Re: how to do something with really small probability?

From: Lee Corbin (
Date: Tue Nov 20 2007 - 23:02:33 MST

Wei Dai originally asked,

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Sent: Thursday, November 01, 2007 12:30 PM

> Suppose an SI wants to do something with a
> very small, but non-zero probability,
> say 0 < p <= 1/3^^^3 (in Knuth's up-arrow
> notation).

(By the way, on an Overcoming Bias pages the
writer seems to assume that Knuth also
invented the single-arrow uparrow notation
---see link below. This is incorrect. Knuth
invented only the double and triple (etc.)
up-arrow notation, the single arrow notation
as in 10^10^10... being as old as keyboards
with carets.)

There is no symbolic way that I can think
of to do this (and I don't think it's possible),
but one may fall back upon our best physics
theories. For example, the probability of
the air rushing out of a small bag, or the
probability of an ocean freezing are rather
small. For extremely small probabilities,
imagine that dust has gravitationally
contracted rather recently to assemble the
Andromeda galaxy, and then calculate the
probability that the reverse could take place,
i.e., that the atoms making up the A galaxy
might begin rushing apart in the time reversed

The SI could base its action on the occurrence
of such an event. But even there, a size
limitation is evident.

> How would it go about doing this? The answer
> seems to be "it can't", because algorithmic
> information theory says there is no way that
> it can generate a sufficiently long random
> number

You probably could stop right there! :-)


> that it can know is truly random. Can anyone
> see a way around this limitation?
> This problem came up in the context of
> Eliezer's "Torture vs. Dust Specks" dilemma(, but I
> think it might be of interest outside that context.

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