- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

From: Wei Dai <weidai.domain.name.hidden>

Date: Mon, 19 Jan 1998 21:03:07 -0800

Before we start talking about philosophical implications, maybe we should

reach a consensus about more basic questions.

What is the set of all possible universes? Max Tegmark says its the set of

all mathematical structures, and Juergen Schmidhuber says its the set of

all Turing machines, but neither gives much justification. I tend to agree

with Schmidhuber, if only because Tegmark's definition does not seem to

lead to an effective theory. For example, what does a uniform distribution

on all mathematical structures mean? However it would be nice to have some

stronger justifications for assuming that only computable universes exist.

If we agree that all Turing machines exist, it's still not clear how they

should be interpreted as physical universes. Schmidhuber suggests that the

output of a Turing machine should be interpreted as the evolution of a

universe. However this is problematic because it does not lead to an easy

way to think about the complexities and probabilities of structures within

a universe. For example, consider a simple Turing machine that enumerates

the natural numbers. The output of this TM includes every possible

configuration of every other universe. What is the probability that I'm

living in this TM? I don't see a straightforward way to answer this

question under Schmidhuber's interpretation. The problem is that even

though this universe is very simple, it would take a lot of extra

information to find anything in it, so the simplicity of the universe as a

whole is deceptive.

As an alternative, I suggest that each Turing machine should be thought of

as taking the coodinates of a region of a universe as input and producing

the content of that region as output. Here region should be broadly

interpreted. It not only refers to volumes of space-time, but may also for

example specify a branch of a quantum superposition.

Let me clarify some terminology. When I say length of a region of a

universe, I mean the length of the Turing machine plus the length of input

specifying the extent of that region. When I say complexity of the content

of region R, I mean the least length of all regions with the same content

as R, which is just the content's Kolmogorov complexity.

To compute the probability that I'm in a particular universe U, I would

first find the universal prior probability PU(m) of my mind state. (I

assume my mind state, which includes my memories and current perceptions,

is the only information I have direct access to) Then I would find the

region R in U that has my mind state as the content (for simplicity let's

assume there is exactly one such region in U). Finally the probability

that I'm in universe U is 2^-l(R)/P, where l(R) is the length of region R.

The Baysian interpretation of the above procedure is that before taking

into account my mind state, the prior probability that I'm any region R is

2^-l(R). After taking into account my mind state, I eliminate all regions

that do not have my mind state as the content, so the posterior

probability is 2^-l(R)/PU(m) if the content of R is m, and 0 if the

content of R is not m.

Received on Mon Jan 19 1998 - 21:03:07 PST

Date: Mon, 19 Jan 1998 21:03:07 -0800

Before we start talking about philosophical implications, maybe we should

reach a consensus about more basic questions.

What is the set of all possible universes? Max Tegmark says its the set of

all mathematical structures, and Juergen Schmidhuber says its the set of

all Turing machines, but neither gives much justification. I tend to agree

with Schmidhuber, if only because Tegmark's definition does not seem to

lead to an effective theory. For example, what does a uniform distribution

on all mathematical structures mean? However it would be nice to have some

stronger justifications for assuming that only computable universes exist.

If we agree that all Turing machines exist, it's still not clear how they

should be interpreted as physical universes. Schmidhuber suggests that the

output of a Turing machine should be interpreted as the evolution of a

universe. However this is problematic because it does not lead to an easy

way to think about the complexities and probabilities of structures within

a universe. For example, consider a simple Turing machine that enumerates

the natural numbers. The output of this TM includes every possible

configuration of every other universe. What is the probability that I'm

living in this TM? I don't see a straightforward way to answer this

question under Schmidhuber's interpretation. The problem is that even

though this universe is very simple, it would take a lot of extra

information to find anything in it, so the simplicity of the universe as a

whole is deceptive.

As an alternative, I suggest that each Turing machine should be thought of

as taking the coodinates of a region of a universe as input and producing

the content of that region as output. Here region should be broadly

interpreted. It not only refers to volumes of space-time, but may also for

example specify a branch of a quantum superposition.

Let me clarify some terminology. When I say length of a region of a

universe, I mean the length of the Turing machine plus the length of input

specifying the extent of that region. When I say complexity of the content

of region R, I mean the least length of all regions with the same content

as R, which is just the content's Kolmogorov complexity.

To compute the probability that I'm in a particular universe U, I would

first find the universal prior probability PU(m) of my mind state. (I

assume my mind state, which includes my memories and current perceptions,

is the only information I have direct access to) Then I would find the

region R in U that has my mind state as the content (for simplicity let's

assume there is exactly one such region in U). Finally the probability

that I'm in universe U is 2^-l(R)/P, where l(R) is the length of region R.

The Baysian interpretation of the above procedure is that before taking

into account my mind state, the prior probability that I'm any region R is

2^-l(R). After taking into account my mind state, I eliminate all regions

that do not have my mind state as the content, so the posterior

probability is 2^-l(R)/PU(m) if the content of R is m, and 0 if the

content of R is not m.

Received on Mon Jan 19 1998 - 21:03:07 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:06 PST
*