OFF : Physics *is* fun.

[M Holmes]

Paul Mather writes:

> On 4 May 2007, at 6:00 PM, M Holmes wrote:
> 
> > Then he went into Omega, the number which
> > answers every question in the universe but which has an infinite  
> > number
> > of digits which are unpredictable and can't be calclated. Except that
> > someone has used a quantum computer to calculate sixty digits,  
> > which is
> > great because that answers *some* of the questions in the universe.
> > unfortunately we don't know which those are.
> 
> That wouldn't happen to have been Chaitin's Omega, would it

I believe so...

> or some  
> other Omega?  (Chaitin of Algorithmic Information Complexity fame,  
> that is, which is an offshoot of Kolmogorov Complexity.)  Dim  
> recollection tells me Chaitin's Omega has the property that it is the  
> most compressed string in existence (mathematically possible), i.e.,  
> no other string contains more information than Omega.

Since this part of the talk was prefaced by information theory
concerning complexity, it does look rather likely.

> Further dim  
> recollection tells me this property springs from the fact that the  
> first N bits of Omega encodes the halting probability of universal  
> Turing Machines (i.e., programs) of N bits or smaller. 

Indeed. This awoke some dim and distant computability lectures which had
rested undisturbed in my brain. I'd forgotten that some of the philosphy
side of computing is a lot less dreary than the day to day side I'm used to. 

> Of course,  
> because the halting problem is unsolvable, it makes computing Omega,  
> um... difficult

In a single universe yes, but if you have a quantum computer then you
have a lot more universes in which to run the calculations...

> but means that Omega encodes a heck of a lot of  
> information

It contains all possible information doesn't it?

> (besides being very useful as an oracle).

As with all oracles, the trick is less getting the answer than it is
getting the question which matches the answer. As Chown pointed out,
Douglas Adams wasn't very far out at all with his "42" and "The Question
of Life, The Universe and Everything". It seems that one of the few
things we do know is that Omega is definitely not 42.

> Dear me, this is taking me back years...

It surely gets the old grey cells awake and running again dunnit?

Cheers

FoFP