10. DM-4, the nature of a rule
So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.
— Richard Feynman
DM-4 is a hypothetical member of a class of typical DM systems, some of which may be suitable for modelling physics. The purpose of this example is not to put forward a candidate for modelling physics, but rather to impart to the reader the flavor of what such a candidate might be like. We will first give a general definition of the kind of RUCA which runs DM-4.
The RUCA for DM-4 is a four-dimensional space-time lattice. In spatial extent, x, y and z, the RUCA can be any size, according to resources. If programmed on a Macintosh, a reasonable size might be 64 x 64 x 64, for a Cray, one might do 256 x 256 x 256, and a specially constructed cellular automaton machine such as the proposed CAM-8 might handle more than 1024 x 1024 x 1024 [20]. Modelling the whole universe would require a very large but finite array which would have to exist somewhere else. While our universe is exactly large enough, it is busy doing its own thing. That large but finite array would have to exist in some other universe.
In the time dimension, the system has a depth of 2. This is because it is a second-order system, which requires both the present and the immediate past in order to calculate the future . There are many different kinds of rules that are possible, but we have chosen one kind for concreteness. The RUCA is a synchronous and deterministic system that operates according to a simple rule. There is a clock that governs the timing of the system, and the clock is a 6-phase clock; instead of going "...tick, tick, tick..." the clock goes "...tick, tock, tack, toock, teck, tuck, tick, tock, tack, toock, teck, tuck, tick...". Various phases of the clock are associated with the fixed coordinate system. There are six directions, north and south, east and west, and up and down. Each phase of the clock is associated with a single direction, in the order east, down, south, west, up, north. This sequence is abbreviated EDSWUN. It is easy to design such systems so that certain properties are either present or absent. However, there is not yet any magical method for designing just what you might want; you have to try it out and see if it does what you want.
The total operation of DM-4 is determined by the geometry and connectedness of the RUCA, the rule, and the initial state. More accurately, the complete time evolution is determined, yet in general it is unknown and unknowable in advance. This is an interesting consequence of such systems; while they are simple and deterministic, there is no shortcut to determining their future; every step must be carried out (in general). This means that there is no possibility of a superman within the system, who could predict any exact future state before it happens. The amount of work to predict the future exactly for any non-trivial DM model is always proportional to the space-time volume. "unknowable determinism" is a good description of what is happening in DM models.
When a RUCA is set to an initial state and put into motion, it is nearly always full of surprises. Of course, a poor choice of rules or initial conditions can quickly result in pure chaos or perfect order, neither of which seems as interesting as the right combination of chaos and order and the right combination of interaction and superposition. We need order to allow for the kind of macroscopic determinism that is embodied in Newton's laws. We need chaos in order to supply the kind of randomness that is a part of QM. We need superposition to allow particles and fields to persist for long periods of time. We need interaction in order to make QED work. We have observed RUCA that simultaneously exhibit order and chaos, interaction and superposition. Many rules have structures that persist, objects that travel in various directions and at various speeds, and interesting interactions when objects collide. One of the most famous of such systems is Conway's Game of Life [2], a simple two-dimensional system that is full of interesting objects. However, Life, in its basic form, is definitely not a RUCA; it can be universal but it is not reversible.
[2] M. Gardner, The fantastic combinations of John Conway's new solitaire game of 'Life', Sci. Am. 223 (April 1970) 120-123.
[20] N. Margolus and T. Toffoli, Cellular automata machines, in: Lattice Gas Methods for Partial Differential Equations, eds. G. Doolen et al. (Addison-Wesley, Reading, MA, 1990) pp.219-248.
