Re: My claim on Omega's defn
From: Arthur Fischer (arthurfischer_at_sym.ca)
Date: 01/30/05
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Date: Sun, 30 Jan 2005 09:38:49 -0500
|-|erc wrote:
> Omega = sum( p halts) 1 / (2 ^ size(p))
>
> [big snip]
>
> Even with infinitesimally small total probability of halting, Omega will not converge and will equal oo.
You seem to be missing the point that the domain of the universal
self-delimiting TM U is taken to be prefix-free --- ie, the encoding of
halting TMs is such that if x is the encoding of some halting TM, then
no proper prefix of x is a encoding. Basically, any branch of the
infinite-binary tree will contain at most one such encoding, and so to
simply say that there are 2^n encoding with n bits is just being
ignorant. If follows via Kraft's inequality that Chaitin's Omega will
be bounded.
> Now try the modified Omega.
>
> If all programs halt
>
> Omega' = 1/4 + 1/4 + 1/16 + 1/16 + 1/64 +1/64 + 1/64 + 1/64 + 1/256 + ..
> = 1/4 + 1/4 + 1/8 + 1/16 + 1/32...
> = 0.75
>
> You could dissalow the 1st program, 0, so it adds to 1.
>
>
> Omega' = sum (p halts) 2 ^ (-2 |p| )
How will your "omega" correspond to the "probability that a program
halts"? This seems to be one of the most important aspects of Chaitin's
constant, though you just seem to be taking a naive approach to ensure
that things will work out.
You do not seem to have invested any time into any sort of worthwhile
literature-search. Even a Google search will lead you to various papers
by Calude, Chaitin, etc. that will show your objections are founded only
on ignorance. Try, perhaps, the first 3 sections of
http://www.cs.auckland.ac.nz/CDMTCS//researchreports/059cris.pdf
__
Arthur
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