Explorations of the definitional basis of a universe and its effect on the
idea of decisions:
First examine a deterministic universe j such that [using notation from a
post by Matthieu Walraet]:
Tj Tj Tj
Sj(0) ----------------> Sj(1) ----------------> Sj(2) ....
----------------> Sj(i)
An interpretation is that all the information needed to get from Sj(0) to
Sj(i) is contained in Sj(0) and the rules of state evolution for that
universe that is Tj.
I see a problem with this interpretation.
Suppose we write an expression for the shortest self delimiting program
able to compute Sj(i) as:
(1) Pj(i) = {Tj[Sj(i - 1)] + DLj(i)} computes Sj(i)
where DLj(i) is the self delimiter.
Compressing Sj(i - 1) it can be written as Pj(i - 1) and this short hand
substituted into (1) to yield:
(2) Pj(i) = {Tj[Pj(i - 1)] + DLj(i)} computes Sj(i)
Note that Pj(i) is always longer than Pj(i - 1) because it contains Pj(i -
1) plus the Tj plus the delimiter so Sj(i) contains more information [using
the program length definition of information] than Sj(i - 1) and thus more
information than Sj(0).
What kind of information is it? I see it as location record keeping
information. The universe is at state i of the recursion and this extra
information is the tag providing that location. The effect has several results:
1) This new information can never be removed from such a universe so its
local "time" has an arrow.
2) New information can manifest as either a decorrelation of the bit
pattern of and/or an increased length of the string representing Sj(i). The
length of the string is interpretable as "space" [a fixed number of bits
say x bits describe the configuration of a small region of that "space" and
there are y regions requiring description so an increase in length of the
string causes y to increase.]. Note that the effect increases as the
recursion progresses since DLj(i) increases monotonically with i. Thus such
a universe should see a long term acceleration in the rate of expansion of
its "space".
3) So how do we define a universe? Suppose many universes are following
the same recursion some at earlier states and some at later states than
universe j. It seems best to define such universes by the state they are
in [which includes Tj and DLj(i)]. Where did the additional information
come from? The additional information is not that a universe can follow
the recursion but rather as stated above the location of a particular
universe in the recursion. Since this information is not in Sj(0) or Tj it
must have come from outside universe j.
Universes that are not deterministic but have rules that allow external
true to enter are easier to analyze in this regard since the current state
seems the only reasonable definition.
4) For a deterministic universe is the additional information true
noise? In at least one sense it is because a particular universe j is
defined by its current state it can not tell which state including the
current state was or is Sj(0) so there is no clue as to what the
information means or if it is somehow even additional or new or which
information is involved. This is the same as the situation for a universe
whose rules allow external origin true noise.
5) This would seem to enhance the case against the idea of "decision" since
noise [chance] of some sort seems to be everywhere.
6) Behavior similar to (2) is found in universes that are sufficiently well
behaved so that it is possible to propose a prior state such that the
universe's rules when stripped of their allowance for external true noise
can deterministicly arrive at the universe's current state. This proposed
prior state need not have been the actual prior state.
Hal
