From: Matt Mahoney (matmahoney@yahoo.com)
Date: Fri Jan 25 2008 - 08:20:58 MST
--- Vladimir Nesov <robotact@gmail.com> wrote:
> On Jan 24, 2008 11:56 PM, Matt Mahoney <matmahoney@yahoo.com> wrote:
> >
> > Our model of the world is one which assigns nonzero probability to all
> > possible observations. One could conceive of a universe where this did
> not
> > hold; that we could model certain events to occur with probability 0 or 1,
> in
> > the same sense that some mathematical statements are absolutely true or
> false.
> > If we could (correctly) model all observations with certainty, it would
> be
> > proof that the universe is not simulated by a Turing machine. In such a
> > universe, learning by induction would be suboptimal.
>
> But how would you know that this is the case? Even if you find a law
> that seemingly works all the time, you would still have finite
> confidence in it keeping on, although this confidence will only grow
> with time, to astronomically huge values.
>
> Even mathematics works only as long as computers or brains that carry
> it out don't fail. If they do, math also stops working. This one is a
> highly counterintuitive point, Eliezer tried to communicate it couple
> of times at Overcoming Bias, and most of commenters don't get it (I
> think it needs better elaboration). It took me some time since his
> first post to figure it out for myself independently. If you copy a
> file to 1000 computers, it doesn't disappear when some of them fail,
> but it does disappear when all of them fail.
What I mean is, if we lived in a universe where certain things worked
perfectly, including our model of it, then that universe would not be Turing
computable. It is hard to imagine such a universe because we don't live in
one. The fact that we can't imagine it (because a perfect model is
impossible) is evidence that the universe is Turing computable.
> > The question is whether an observer (a computer) in a universe can model
> > (predict) the universe (including itself) exactly. If the universe is
> > simulated by a Turing machine, then it is not possible. A Turing machine
> > state can be described by a map: N -> {0,1}. There may be more powerful
> > machines that can model themselves in this sense, for example, R -> {0,1},
> N
> > -> R, R -> R, (R -> R) -> R, etc. But because we lack non probabilistic
> > models of the universe, there is no evidence that anything more powerful
> than
> > a Turing machine is required.
> >
>
> You didn't describe a model. Anyway, if the goal is to simulate
> yourself, you can destroy the rest of the universe (so that you'll
> become the only object in the universe), and you'll have enough
> computational resources to simulate this small blob of matter. What
> poses a problem here?
A whole universe simulating itself isn't really a simulation. But yes, I did
not fully describe a model, which would include operations on real numbers or
real valued functions, possibly in discrete or continuous time. There are
many such models. In some of them, it is possible for a universe to model
itself exactly. A simple example would be a fractal where each region of
space is a scaled down copy of a larger region containing it. Perfect
fractals are not Turing computable, and do not exist in our universe.
-- Matt Mahoney, matmahoney@yahoo.com
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