random

[Alex Martelli]

"Darren New" <dnew at san.rr.com> wrote in message
news:3B183CFF.A90213BA at san.rr.com...
> > The quote above contains the premise "even given _complete_
> > information".
>
> Except you snipped the end of the sentence, which is
> "complete information about how they're being generated."
>
> I can have complete information about exactly how many atoms of uranium
> I have in my sample, and still not be able to predict when the next atom
> will pop. I have complete information about how the numbers are being
> generated (watching a radioactive sample) without any information about
> what the next number will be.

_Complete_ information about how the numbers are generated
requires complete information about the state of their generator.
This is a tautology, of course, but you seem to be denying it.

"Exactly how many atoms" is NOT complete information about
the physical state of a system -- isn't that obvious, and, if so,
why are you making statements that appear to imply the reverse?

Position and velocity vector for each of these atoms are part of
system's state.  So, if Heisenber Principle holds, how are you
going to have complete information about this state?


> > > >A physical system that is macroscopic enough to escape
> > > >Heisenbergian conumdrums
> > >
> > > There is no such system.
> >
> > In this case, no physical computer is predictable, and all
> > we study about algorithms are abstractions -- since a
> > computer system IS a physical system, then ALL bits
> > coming out from it must be random in your sense of
> > the word.
>
> Nope. Because you have an infinite number of possible states, and you
> collapse some of those into a 0 bit and some into a 1 bit. It's because
> you're *classifying* states in your brain as 1 or 0, ignoring the
> reality that there's an infinite number of voltages that represent 0.

So the system becomes "macroscopic enough" - yet the
discussant asserted there was no such system.  Will you
guys please agree to ONE party line, or is Heisenberg
uncertainty affecting you too badly?

If my physical system is designed and considered in ways
that are macroscopic enough to escape uncertainty effects,
then (given some possibly huge amount of bits of
information, which it may not be easy to obtain, of
course) its trajectory in state-space is predictable.

Is your claim, then, that "randomness" MUST be used ONLY
for systems which, at the level of observation under discussion,
ARE inevitably Heisenbergian?  A roulette wheel, dice being
rolled, cards being well-shuffled, balls being drawn from an
urn -- we can enumerate all the classic devices used to obtain
randomness "from physical systems" without necessarily
encountering quantum effects in action.  Chaotic effects may
well ensure that the number of bits of information that would
be needed to obtain prediction is higher than any practical
measure.  This, apart from the mention of bits, is the classic
way probability theorists frame 'randomness'.  I see nothing
wrong with it, nor any need to demand that the amount of
information needed for prediction be "actually infinite" (if that
sentence makes any sense:-) before the word "randomness"
can be used.  Use of that word has never demanded such
"infiniteness", so why should it suddenly impose such demands
today?  I consider it perfectly correct and proper to treat (e.g.)
the results of good card-shuffling as random.

So what is so special, that distinguishes a computer system
from a deck of cards -- considering both systems at a
macroscopic enough level that quantum effects can be
ignored, which is easier for the deck of cards I think but
is commonly done for well-designed computers too:-) --
so that I *could* get randomness from one and not the
other?  It seems more fruitful to me to consider the amount
of information, i.e., a quantitative assessment of randomness,
rather than claim that only if that amount is 'infinite' can
the word 'randomness' be properly used.  The information
I _do not_ have, whether I don't because I _can't in
principle have it_ or, as is more likely at a macroscopic
level, because it's _impractical enough to determine it,
transmit it, store it, process it_.

Card-shuffling can be treated this way, in finite terms, and
interesting results are obtained on how best to shuffle a
deck of cards -- what happens to the deck if you shuffle
algorithmically according to certain rules, and what one
player (seeing 1 card out of every 4 from the shuffled
deck) can determine about the other unseen hands.  As
the _purpose_ of shuffling the cards at all can be framed
as minimizing the amount of information thus leaked, this
is theoretically important in studying card-play -- theoretically
only, these days, because in every important championship
the 'shuffling' is in fact obtained by computer instead, but
there is a separate theory about such shuffling and some
interest in trying to get roughly as much randomness as a
deck being shuffled would contain (which is less than a
perfect shuffling affords: players new to computer shuffling
used to complain all the time about the 'freak distributions'
that kept coming up:-).

This sort of application requires finite-mathematics theories,
and I don't see why they shouldn't be, as they are, called
theories of randomness.  The concept of "HOW MUCH"
randomness is practically and psychologically important,
and becomes useless if the only possible answers are
'infinite' or 'none'.


> > If, on the other hand, you maintain that the output
> > from a physical system CAN be predictable if that
> > system is a computer running your program, then
> > what makes you SO certain that NO other physical
> > systems can also be predictable in the same sense?-)
>
> Again, you're talking two diferent things. In the first, you're talking
> about "the output will be somewhere between 4.5v and 5.5v on this
> wire."  The other you're talking about an exact number.

Taking a set of states and deeming them equivalent can
perfectly well be done -- who ever gets an "exact" number
from physical measurement anyway?!  It applies just as
well to the finite definition of randomness.  For example,
when cards are dealt, we only deem significant the cards
that are received and not the sequence in which they
are received -- we collapse several possible states into
one significant state.  So, 13! states are collapsed into
one (assuming the shuffling is such that all sequences are
possible -- a big assumption in fact, but I don't think
shuffling theory is what we're discussing here).  Fine, no
problem, and an important consideration when measuring
_quantity_ of randomness of course.  But of course we do
have exact (integer) numbers in finite mathematics -- why
shouldn't we?


> > quote talks generically about "random", without using
> > the word "perfect" you now insert, and without any
> > specification of infiniteness as a prerequisite, and, as
> > such and without the needed qualifications, it is not
> > correct.  _with_ the qualifications you want to place,
> > it seems to become tautological ("I define X as being
> > something that requires infinities: no finite system can
> > reach X, it can only approach it") and therefore sterile
> > and uninteresting, unless there are hidden layers of
> > exoteric meaning nesting in it, I guess.
>
> I think "random" here means "unpredictable". Tossing dice will lead to
> random/unpredictable results, even if you know everything there is to
> know about the dice and the table. Generating numbers between 1 and 6
> will not be unpredictable if you know the seed and algorithm.

If you have COMPLETE information about the dice &c (taking
a HUGE number of bits, no doubt) you can predict their
trajectories, bounces (if allowed), rolls, AND final results.
Why should these macroscopic-physics effect be inherently
unpredictable as you claim?  VERY HARD to predict, yes
(LOTS AND LOTS of bits), but the dice are VERY massive --
it seems to me a fine measurement system could give me
(in principle) the information I need for prediction, CONSIDERING
inter alia how many final states I collapse into one significant
one ('infinite' number of minutely different dice positions at
rest are all considered "a 3 throw" in the end -- just like the
computer-system example you posed).

Among classical randomness-generating physical systems,
the ones of most interest to me are deck of cards, because
card-playing (bridge specifically) is my passion, and in that
case the finite-mathematics effects are even more in
evidence, of course -- there are only 52! distinct states
even before collapsing, after all.  Yet randomness is there...

If you have complete information about the algorithm and
the relevant part of the state of the computer where it
runs, again you have predictability.

The only important issue is -- HOW MUCH information do
you need for prediction.  If too little info suffices, then the
system is not random ENOUGH for a given application --
fine, get a better system, one requiring more information
for prediction.  It's important to quantify this, as I see
things.  Trying to forbid the use of the word 'randomness'
whenever information amount is finite (deck of cards,
rolled dice, roulette wheels, algorithms...) seems totally
unproductive and fruitless.


Alex