NewsGroup: comp.ai.philosophy, comp.society.futures, rec.arts.sf.science,
rec.arts.sf.written
Author: Simon Laub (silanian@mail.tele.dk)
Subject: Time, Omega and SETI.
In a previous post I claimed that clever aliens
could send us the Omega number - and we would
have a key to infinite-like knowledge. Perhaps
not so surprisingly a couple of guys wanted me
to elaborate on that one:
Omega = halting probability.
According to Alan Turing you can't decide whether
a (computer) program will halt. There is no mechanical
procedure for doing that. As most problems can be recast
as halting problems, this is pretty universal.
The real number Omega is then the probability
that a program generated by tossing a coin halts.
quoting Calude:
"Omega embodies an enormous amount of wisdom in
a very small space... its first thousands digits,
which could be written on a small piece of paper,
contain the answer to more mathematical questions,
than could be written down in the entire universe.
Thoughout history mystics and philosophers have
thought out a compact key to universal
wisdom, a finite formula or text which, when known
and understood, would provide the answer to every question.
The use of the Bible, The Koran and the I Ching
exemplify this belief or hope...
Omega is in many senses a cabalistic number.
It can be known of, but not known through human reason.
To know it it detail, one would have to accept its
uncomputable digit sequence on faith, like
words of a sacred text."
However, to get Omega in the limit from below, you just look
at more and more programs for more and time, and everytime
you see that a K-bit program halts, that contributes
1/2^K to the halting probability. But this tactics
would never tell you how close you are to the real Omega.
As Omega is irreducible information - which tastes like
Chaitins own description of random:
"Something is random if it can't be compressed into a
shorter description. In other words there is no (shorter)
concise theory that produces it."
So, the n first digits of Omega represent the probability
that a program thats less than n bits long
(when translated into binary from whatever language)
will halt.
Still, Cristian Calude found a structure to all halting programs
longer than 84 bits. Apparently they all
have to start with a particular sequence of bits.
This limits the contribution that the remaining
infinite set of halting programs can make to the first bits
of Omega. This set cannot influence the first
69 bits. Thus, Calude has calculated the first 64
bits (for technical reasons) of Omega:
0000001000000100000110001000011010001111110010 ...
And given the first n bits of Omega one can decide
whether a Turing Machine of length n
halts or not.
KRamsay (kramsay@aol.com) wrote in:
Re: Uncomputable Probabilities (was: Turing Machines)
NewsGroup:sci.math Date:1998/06/04
>And then, one can solve a halting problem by the following tedious
>method. One can enumerate all halting Turing machines in order by
>taking them in order of size+execution time (say). Omega is the sum
>over these halting instances of the probability of generating that
>instance at random. (Chaitin carefully defined what sort of random
>generation he had in mind....) This is a converging increasing
>sequence. For each given Turing machine X, eventually either X will
>show up as halting, or you will find that what is left of Omega after
>subtracting the already discovered halting probabilities is smaller
>than the probability of generating X, which implies X does not halt.
>It is very tedious since it involves waiting for all the shorter
>Turing machines which do halt to halt.
Still, now you can all of a sudden solve halting problems.
and surely friendly aliens would like to send us
Omega! Hope the SETI guys are looking for it!
-Simon
----------------
links on Omega:
The Omega man
http://www.dc.uba.ar/people/profesores/becher/ns.html
Computing a glimpse of Randomness:
http://www.arxiv.org/PS_cache/nlin/pdf/0112/0112022.pdf
Quantum measurements:
http://www.arxiv.org/PS_cache/quant-ph/pdf/0112/0112087.pdf
Incompleteness and beyond:
http://www.arxiv.org/PS_cache/quant-ph/pdf/0111/0111118.pdf
Omega and quantum physics:
http://www.arxiv.org/PS_cache/quant-ph/pdf/0111/0111062.pdf
more Omega:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/cmu.html
Randomness everywhere:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/nature.html
Cristian Caludes homepage:
http://www.cs.auckland.ac.nz/~cristian/
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My original post on Time according to Stephen Baxter,
where I dangerously hinted at a Omega solution:
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It seems to me that science fiction
guru Stephen Baxter is obsessed with
time, trying to get a handle on it. Figuring out
what it really is. But a heroic attempt it is amidst
frail human complacency.
Stephen Baxters "Time" has one of the best time-
jumping sequences in recent SF, during which the
protagonist witness the entire future of the universe.
Even better though is the hints
concerning how "our unimaginably distant descendants".
The far downstreamers (= our god-like descendants),
who are reaching back through time, shaping our current world
in order to create more mind and more intelligence.
For I take it that Baxter thinks thats the purpose of Time and Space,
setting the stage for intelligence. The more of it the better.
In the Manifold series Baxter
describes the future of a number of universes (the multiverse)
- in fact all our universe.
An infinite number of universes
connected by a common origin.
In manifold 1, Time, genius kids, receives
knowledge from "our unimaginably distant descendants",
the far downstreamers (through a Feynman "radio"
where you listen to the future and,
by detecting coded quantum waves traveling back through time,
divine the fate and voice of human "downstreamers"?)
I.e. Eventually the kids transform the structure of space,
via the vacuum collapse, where they destroy the current reality but
produce a universe where more intelligent species exist.
Creating a storm of mind.
Manifold 3, Origin, is a little different.
The idea is the same though?
Again it seems to me that
Baxter imagines that humans survives indefinitely
and that our unimaginably distant descendents,
distressed with the finitude of time,
eventually reaches back and triggers
changes which results in a universe
with an infinite number of parallel branches,
yielding an infinite amount of subjective time?
Pretty close to saying G*d?
The idea is the same though.
More space and more time equals more mind.
If the purpose of time and space is more
mind we are led to the Fermi's Paradox,
which asks the simple question, if there are other creatures
in the universe like us, where are they?
In Manifold 1, Time, the answer was that we are alone
- with these genetically engineered intelligent squids.
The genius kids "enlarges" our universe though, with
the possibility of more mind as a result.
In Manifold 2, Space, it turned out that the other
technological civilizations just hadn't gottan here yet,
Finally, Manifold 3, Origin, states that our universe
was made without aliens, by ourselfes/downstreamers,
for protection.
But always with the far downstreamers sitting there
in the future making sure that we arive at the optimum
condition for mind.
Thus, the goal seems clear, something like
the Omega Point of Chardin. The trip less clear -
but something like a handshake between them,
our children, the far downstreamers and us across time
with a mind explosion as the hoped for result.
I feel that Stephen Baxters new book "Deep Future"
(published 2002 in Great Britain) works as
a kind of background information to the Manifold
series.
Starting with the close future. Stephen
Baxters starts out easy with some neat gadgets.
E.g. the "recording angel" -
that sits on your shoulder and transcribes your meetings,
the evolution of your thoughts etc. - It will of course
be a brilliant thing for the authorities - Later
individuals are joined together in an "internet of the mind",
to make an extension of our present selfes,-
To me, that seems rather inevitable.
So, actually I don't think we are presented with that
many surprises by Stephen Baxter in the immediate future.
The deep future is that more exciting. Great nations
(companies?) should afford something more than welfare
programmes (if they want to survive). And surely if the dinosaurs
had had a spaceprograme they might have been around today.
So space bound we must be....
It might even be, as David Brin has suggested, that
humanity is currently being farmed by alien "space-friends"
for an interstellar future. Though, I imagine,
Stephen Baxter would say that our own children,
the far downstreamers, are doing the farming.
But off we go to the very deep future and the heath
death of the universe. Where our very distant descendants
will try to survive there in a "universe in ruins",
when the stars are long gone.
Here Stephen Baxters presentation also seem rather inevitable.
Rather disappointing though, Stephen Baxter doesn't
attempt to describe the far downstreamers in any detail.
Personally I think there are a number of possibilities
for creating these godlike children.
- Where, In Western religions God is
described as omniscient and omnipotent. -
Moore's law of computer power tells us
that computer power is doubling every
18 months. According to Ray Kurzweil, in the book
"the age of spiritual machines", computational power
will appear nearly infinite (i.e. omniscient)
as early as 2050.
When, if, this happens it will make the
previous 100.000 years of evolution seem
like something going at a snails pace.
Somehow approaching science to religion ?
At least these beings would appear
to us as omniscient and omnipotent.
A neutron star is the superdense remnant of
a supernova explosion. The original star collapses
to a state where gravity overcomes repulsion
between electrons and protons, forming
a sea of neutrons.
According to Hans Moravec life there could exist as a pattern
of bounded neutrons. With a breakneck
speed of metabolism. Where organisms live and die
within 10e-15 seconds. Entire civilisations
might be formed within a fraction of a second.
Advanced civilisations might create such neutron
stars in order to use them as computers. Some
10e30 more powerful than the human brain.
My own addendum to this is that it would be neat to
press things even further and install
computers in the fabric of spacetime itself.
Smash the neutron star a bit further and create a black hole,
which explodes into a new universe (big bang)
with the new order (computer) installed in its
very fabric of space and time.
Dude Blion (wierdloc@yahoo.com) thought I should
have quoted Moravec differently - and why not:
> 'The very moment we are now experiencing
> may actually be (almost certainly is) such a distributed
> mental event, and most likely is a complete fabrication
> that never happened physically."
FROM:
> 'If these minds spend only an infinitesimal fraction of
> their energy contemplating the human past, their sheer
> power should ensure that eventually our entire history
> is replayed many times in many places, and in many
> variations. The very moment we are now experiencing
> may actually be (almost certainly is) such a distributed
> mental event, and most likely is a complete fabrication
> that never happened physically. Alas, there is no way
> to sort it out from our perspective: we can only wallow
> in the scenery.'
> Hans Moravec
> Date: Sun, 4 Oct 1992 08:40:15 GMT
> Message-ID: <2siae1$6q0@nbc.ksu.ksu.edu>
> Message-ID:
But where are we now ? :-)
According to Stephen Baxters "Time" you would be
listening to the future, by detecting coded quantum waves
traveling back through time in your Feynman radio.
And our distant descendants (?) would inform you that
a vacuum collapse is what is needed
(at least that is, according to Stephen Baxter)?
This seemed a bit "easy" to me though - wouldn't
you rather get something like the Omega digits?
Where the first n digits of Omega represent
the probability that a program that's less than
n bits long will stop (when translated
into binary from whatever programming language
being used). As most problems can be recast as halting
problems, Omegas first few thousand digits contains
the answer to more (mathematical) questions than can be written
down in the entire universe. Problems in math, physics
and elsewhere would be settled by knowing Omega.
But of course knowing Omega might still prompt you
to make a vacuum collapse :-)
Still, maybe we could actually be a bit practical
about it here - why not ask the SETI guys to listen
for Omega now? Maybe our friends out there have started the
Omega broadcast?
The only problem is of course that the
random Omega digits would look a whole lot like random noise!
The prize is huge though - Omega will tell you all you want to know
about time!
Simon Laub