random

David C. Ullrich ullrich at math.okstate.edu
Sat Jun 2 16:22:54 CEST 2001


On Sat, 2 Jun 2001 09:39:23 +0200, "Alex Martelli" <aleaxit at yahoo.com>
wrote:

>"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?

I don't know why you assume that when two people are
commenting on things you said they must be in complete
agreement with each other.

As far as what _I've_ said is concerned, the reason you
seem to think I'm contradicting myself is that you seem
to think I regard an _algorithm_ as a physical system.

>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.

There _is_ no such amount of information. It's _very_
unlikely, but regardless of _how_ much you know about
your physical system it's always _possible_ that there
a bunch of virtual particles is going to just happen
to appear, all in just exactly the right places,
so as to totally randomize the output of your machine.

I mean things like transistors are not guaranteed to
work. They work most of the time.

>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? 

Who _has_ made such a demand?

> I consider it perfectly correct and proper to treat (e.g.)
>the results of good card-shuffling as random.

Of course it is! That's because the word "random" means
so many different things, in different contexts.

>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,

But in fact the deck of cards is not macroscopic enough
for this. It _could_ happen that at a given instant one
of the cards just disappears and reappears on the surface
of the Moon. Not likely, I'm not holding my breath. But
particles do not have positions, they have _probable_
positions.

> 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'.

Nobody said that the word "random" must only be used
in this sense. The fact that someone asks a question in
which he thought he made clear that he was using the
word in some absolute sense does not imply that he's
saying that that's the only way the word shall be used.

>> > 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.

This is simply not true. No, I'm not claiming to have
perfect knowledge here, as you implied elsewhere - I've
inserted the appropriate qualifiers several times:

Unless physics is simply _all_ _wrong_ it is simply
not true that there is such a thing as enough knowledge
of the initial state of those dice to be able to
predict which number comes up with certainty.

(That's assuming we're talking about actual physical
dice, not the abstract dice in some discussion of
proability.)

>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.

I wish you'd clarify who's been trying to forbid this,
and where he did so. Again, I inserted the appropriate 
qualifiers many times: I was asking about "perfect"
randomness - when I ask a question about that it
does not follow that I think that that's the only
sense in which the word may be used.

>Alex
>
>
>



David C. Ullrich
*********************
"Sometimes you can have access violations all the 
time and the program still works." (Michael Caracena, 
comp.lang.pascal.delphi.misc 5/1/01)



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