On Wed, Jan 21, 1998 at 10:33:27AM -0800, Hal Finney wrote:
> I'm not sure I follow you here.  Are you suggesting that our observations
> of the universe's history might be in error, that we might be instantiated
> for a single instant with all of our memories of the past being illusions?
> We could write down the state of the universe at this instant as a long
> string using a simple mapping, and at some point the counting TM will
> emit this string.  At that instant we will all exist with our memories
> of the past, but none of that past will actually have happened.  Is this
> right?

Yes, that's what I'm saying.

> The problem then is not so much with the objective question of whether we
> live in the counting universe, which an outside observer could easily
> answer, but rather with the difficulty of us as residents in the universe
> knowing the answer.
> 
> A more extreme case occurs if the counting program works in trinary,
> using the 0/1/comma alphabet.  Now it outputs not only all numbers, it
> outputs all possible comma-separated sequences of numbers.  It eventually
> outputs not only every possible state, but every possible sequence of
> states, including the entire history of our universe.  This history would
> be a subset of an enormously larger output string, all created with a very
> short (and therefore a priori probable) input program.
> 
> All I can suggest about these problems is that they would imply that the
> universe will shortly cease to behave lawfully.  When we don't observe
> this, we can reject this possibility.  Otherwise we are forced to say
> that all our perceptions are illusions, that our memories are false,
> that we ourselves may be simply memories of some future self.  Logically
> we can't rule this out, but it does not seem reasonable.  If we lived in
> the counting universe, there is no reason why the universe should seem
> lawful in the sense we see.

Logically we can't rule this out, but we should be able to rule it out
probabilisticly. My point is that I don't see how to do it under
Schmidhuber's interpretation. 

> You solution is somewhat different in nature from Schmidhuber's.
> Your set of possible universes is more structured than his.  You have
> a coordinate system, you have regions (which may imply a certain amount
> of continuity in the coordinates).  He had binary strings to represent
> the universe state, which would allow for a wider set of possible
> universes.

The coordinate systems and regions are just interpretations to help our
intuition. The procedure that I gave for computing probabilities does not
depend on any structure in the coordinate system. If its not easy to
interpret the input to a TM as coordinates in a universe, then it would be
hard for us to think about that TM as the sort of universe we're familiar
with, but it doesn't matter in the probability computations. 

> I'm not sure you can fully avoid the mapping problem with your approach.
> There is still a mapping involved in the choice of coordinates.
> With a sufficiently exotic coordinate system, I suspect we could map
> our universe's state to the output of a trivial program.

Remember that the prior probability of a region is related to the
program length plus the coordinate length. If you have a universe with an
exotic coordinate system, the lengths of the coordinates would be long and
so the regions in that universe would have small priors.

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