Re: everything priors

From: Max Tegmark <>
Date: Sun, 31 Oct 1999 14:51:33 -0500 (EST)

Hi Juergen,

Thanks for your note!
Inspired by your message, I finally got around to downloading your nice paper
- something I'd been meaning to do for ages.

Yes, I think our ideas are quite similar in spirit.
I noticed a few interesting differences which I think reflect our
different backgrounds: yours as a computer scientist and mine as a physicist.
I agree completely with your point that
"All Universes are Cheaper Than Just One", and you'll find that
echoed also in my older paper "Does the Universe in fact contain almost no
information" (at
To me, the interesting difference is that your starting point is
(universal) computer programs whereas mine is mathematical structures.
I think the latter are harder to equip with a prior, but I certainly have
nothing againt some form of complexity-based prior as long as it can
be justified in some natural way.

As a non-CS person, I think of a computer program as
merely a special case of a mathematical structure (albeit
a very interesting special case):
as a function from the set of integers to the set of integers.
I simply interpret the memory contents of a finite computer (which
includes both the program itself and whatever variables it uses)
as the binary digits of the integer. Running a program on this
computer then corresponds to iterating the
function f: i -> f(i) -> f(f(i)), etc.

If you need truly infinite computer memory to do the equivalent of
a Turing machine (you do, right?), then you can take f to be a mapping
on the unit interval of the real line instead so that it iterates
real numbers like 0.10010110101.

So here are some questions for you:
 1) Has is been proven that such an f can compute anything that a
    TM can? I'm assuming that someone has proven that the two are
    in fact equally powerful/universal.
 2) If so, should we really limit ourself to this particular kind of
    mathematical structures? My concern is that we may be a bit too
    narrow-minded if we do. For instance, this would automatically
    give our world a causal one-dimensional (discrete) time, even though we
    know that general relativity is perfectly consistent with
    having more than one time-dimension. My concern is that
    we're limiting ourselves to such "1-dimensional" computations
    simply because our world happens to have one time-dimension.

> From Mon Oct 25 04:55 EDT 1999
> To:
> Subject: everything priors
> Hello Max,
> through Wei Dai's everything mailing list I became aware of:
> M. Tegmark. Is "the theory of everything" merely the ultimate ensemble
> theory? Annals of Physics, 270:1-51, 1998.
> And perhaps you are aware of a related paper:
> J.Schmidhuber. A computer scientist's view of life, the universe, and
> everything. In C.Freksa, M.Jantzen, and R.Valk, editors, Foundations
> of Computer Science: Theory, Cognition, Applications, volume 1337, pages
> 201-208. Lecture Notes in Computer Science, Springer, Berlin, 1997.
> It seems to me that a major conceptual difference is that you assume
> "... all mathematical structures are a priori given equal statistical
> weight", while I am focusing on complexity-based weightings based
> on "optimal universal priors" or Solomonoff-Levin distributions.
> Would you agree? Do you see any additional important differences?
> All the best,
> Juergen
> ____________________________________________________________________
> Juergen Schmidhuber

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| Prof. Max Tegmark |
| Dept. of Physics |
| Univ. of Pennsylvania |
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Received on Sun Oct 31 1999 - 11:56:49 PST

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