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Essays and Articles

Discrete Theoretical Processes

I agree that computation universality by itself is no guarantee that a CA universe will be interesting except on a tiny subset of initial states: those that simulate more interesting systems! So I think a very good question in general is, What additional properties can we include in simple spatial models, to capture more of the richness of real physical dynamics?

Of course classical lattice gas models have been used since the dawn of QM in statistical mechanics, and dynamical versions of these are interesting. Some of the earliest physical properties captured in dynamical lattice gases were conservation laws: this led to the field of lattice gas hydrodynamics. With just a few discrete conservations and a few discrete values of microscopic momentum, realistic sound propagation and fluid flow are recovered in the macroscopic limit. Add in a few more properties and we also get thermodynamics and chemistry. With a bit more elaboration, crystallization and elastic materials. Invertible long range forces are an open problem.

Since much of the interesting complexity in our world comes from evolution, you might ask specifically what physics-like properties in a CA are essential for a robust darwinian evolution? One answer to this seem to be: special relativity. You have to be able to set complex large-scale structures in motion, without disrupting their internal chemistry and dynamics. Large-scale dynamics can be made independent of inertial motion by just incorporating appropriate discrete relativistic conservations into the microscopic lattice gas rule.

Then there are questions about hierarchical structures and separation of macroscopic and microscopic dynamics. To address the original problem we started with, about whether we can model macroscopic control of microscopic dynamics in a CA, we should really use a CA that naturally develops multiple levels of hierarchical structure. I think this is tremendously interesting because much of the apparatus of physics is designed to deal with this situation (hamiltonians, thermodynamics, etc.). It’s worth pointing out, in this context, that classical lattice gases are special cases of quantum lattice gases, and so can be analyzed using the quantum formalism.

If you believe that the universe can be viewed as just a big quantum computation, then fundamental physical and computational quantities and concepts are really the same thing. One might learn a lot from studying classical special cases of spatially organized computation.

]]>Time is divided into phases, so that in one phase, the cells in the ‘real’ plane will cause the cells in the ‘imaginary’ plane to change state, and then in the next phase, the imaginary cells cause the real cells to transition. This follows Fredkin’s principle of reversibility, whereby the ‘present’ transforms the ‘past’ into the ‘future’. He has a nice description of building a reversible formulation of the Schroedinger wave equation in this manner.

In the SALT model, these planes are implemented as the set of ‘even’ cells (those whose coordinates add to an even number) and ‘odd’ cells. In the 3D SALT model, these cells interdigitate like the atoms of a Sodium-Chloride crystal. He then divides time into six phases, and in Miller’s BusyBox rule, each plane (XY, XZ, YZ) is processed with a 2D rule. This provides a three dimensional rule which provides time and parity reversal. (see the reference above).

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Circular Motion of Strings in Cellular Automata, and Other Surprises (arxiv) http://64.78.31.152/wp-content/uploads/2012/08/CircularMotion.pdf

Two-state, Reversible, Universal Cellular Automata In Three Dimensions http://64.78.31.152/wp-content/uploads/2012/08/2stateRevCAin3D.pdf

The “SALT model” framework was developed by Fredkin to try to design in analogs of some of the basic conservation laws known in physics, and to preserve CPT symmetry (time reversibility, parity, and something like charge although that is a work in progress).

Miller has a nice implementation of a simple rule developed for the SALT model, which he called “BusyBox” which exhibits some very thought provoking behaviors http://busyboxes.org/?hash=PeciXwDuTA0

Among the surprises were an asymptotically circular motion path of a particular glider configuration, and some behaviors that have parallels perhaps to nuclear decay.

They also found interesting wave-like gliders which seem to have harmonics of oscillation.

I did some experiments I call “desktop particle collider” ,colliding the circular and linear gliders in the BusyBox model. I saw some interesting behaviors that perhaps are precursors to models of momentum transfer and photon-electron interactions (where two gliders will collide, form an orbit for a fixed number of cycles, and then decay off again into gliders).

I wrote some notes on those ‘collider’ experiments at http://www.digitalphilosophy.org/wp-content/uploads/2014/02/GlidersWithCircularandLinearMotion-2.pdf

I recently did a reimplementation of Miller’s BusyBox system in Java, to see if I could get some speedup, and to make it a little easier to experiment with new rules. (It turns out Java is not much faster than Javascript these days!)

Just as an experiment, I ran Miller’s BusyBox rule, starting from a mostly symmetric ‘cube’ of cells, with a small asymmetry inside, and it produced an interesting ‘diffusion’-like ‘big-bang’ pattern similar to the ones described in Luke’s paper:

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