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From: <juergen.domain.name.hidden>

Date: Thu, 22 Mar 2001 09:40:27 +0100

Where does all the randomness come from?

Many physicists would be content with a statistical theory of everything

(TOE) based on simple probabilistic physical laws allowing for stochastic

predictions such as "We do not know where this particular electron will

be in the next nanosecond, but with probability 0.04 we will find it in a

certain volume V".

Any source of randomness, however, leaves us with an unsatisfactory

TOE, since randomness does not have a compact or simple explanation, by

definition. Where does the enormous information conveyed by a particular

history of random events come from? A TOE that cannot explain this

is incomplete.

The in hindsight obvious solution is an "ensemble TOE" which covers all

possible universe histories. The ensemble conveys less information than

most particular histories - one main motivation of this mailing list.

Which are the possible histories? Let us focus on well-defined ensembles

only, and ignore those that cannot be sufficiently specified to permit

reconstruction through a formally describable computer. In particular,

we may ignore uncountable ensembles such as continua, or other ensembles

including histories without finite descriptions.

Is there an optimally efficient way of computing all the "randomness" in

all the describable (possibly infinite) universe histories? Yes, there

is. There exists a machine-independent ensemble-generating algorithm

called FAST that computes any history essentially as quickly as this

history's fastest algorithm. Somewhat surprisingly, FAST is not slowed

down much by the simultaneous computation of all the other histories.

It turns out, however, that for any given history there is a speed

limit which greatly depends on the history's degree of randomness.

Highly random histories are extremely hard to compute, even by the optimal

algorithm FAST. Each new bit of a truly random history requires at least

twice the time required for computing the entire previous history.

As history size grows, the speed of highly random histories (and most

histories are random indeed) vanishes very quickly, no matter which

computer we use (side note: infinite random histories would even require

uncountable time, which does not make any sense). On the other hand,

FAST keeps generating numerous nonrandom histories very quickly; the

fastest ones come out at a rate of a constant number of bits per fixed

time interval.

Now consider an observer evolving in some universe history. He does not

know in which, but as history size increases it becomes less and less

likely that he is located in one of the slowly computable, highly random

universes: after sufficient time most long histories involving him will

be fast ones.

Some consequences are discussed in

http://www.idsia.ch/~juergen/toesv2/node39.html

Juergen Schmidhuber

Received on Thu Mar 22 2001 - 00:48:32 PST

Date: Thu, 22 Mar 2001 09:40:27 +0100

Where does all the randomness come from?

Many physicists would be content with a statistical theory of everything

(TOE) based on simple probabilistic physical laws allowing for stochastic

predictions such as "We do not know where this particular electron will

be in the next nanosecond, but with probability 0.04 we will find it in a

certain volume V".

Any source of randomness, however, leaves us with an unsatisfactory

TOE, since randomness does not have a compact or simple explanation, by

definition. Where does the enormous information conveyed by a particular

history of random events come from? A TOE that cannot explain this

is incomplete.

The in hindsight obvious solution is an "ensemble TOE" which covers all

possible universe histories. The ensemble conveys less information than

most particular histories - one main motivation of this mailing list.

Which are the possible histories? Let us focus on well-defined ensembles

only, and ignore those that cannot be sufficiently specified to permit

reconstruction through a formally describable computer. In particular,

we may ignore uncountable ensembles such as continua, or other ensembles

including histories without finite descriptions.

Is there an optimally efficient way of computing all the "randomness" in

all the describable (possibly infinite) universe histories? Yes, there

is. There exists a machine-independent ensemble-generating algorithm

called FAST that computes any history essentially as quickly as this

history's fastest algorithm. Somewhat surprisingly, FAST is not slowed

down much by the simultaneous computation of all the other histories.

It turns out, however, that for any given history there is a speed

limit which greatly depends on the history's degree of randomness.

Highly random histories are extremely hard to compute, even by the optimal

algorithm FAST. Each new bit of a truly random history requires at least

twice the time required for computing the entire previous history.

As history size grows, the speed of highly random histories (and most

histories are random indeed) vanishes very quickly, no matter which

computer we use (side note: infinite random histories would even require

uncountable time, which does not make any sense). On the other hand,

FAST keeps generating numerous nonrandom histories very quickly; the

fastest ones come out at a rate of a constant number of bits per fixed

time interval.

Now consider an observer evolving in some universe history. He does not

know in which, but as history size increases it becomes less and less

likely that he is located in one of the slowly computable, highly random

universes: after sufficient time most long histories involving him will

be fast ones.

Some consequences are discussed in

http://www.idsia.ch/~juergen/toesv2/node39.html

Juergen Schmidhuber

Received on Thu Mar 22 2001 - 00:48:32 PST

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