On Saturday, October 12, 2013 10:08:05 AM UTC+11, Brent wrote:
>  On 10/11/2013 3:44 PM, Russell Standish wrote:
> On Fri, Oct 11, 2013 at 03:08:30PM -0700, meekerdb wrote:
>  UD* (trace of the universal dovetailer) is a continuum, AFAICT. It has
> the cardinality of the reals, and a natural metric (d(x,y) = 2^{-n}, where n 
> is
> the number of leading bits in common between x and y).
>  Hmm? So 1000 is the same distance from 10 and 111?  What's the measure on 
> this space?
>  1000... and 101... are 0.25 apart. 1000.. and 111... are 0.5
> apart. (the ... refers to an infinite number of bits that are not
> relevant to the computation). So the answer to your question is that
> these these three strings are not the same distance from each other.
> The measure over a set of these things would be something like the
> supremum over the distance between any two pairs drawn from the
> set. Of course, that assumes that only sets defined by finite length
> prefixes, and countable unions and intersections thereof are
> considered. My maths chops aren't quite up to generalising this for
> arbitrary sets of binary strings.
> Maybe I'm not clear on what UD* means.  I took it to be, at a given state 
> of the UD, the last bit output by the 1st prog, the last bit output by the 
> 2nd program,...up to the last prog that the UD has started.  Right?
> Brent

But Russell just said there *is* no output. There are only machine states 
(computation X is at step Y and so on). I thought the UD* was the entire 
history of computational states the the UD passes through from the moment 
it starts up to ... well, forever. 

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