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From: Russell Standish <r.standish.domain.name.hidden>

Date: Tue, 7 Jun 2005 10:19:49 +1000

On Mon, Jun 06, 2005 at 01:51:36PM -0700, "Hal Finney" wrote:

*>
*

*> Another area I had trouble with in Russell's answer was the concept of
*

*> a prefix map. I understand that a prefix map is defined as a mapping
*

*> whose domain is finite bit strings such that none of them are a prefix
*

*> of any other. But I'm not sure how to relate this to the infinite bit
*

*> strings that are "descriptions".
*

A prefix map attaches the same output to all strings that share a

common finite length prefix.

*>
*

*> In particular, if "an observer attaches sequences of meanings to sequences
*

*> of prefixes of one of these strings", then it seems that he must have a
*

*> domain which does allow some inputs to be prefixes of others. Isn't that
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*> what "sequences of prefixes" would mean? That is, if the infinite string
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*> is 01011011100101110111..., then a sequence of prefixes might be 0, 01,
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*> 010, 0101, 01011, .... Does O() apply to this sequence of prefixes? If
*

*> so then I don't think it is a prefix map.
*

*>
*

Yes I agree this is vague, and seemingly contradictory. I'm not sure

how to make this more precise, but one way to read the paper is to

treat observers as prefix maps for section 2 (Occam's razor), and then

for section 3 (White Rabbit problem) ignore the prefix property.

It could be that the way of making this more precise is to assume

observers have some internal state that is constantly updated (a time

counter perhaps), so actually going through a sequence of prefix maps

in (psychological) time, but at this stage I don't have an answer.

*> I want to make it clear by the way that my somewhat pedantic and labored
*

*> examination of this page is not an attempt to be difficult or
*

*> stubborn.
*

Even being difficult and stubborn has its place (to help winkle out

subtle errors of logic eg), so long as you relax enough at other times to

obtain understanding. I appreciate the effort in any case.

*> Rather, I find that by the third page, I don't understand what is going
*

*> on at all! Even the very first sentence, "In the previous sections, I
*

*> demonstrate that formal mathematical systems are the most compressible,
*

*> and have highest measure amongst all members of the Schmidhuber ensemble,"
*

*> has me looking to see if I skipped a page! I don't see where this is
*

*> discussed in any way.
*

This is pretty much a tautology. Formal mathematical systems are a

means of compressing data in the form of facts about numbers. If one

were to include all such possible compression schemes, rather than

just the systems studies by mathematicians to date, one would end up

with the set of Turing machines, or equivalently of computable

functions. The Occam's razor result clearly relates measure to the

amount of compressibility in the description.

Perhaps such a view of mathematics is strange. Certainly I find it

strange when Stephen Wolfram says mathematics is incapable of

understanding complex phenomena, and one should cellular automata

instead. To me, cellular automata are just another example of a

mathematical system.

*> So I hope that by pinning down and crystalizing
*

*> exactly what the first page is claiming, it will help me to see what
*

*> the more interesting third page is actually able to establish. I think
*

*> Paddy is in much the same situation.
*

*>
*

*> Hal Finney
*

I hope so too.

Received on Mon Jun 06 2005 - 21:20:44 PDT

Date: Tue, 7 Jun 2005 10:19:49 +1000

On Mon, Jun 06, 2005 at 01:51:36PM -0700, "Hal Finney" wrote:

A prefix map attaches the same output to all strings that share a

common finite length prefix.

Yes I agree this is vague, and seemingly contradictory. I'm not sure

how to make this more precise, but one way to read the paper is to

treat observers as prefix maps for section 2 (Occam's razor), and then

for section 3 (White Rabbit problem) ignore the prefix property.

It could be that the way of making this more precise is to assume

observers have some internal state that is constantly updated (a time

counter perhaps), so actually going through a sequence of prefix maps

in (psychological) time, but at this stage I don't have an answer.

Even being difficult and stubborn has its place (to help winkle out

subtle errors of logic eg), so long as you relax enough at other times to

obtain understanding. I appreciate the effort in any case.

This is pretty much a tautology. Formal mathematical systems are a

means of compressing data in the form of facts about numbers. If one

were to include all such possible compression schemes, rather than

just the systems studies by mathematicians to date, one would end up

with the set of Turing machines, or equivalently of computable

functions. The Occam's razor result clearly relates measure to the

amount of compressibility in the description.

Perhaps such a view of mathematics is strange. Certainly I find it

strange when Stephen Wolfram says mathematics is incapable of

understanding complex phenomena, and one should cellular automata

instead. To me, cellular automata are just another example of a

mathematical system.

I hope so too.

-- *PS: A number of people ask me about the attachment to my email, which is of type "application/pgp-signature". Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 (") UNSW SYDNEY 2052 R.Standish.domain.name.hidden Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 ----------------------------------------------------------------------------

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