If I understand the Measure Problem correctly, we wonder why we find
ourselves in a "Goldilocks Universe" of stars and galaxies rather than
a simpler universe consisting solely of blackbody radiation, or a more
complex, unpredictable Harry Potter universe.
1. An attempt at the solution was that more complex universes are less
probable; they are less likely to be produced by a random UTM. This
explains why induction works, why we don't live in a Harry Potter
universe. But this also means a simple blackbody radiation universe is
more probable than a Goldilocks Universe.
2. So we say, "There are more observers in a Goldilocks Universe,
where observers evolve through natural selection, than in a blackbody
radiation universe, where observers can only occasionally emerge
through extremely infrequent statistical anomalies." But if both the
Goldilocks Universe and the blackbody radiation universe are infinite
in size, then both have an infinite number of observers.
3. Maybe we say, "The Goldilocks Universe produces more observers per
cubic meter." But then someone can propose the Tiny Blackbody
Universe, which looks like an infinite region of blackbody radiation
in our universe, but where everything is smaller by a factor of 10 to
power of 1000. Why don't we live in the Tiny Universe? And at the very
least, why does the universe waste all this empty space around us?
Here is one possible solution: the UTM instead directly produces a
qualia (or, if you prefer, substitute "observer moment" or whatever
terminology you deem appropriate). We'll use a broad definition of
"qualia" that can encompass complex observations like "Rolf sits at
his keyboard, reflecting on past observations and wondering why he
seems to live in a Goldilocks Universe", since that's exactly the type
of observation that we're trying to explain when we ponder the Measure
Problem.
Each qualia, in the proposed model, is a long, finite-length string
that is output by a UTM running every possible random program. (This
is the same type of UTM that some of you have been proposing, but it
outputs an attempt at a single qualia, rather than outputting an
entire universe.) Very few strings are qualia; most UTM programs fail
to produce qualia. The proposed model additionally postulates that
many qualia are compressible in a certain interesting way, such that
the World-Index-Compression Postulate (below) is true.
World-Index-Compression Postulate: The most probable way for the
output of a random UTM program to be a single qualia, is through
having a part of the program calculate a Universe, U, that is similar
to the universe we currently are observing; and then having another
part of the program search through the universe and pick out a
substring by using an search algorithm SA(U) that tries to find a
random sentient being in U and emit his qualia as the final output.
As an example, take two qualia, that we will call Q(Goldilocks) and
Q(Potter):
Q(Goldilocks): "All my life I have read that all swans are white. And
indeed, today I just saw a white swan."
Q(Potter): "All my life I have read that all swans are white. But,
today I just saw a black swan."
The measure (or "probability", if you prefer) of Q(Goldilocks) is
greater than the measure of Q(Potter). Why? Perhaps Q(Goldilocks) and
Q(Potter) are both 5000 bits long; it's almost impossible that any
random UTM program would output either qualia directly by using a 5000
bit program. But postulate that there are shorter programs to emit
these types of qualia:
Smallest Goldilocks Program's code: (outputs Q(Goldilocks))
1. Execute subroutine to internally generate the Goldilocks Universe.
(This subroutine is 1000 bits long)
2. Search through the infinite universe you generated, until you find
something that has a head and seems to react to swans. Look inside its
head, and output the contents of whatever is inside its head. (This
subroutine is 500 bits long)
Total program size: 1500 bits.
Smallest Potter Program's code: (outputs Q(Potter))
1. Execute subroutine to internally generate the Potter Universe.
(This subroutine is 1020 bits long, since Potter is more complex than
Goldilocks)
2. Search through the infinite universe you generated, until you find
something that has a head and seems to react to swans. Look inside
its head, and output the contents of whatever is inside its head.
(This subroutine is 500 bits long)
Total program size: 1520 bits. The Potter qualia is 20 bits longer,
and therefore is literally a million times less likely than the
Goldilocks qualia in this scenario.
What about the blackbody radiation universe? Because the universe is
totally random, you can't efficiently use the same trick as in the
more orderly universes. Your Search Algorithm will eventually find a
thermal fluctuation that looks like "something reacting to a swan" and
will output the contents of its head; however, the head is just going
to contain an uncorrelated sea of thermal radiation, rather than a
qualia.
Q(Goldilocks) is a more likely output than Q(Potter), and is also more
likely than a similar qualia output from a blackbody radiation
universe. Therefore, the World-Index Compression Postulate, if true,
provides one possible solution to the Measure Problem.
-Rolf
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Received on Sat Sep 15 2007 - 15:00:19 PDT