Comparators vs computers etc.

From: Hal Ruhl <hjr.domain.name.hidden>
Date: Wed, 29 Nov 2000 22:56:36 -0800

1) The way in which I described my model in the several posts where I
accumulate things did not, upon reflection, explain the comparator approach
to my Rj very well so I have fixed this and attach the applicable revised
material.

2) Compatators vs computers:

Can a computer act as my comparator?

The comparator examines the current string Uj(i - 1) and forms - by some
means - a template with which randomly encountered numbers are tested. The
first encountered number that fits the template is accepted as the next
string Uj(i). If the template allows for more than one number to fit then
there is no prior knowledge of the actual complete resulting Uj(i). This
amounts to at least some "true noise" in the sequence Uj(i) as i counts up.

If the template is such that only one number will fit it then it can be
computed and it is equivalent to computing Uj(i). This is a no "true
noise" situation and at best a minor class of comparator.

3) The consequent of fixed rules:

In my equation:

(1) Pj(i) = {Rj(Pj(i - 1)) + PLj(i)} is the compressed form of Uj(i).

If Rj is fixed and of course Pj(i - 1) is fixed any new information - the
result of "true noise" - in Uj(i) must be represented in PLj(i). But
PLj(i) is just a measure of the length of Pj(i). So this indicates that Rj
runs beyond its data i.e. Pj(i - 1) if there is a "true noise" content in
the step Uj(i - 1) to Uj(i). In my model this is just the addition of more
u to complete the formation of Uj(i) from Uj(i - 1).

4) Thus there are two sources of "true noise" in my approach. Rj may
contribute and the necessary increases in the complexity of the fc-FAS will
contribute.

5) Axioms and alphabets:

I have said that the necessary ongoing increases in complexity to the
underlying fc-FAS take place as additions to its alphabet. But my equation
is highly compressed. These additions can also be viewed as restarting the
recursion with a new axiom.

6) It is my opinion that life exists best in universes that have an Rj that
is at least a bit non deterministic.

--------------------------------

REVISIONS


POSTULATE:

All counting numbers exist with countably infinite copies in a seething
foamy fractal.

Call this fractal Everything or as I prefer to call it the Superverse S.

In this fractal every universe Uj is isomorphic to a non deterministicly
self sorting sequence Uj(i) of these numbers that, upon chance encounters
between numbers, they self sort according to:

(1) Pj(i) = {Rj(Pj(i - 1)) + PLj(i)} is the compressed form of Uj(i).

where Rj(Pj(i - 1)) is the fixed rule set of a particular universe acting
as a comparator [it may be non deterministic that is a source of "true
noise" - also see (2) below] to the previous Uj(i) and the underlying
N-bit, finite, consistent FAS [N-bit, fc-FAS} structure is:

a) It has a single axiom Aj unique to a particular family of universes
which is in Pj(1), it serves as Pj(0), it is thus the compressed form of
Uj(0) and initiates the recursion.

b) It has a fixed set of rules = Rj the comparator.

c) It has an alphabet = differently sequenced strings of bits I call types
of "regions". Each region type is isomorphic to a physical structure that
has existence such that the location of a discrete space point inside a
small fixed portion of a grid is coded by each region type. [In our case I
believe the grid to be 3D and most likely Face Centered Cubic.] The axiom
Aj contains the entire initial alphabet for a particular universe.

So the equation (1) contains the entire N-bit, fc-FAS.

Uj(i) contains a theorem of this FAS giving a specific arrangement of
regions - a configuration of a particular universe.

(2) Uj(i) grows in length [number of bits] randomly to avoid Chaitin's
limit on how much information you can put into an N-bit number. Since
Pj(i) can not yield Pj(i) unless Pj(i -1) is empty and each Pj(i) is
clearly longer than Pj(i - 1) the sequence can not stop on a finite
number. To sustain this growth in complexity of Uj(i) the fc-FAS - again
as per Chaitin - must also increase in complexity - randomly adding
additional alphabet elements - new types of regions. This is equivalent to
"true noise" and to starting the sequence over with a new axiom which is
now just Uj(i).

Thus there are two sources of "true noise" in the model.

(3) When Uj(i) gets to have a countably infinite number of bits the
sequence ends. From Turing [limits to FAS size] and Hilbert [Mathematical
existence]- See Tegmark Section II.

The numbers in the fractal non deterministicly self sort upon encounters
because of (1) and (2). Basically from the perspective of Uj: Uj(i -1)
contains Rj and recognizes any PLj(i) suitably higher valued than its own
PLj(i -1) i.e. Uj(i -1) picks the first good number it encounters out of a
countably infinite set of potential successors.

(2) and (3) may not be independent of (1).

The fractal contains all universes potentially an infinite number of times.

-------------------------------------

Hal
Received on Wed Nov 29 2000 - 20:12:03 PST