Re: My claim on Omega's defn
From: |-|erc (H_at_r.c)
Date: 01/30/05
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Date: Mon, 31 Jan 2005 09:45:38 +1000
"Arthur Fischer" <arthurfischer@sym.ca> wrote in
> |-|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.
So all Omega means is there exists 1 program of that size that halts.
What a load of crap, I could design an arbirtrary UTM where Omega = 1.
It skips 100% of programs without any reason.
Herc
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