One Small Answer
Stan Kugell
Abstract
Background independence can be directly harmonized with cellular automata-like models for
discrete physics, without resort to computational universality arguments. The problem
addressed is that discrete finite models of space-time and matter, consistent with information
theory and the Finite Nature hypothesis, such as those based on cellular automata and the work
of Fredkin and others, all rely on a fixed reference system inherent in cellular automata, and
these appear to be difficult to conform to general relativity. While computational equivalences
with general relativity’s curved space-time can be constructed in cellular automata, they are
awkward constructions at best. A modification to classical cellular automata is proposed,
consistent with the Finite Nature hypothesis, to resolve tensions between cellular automata’s
fixed notion of space-time on the one hand, and general relativity’s distaste for a fixed
reference system, desire for a way to transform space-time dynamically, and quantum
mechanics’ appetite for non-local effects on the other.
Introduction
One mathematical/computational foundation for developing the Finite Nature hypothesis (FN)
has been cellular automata (CA), as a model for how finite nature might work. However
cellular automata may not be the best model for the real structure of a finite universe. There are
strong reasons to suppose that cellular automata are a good starting point, yet need some
generalization to provide a better fit for experimentally observed physics. Specifically, this
response proposes a new kind of CA, conforming CA’s absolute kind of space-time to relativity’
s pliable kind of space-time.
Fredkin says that “the whole point of this paper is to suggest there must be another branch of
mathematics that deals with … FN” and “[t]he new mathematics is rooted in automata theory,
but that fact doesn’t give a clue as to what the new math is.” [1] (emphasis added). One
possible clue is proposed.
The Problem
A troubling aspect of cellular automata as a useful model for physics is CA’s reliance on a fixed
reference system, i.e. a fixed space-time lattice. General relativity and quantum mechanics
(QM) both seem to point away from a fixed reference system. Fredkin addresses this by
observing CA’s computational universality. However universality merely shows that a
transformation from FN to CA is possible, but not that it is irreducible. Indeed, the tricks and
loops a CA might have to jump through to produce the space-time effects wanted by known
physics strongly suggest that classical CA alone are not the best mathematics to use.
A Solution
Cellular automata start with a fixed lattice. Let’s discard that, and create something more
pliable, while retaining CA’s relevant characteristics: discrete, rule-based, universal and able to
be made reversible. Retaining those characteristics, there is no reason to assume that CA’s
strict separation between fixed machinery (the rule and the graph of cells) and variable data (the
state within the cells) is needed or helpful. FN does not require it. Moreover, the observed
nature of both QM (non-local effects) and general relativity (curved space-time) point in the
opposite direction, towards a merger of the two, as follows.
A useful FN model needs to gracefully accommodate these two hints:
(1) General relativity suggests a reference system of variable space-time, not merely a
computational equivalence for it. There needs to be a good clean mechanism to curve space-
time.
(2) Quantum mechanics suggest granular and non-local (in the Cartesian sense) action in
space-time.
Fortunately, FN, if true, requires only that a space-time lattice system be discrete, but it does
not require that it be fixed or absolute. These two hints taken together, relativity’s need for
variable space-time and QM’s need for granular non-local action, point in a coherent direction:
the organization of the space-time reference system itself is part of the variable information
content of an FN universe.
A new computation model for FN might look a lot like a classical CA, with a difference. A
cellular engine is proposed as a modified CA, with an interconnectivity graph more like the
observed space-time structure. The graph of a cellular engine has the following properties:
- Locally and instantaneously fixed (i.e. at a point in x,y,z, and t);
- Globally and timely variable (i.e. can look different elsewhere in x,y,z or t);
- Varies according to a rule; and
- Varies in a reversible, symmetric and discrete manner.
While classical CA strictly separate structure from data, a cellular engine’s information content
is both the data contents and the graph of connections among cells. As with CA, a rule
deterministically operates on the content and connectivity of a cell. A useful rule would need to
be universal, reversible and have other needed symmetries.
An Example
How might CE cells reorganize without a fixed reference system? One way is that each cell
might reorganize on the reference system as they find it at their point in space-time. If at time
t, cell A connects to B and B connects to C, then, according the operations of a rule, at time
t+1, a direct connection between A and C may form. Perhaps connectivity changes in response
to mass, acceleration, and gravity, or perhaps changed connectivity is gravity. Rules must be
found that can modify the graph reversibly, and with wanted space-time symmetries. Nothing
in principle excludes the possibility of finding such rules.
FN requires only that the connection graph be finite in a bounded region of space-time, but
cellular engines are free within the FN hypothesis: to increase or decrease in connections or
cells (according to finite rules operating on finite quantities of data), grow arbitrarily large
(given an infinite motion in time, the graph could grow infinitely large consistent with FN),
contract arbitrarily small, operate over long Cartesian distances, and create and destroy highly
isolated regions of space-time, such as black holes.
Imagine a rule that grows and bends space over time. T = 0 might include only a single cell and
rule, one capable of growing space from its initial conditions.
Conclusion
Finite nature leaves lots of room to roam beyond classical cellular automata. Fredkin observes
that the mathematics needed for FN has not been described. CA seems a useful stepping stone,
but appears to have important limitations. Perhaps by relaxing some of CA’s premises into
something new, its utility as a tool for exploring FN could be extended.
So long as a transformation rule can be defined in a finite series of discrete steps, and can
preserve universality, reversibility, and other symmetries, there is no reason to exclude that a
useful cellular rule could operate on the connection graph, or on the rule itself, as well as on cell-
stored content.
References
[1] E. Fredkin, “Five big questions with pretty simple answers”, IBM J. Res. & Dev., Vol
48. No. 1, 31-44 (2004), http://www.research.ibm.com/journal/rd/481/fredkin.pdf.
Stan Kugell
Revised as of February 17, 2004
Original edition February 10, 2004 http://kugell.com/public/onesmallanswer.rtf
For submission to Usenet sci.physics.discrete when inaugurated
Untitled