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From: Max Tegmark <max.domain.name.hidden>

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 www.sns.ias.edu/~max/nihilo.html).

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.

Cheers,

Max

;-)

*> From juergen.domain.name.hidden Mon Oct 25 04:55 EDT 1999
*

*> To: max.domain.name.hidden
*

*> 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 www.idsia.ch
*

/////

( O O )

| " |

|--------.oooO---------Oooo.---------|

| Prof. Max Tegmark |

| Dept. of Physics |

| Univ. of Pennsylvania |

| Philadelphia, PA 19104 |

| http://www.physics.upenn.edu/~max/ |

|____________________________________|

| | Oooo.

.oooO ( )

( ) ) /

\ ( (_/

\_)

Received on Sun Oct 31 1999 - 11:56:49 PST

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 www.sns.ias.edu/~max/nihilo.html).

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.

Cheers,

Max

;-)

/////

( O O )

| " |

|--------.oooO---------Oooo.---------|

| Prof. Max Tegmark |

| Dept. of Physics |

| Univ. of Pennsylvania |

| Philadelphia, PA 19104 |

| http://www.physics.upenn.edu/~max/ |

|____________________________________|

| | Oooo.

.oooO ( )

( ) ) /

\ ( (_/

\_)

Received on Sun Oct 31 1999 - 11:56:49 PST

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