Revised Computing Randomness

From: Hal Ruhl <hjr.domain.name.hidden>
Date: Wed, 18 Apr 2001 20:57:02 -0700

Here is a revised version of my comments on this subject. I think it fixes
several aspects of what I have had to say earlier.

Standalone deterministic evolving universes:

Such a universe is describable as a concatenation of single output programs
of the form:

          Rj(Aj) -> B; Rj(B) -> C; ... Rj(F) -> G; Rj(G) -> H; Rj(H) -> I;
....

"Rj" is a subset of the rules of an N-bit FAS. "Aj" is an axiom of that
FAS. There are no potential branches in or out. It is an elegant proof
and defines the complexity of each of its objects B,C,...F,G,H,I ...

Because the proof is everywhere elegant each successive object is more
complex than its preimage. This is the type of theorem cascade I was
trying to identify.

It has a problem: It will - if it does not stop - eventually outrun the
complexity of its N-bit FAS. Since the proof is known to be elegant this
is [as I understand it] a contradiction of Chaitin's result.

However, if it stopped at complexity N + c as required by Chaitin [again I
think] the end state would be highly but finitely complex. Highly but
finitely complex states are going to have some consequent under the rules
of the cascade. It can not be both "provably highly but finitely complex"
and "absent operative correlations" under the same set of finite rules. It
might reach an absence of operative correlations when it becomes infinitely
complex and the domains of any correlations fall below the resolution of
the still finite Rj of the cascade. So the cascade can not stop prior to
an infinity complex end state.

This seems to be a contradiction. It can be cured if the FAS is able to
spontaneously take in additional complexity from an outside source. I call
such a source a Superverse. The new information would act to replace the
latest object of the cascade with a new axiom acceptable to the Rj and the
cascade proceeds.

I use this to argue in favor of non deterministic standalone universes.

Now this universe is actually an isomorphism linked to the successive
strings [objects] B, C etc. The isomorphism has the same Rj as its laws of
physics.

This should continue to hold even if one shifts to a universal dovetailer
running all universes simultaneously. Individual isomorphisms would still
jump to the string currently containing as the latest sub string [string
structure: ....;sub string;sub string;latest substring] that is the
successor state of that isomorphism under its individual Rj. Thus each
individual isomorphism remains a single output program or elegant proof
under its Rj. A non deterministic result is sustained.

Random histories are just isomorphisms with Rj = "do not care" and they
jump as easily as any isomorphism.

Well that is the current state of the argument.

Hal
          
Received on Wed Apr 18 2001 - 18:43:40 PDT