On 12 Oct 2013, at 01:08, meekerdb 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?
Imagine a universal (and thus finite) game of life pattern. Then you
can look at UD* as the infinite cone obtained by adding all the planes
describing its evolution. That gives a static view of UD* as a
discrete infinite 3D conic object.
It is, as Russell and Liz said, the trace of the programs which runs
all computations in that parallel -dovetailing manner.
But physics and theology are emerging from the internal relative
machines points of views, and that gives a richer structure, relying
on the continuum. The 1/3 distinction has a key role here.
Bruno
Brent
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