**From:** William Pearson (*wil.pearson@gmail.com*)

**Date:** Wed Mar 01 2006 - 20:32:58 MST

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On 01/03/06, Philip Goetz <philgoetz@gmail.com> wrote:

*> On 2/28/06, William Pearson <wil.pearson@gmail.com> wrote:
*

*> > With experimentation, by definition the bits of omega or equivalent
*

*> > number are random, so knowing one bit will have no bearing on the
*

*> > others so I wouldn't expect any improvement.
*

*>
*

*> The definition of omega says nothing about randomness, therefore the
*

*> bits are not random by definition. I suspect you are assuming that
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*> the probability of one program halting is independent of the
*

*> probability of any other program halting. This seems unlikely to me.
*

Okay maybe not quite by definition, but it is not far off. Take your

doubt of the randomness of Omega up with Chaitin.

"And what is the most surprising property of Omega? It's the fact that

it is irreducible, or algorithmically random, and that it is

infinitely complex."

http://plus.maths.org/issue37/features/omega/

I'll grant you there might be patterns within subsets of omega, but

within the whole thing I doubt it.

By this I mean I can see why the function X+1 would halt in the same

way that the function X+2 would up to X+infinity. But I don't see why,

whether word crashes or not should have anything to do with whether a

linux kernel panics.

A side note I found Verónica Becher's homepage with analogues of

kolmogorov complexity and omega for infinite running programs.

http://www.dc.uba.ar/people/profesores/becher/

*>Can you give a more precise statement, such as one would see at the
*

*>beginning of a proof?
*

I will have to give the becher papers a look before I state the exact

thing I wish to prove because I am not entirely sure of how to

formulate infinite programs at the moment. I also want to precisely

formulate what sort of change of program I am interested in. Ideally I

am looking for something that captures the idea of a change in

functionality so that the self-improved program can be said to be

better.

Will Pearson

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