <aebr@win.tue.nl> writes:> > >> > > entropy(x) >= entropy(x xor y)> >> > > entropy(y) >= entropy(x xor y)> >> > > >> > Is this trolling? Are you serious?> >> > >> These lemma are absolutely true.> >> > David, did you read this line:> >> >> > Try to put z = x xor y and apply your insight to the strings x and z.> >> > Let us do it. Let z be an abbreviation for x xor y.> >> > The lemma that you believe in, applied to x and z, says> >> >  entropy(x) >= entropy(x xor z)> >  entropy(z) >= entropy(x xor z)> >> > But x xor z equals y, so you believe for arbitrary strings x and y that> >> >  entropy(x) >= entropy(y)> >  entropy(x xor y) >= entropy(y).> >> > This "lemma", formulated in this generality, is just plain nonsense.> > Not quite non-sense, but it would mean that for any strings x and y, > >   entropy(x) == entropy(y),> > which seems incorrect.No, it's a premise stated at the beginning of the thread. We'reassuming perfect distribution for x and y. The problem here is that xand y can be dependent or independent. If they're independent, thenthere's no issue. If they're dependent (for instance correlated oranticorrelated) then x^y biases toward zero or one. Which clearly hasless entropy.-- Matt Mackall : http://www.selenic.com : of or relating to the moon-To unsubscribe from this list: send the line "unsubscribe linux-kernel" inthe body of a message to majordomo@vger.kernel.orgMore majordomo info at  http://vger.kernel.org/majordomo-info.htmlPlease read the FAQ at  http://www.tux.org/lkml/`