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From: Wei Dai <weidai.domain.name.hidden>

Date: Tue, 22 Jan 2002 16:48:07 -0800

On Fri, Jan 18, 2002 at 09:22:57PM -0800, hal.domain.name.hidden wrote:

*> I'm not convinced about the models of computation involving GTMs and
*

*> such in Juergen Schmidhuber's paper. Basically these kinds of TMs can
*

*> change their mind about the output, and the machine doesn't know when
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*> it is through changing its mind. So there is never any time you can
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*> point to the output or even a prefix and say that part is done. It is
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*> questionable to me whether this ought to count as computation. I will
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*> write some more about his paper tomorrow, I hope.
*

I'm no longer sure that computation is a necessary ingredient. Why assume

that a universe must be computable (by whatever definition of computation)

in order for it to exist?

To me, the attraction of GTM was that it let's you define a more dominant

prior, so that you don't have to rule out (i.e. not care about) universes

with things like halting oracles a priori. However I now realize that even

the GTM-based prior is still not dominant enough, because it rules out

things like convergence oracles (i.e. an oracle that tells you whether the

output of a GTM will converge).

I think we need an even more dominant prior and associated notion of

complexity, based on a concept of definitional description rather than

computational description. Maybe it can be based on second-order logic. I

just finished reading Steward Shapiro's _Philosophy of Mathematics :

Structure and Ontology_, where he argues that all mathematical strucutures

that can be defined by second-order theories exist. (This seems very

similar to Max Tegmark's position, but more clearly defined.) I'm going to

read his _Foundations Without Foundationalism : A Case for Second-Order

Logic_ to learn more about second-order logic and see if it's possible to

obtain a definition of complexity and a measure from it.

*> I wonder if a better term than "objective measure" is "probability".
*

*> That carries the connotation that it represents the likelihood that
*

*> something happens. Then you could have an objective probability which
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*> told how likely each universe was (its chance of being selected at
*

*> random from the multiverse), and a subjective measure that told how
*

*> much you cared about universes.
*

No, we need to reserve the word "probability" for the subjective level of

confidence that some statement is true. This is the classical baysian

definition, and I think it's still needed in the new decision theory, even

though I don't know how it will be used exactly. Basicly we still need a

theory that takes into account computational limitations, and perhaps in

order to do that we need to assign probabilities to mathmatical statements

so that you can say things like "P(3.14 < pi < 3.15) > .99999999" or

"P(event x happens in universe y) = z".

Received on Tue Jan 22 2002 - 16:51:53 PST

Date: Tue, 22 Jan 2002 16:48:07 -0800

On Fri, Jan 18, 2002 at 09:22:57PM -0800, hal.domain.name.hidden wrote:

I'm no longer sure that computation is a necessary ingredient. Why assume

that a universe must be computable (by whatever definition of computation)

in order for it to exist?

To me, the attraction of GTM was that it let's you define a more dominant

prior, so that you don't have to rule out (i.e. not care about) universes

with things like halting oracles a priori. However I now realize that even

the GTM-based prior is still not dominant enough, because it rules out

things like convergence oracles (i.e. an oracle that tells you whether the

output of a GTM will converge).

I think we need an even more dominant prior and associated notion of

complexity, based on a concept of definitional description rather than

computational description. Maybe it can be based on second-order logic. I

just finished reading Steward Shapiro's _Philosophy of Mathematics :

Structure and Ontology_, where he argues that all mathematical strucutures

that can be defined by second-order theories exist. (This seems very

similar to Max Tegmark's position, but more clearly defined.) I'm going to

read his _Foundations Without Foundationalism : A Case for Second-Order

Logic_ to learn more about second-order logic and see if it's possible to

obtain a definition of complexity and a measure from it.

No, we need to reserve the word "probability" for the subjective level of

confidence that some statement is true. This is the classical baysian

definition, and I think it's still needed in the new decision theory, even

though I don't know how it will be used exactly. Basicly we still need a

theory that takes into account computational limitations, and perhaps in

order to do that we need to assign probabilities to mathmatical statements

so that you can say things like "P(3.14 < pi < 3.15) > .99999999" or

"P(event x happens in universe y) = z".

Received on Tue Jan 22 2002 - 16:51:53 PST

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