random

[Alex Martelli]

"Darren New" <dnew at san.rr.com> wrote in message
news:3B1A82DD.C4D78472 at san.rr.com...
> Alex Martelli wrote:
> > So what does the "GIVEN _complete_ information" above
> > is, if not a statement that you CAN be given complete
> > information about
>
> I interpreted it as meaning "given that you know everything about the
> system that it's possible to know." If you have complete information,
> then you know the exact probabilities that any particular item is in any
> particular state. Just like if you have "perfect dice", it means that
> all the probabilities are exactly 1/6. This doesn't tell you which
> number a perfect die will roll next.

But why should "perfect information" about the dice tell me only
that they're perfect, and not their state?  That's like saying that
"perfect information" about an algorithm tells me only it's a good
one, and gives me no information about the state it starts in.  What
can I predict then?


> > I still haven't heard your definition of
> > 'perfect' randomness, for example.
>
> Generally, this is something along the lines of:
> Given all available information about the mechanism used to generate the

"available" seems to play a rather crucial role here, then.

> bits, and an arbitrarily long sequence of generated bits, it's
> impossible to predict with >50% accuracy what the next bit in the
> sequence will be.
>
> I'm not sure it always includes the first condition.

That would only leave the arbitrarily long sequence of generated
bits and no other info?  Fine, then:

> I don't think Chaitin's Omega meets this definition, since it is by
> definition a single number than anyone can calculate equally well.

But from Omega's first N bits, you get no hint at all that lets
you predict its N+1-th bit "with > 50% accuracy".  So, if the
definition does not include the first condition, from the bits
only you're not going to get any farther than from N tosses
of an idealized coin.


Alex