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From: Hal Finney <hal.domain.name.hidden>

Date: Wed, 19 Jun 2002 12:59:05 -0700

Saibal Mitra writes:

*> Shouldn't the probability go to zero faster than 1/2^n ? If you consider the
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*> sequence of programs p_{k} were p_{k} will run k idintical copies of a
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*> certain observer. The probability that the observer finds himself in p_{i}
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*> should be i times the measure of P_{i}. I conclude that the measure of p{i}
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*> should go to zero faster than 1/i. The length of the program is some
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*> constant plus Log(i)/Log(2), therefore, if the measure depends only on
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*> program length, it should go to zero faster than 1/2^n.
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It is true, adding a counter of length Log(k)/Log(2) would allow a program

to create k duplicates. Increasing the program by this length (which is

log to the base 2 of k) will reduce its measure by a factor of 2^length

which is k. So these factors balance out.

It is similar to extending the length of a program by n unused bits.

This creates 2^n functionally identical programs corresponding to the

2^n possible values of the n unused bits. But it reduces the measure

of each one by a factor of 2^n, which balances.

These considerations are what lead to the factor of 2^n for comparing

the measure of two programs whose length differs by n.

Hal Finney

Received on Wed Jun 19 2002 - 13:12:24 PDT

Date: Wed, 19 Jun 2002 12:59:05 -0700

Saibal Mitra writes:

It is true, adding a counter of length Log(k)/Log(2) would allow a program

to create k duplicates. Increasing the program by this length (which is

log to the base 2 of k) will reduce its measure by a factor of 2^length

which is k. So these factors balance out.

It is similar to extending the length of a program by n unused bits.

This creates 2^n functionally identical programs corresponding to the

2^n possible values of the n unused bits. But it reduces the measure

of each one by a factor of 2^n, which balances.

These considerations are what lead to the factor of 2^n for comparing

the measure of two programs whose length differs by n.

Hal Finney

Received on Wed Jun 19 2002 - 13:12:24 PDT

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