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.
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Visiting Professor of Mathematics [email protected]
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