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Totalistic rules

A totalistic rule is one for which the transition depends upon the sum of the weights of the neighborhood. Such a rule, for a automaton, would take the form

where is a function of the integers modulo k. In fact, the simple modular sum is a good example of a totalistic rule, yielding a rule for which the computations of linear algebra suffice to determine its evolution.

While the choice of a totalistic rule simplifies evolutionary calculations, it is clear that a given sum can be formed in many different ways. Thus every neighborhood possessing a fixed sum must necessarily evolve into the same state. This should not cause undue concern; since there are always more neighborhoods than states, any rule will necessarily have clusters of neighborhoods mapping into the same state. It is just that the multiplicity follows the degeneracy of sum formation for totalistic rules, giving them a statistical property which is useful.

There are more ways to form sums in the middle range than for the extremes; one might think in terms of the binomial distribution. Thus the values assigned the middle range will be relatively influential in determining the overall behavior of the automaton, while the extremes can be used for fine adjustments. One extreme determines what happens to long sequences of zeroes, the other to long sequences of k-1's, and in both cases to sequences in which these extremes dominate.

In practice, a sum of zero can only arise in one way, from a string of zeroes. Equally, the high sum can only be formed from a string of cells of the highest weight. Whatever value is assigned to these extremes will influence the color of the background, just as will the values assigned to the evolution of any other constant strings. Thus the sequencing of long strings of solid color can be read off directly from the totalistic rule. In prompting for the rule, the programs mark off the sums which would correspond to constant values.

A bit more effort is required to work out the sequencing of repeated pairs, but it is not too hard to do mentally. It is an interesting observation that for totalistic rules, the sequencing of iterated triples follows the same sequencing as for constant sequences, but in terms of the sum corresponding to the triple.

In conclusion, the values assigned the extreme sums influence the cycling of background colors and can be deliberately chosen for this effect, but the drama of this change should not overshadow the fact that the remainder of the evolution is little changed and that the change will hardly be noticeable unless the rule tends toward large areas of constant color.

The next-to-extreme sums tend to determine what happens to the fringes of regions dominated by the middle values, and it is often preferable to step through these values rather than the extremes themselves to get small changes in the patterns of evolution.

Although totalistic rules form a special class of rule, they are general enough to be representative in the sense that any other rule can be transformed into a totalistic rule, although the new rule may have a considerably larger number of states. Albert and Culick have demonstrated [3] that all that is really required is to express the neighborhood as a r-digit number relative to the base k+1, but in such a way that the overlap between neighborhoods is taken into account. Thus, suppose that the configuration

is rewritten

Now, suppose that the sum U of three four digit numbers X + Y + Z = U has the base k+1 representation U = wxyz. The reason for changing to k+1 is that the new representation needs a zero which cannot be confused with one of the states of the original automaton. If the original states were numerical, they should be shifted to accommodate the new base; if they were not numerical, they should now be assigned numerical values. Then, define the auxiliary transition function to be

and finally, the new transition function by

With this definition, we have embedded a automaton with an arbitrary rule in a automaton with a totalistic rule, although it is necessary to code the cells before making the simulation and to decode them afterwards. If strict adherence is made to only necessary values for cell states, one non-zero digit in the four-digit base k+1 expansion, the number of states could be considerably reduced, improving the state economy. Otherwise, transitions for the unused combinations get arbitrary definitions.

next up previous contents
Next: Two-cell neighborhoods Up: What to look Previous: Membranes and macrocells

Harold V. McIntosh