If one restricts oneself to a fixed number of elementary mathematical Nested patterns constructed using arithmetic operations. … For the top picture the pattern is what would be generated by an additive cellular automaton following rule 90; for the bottom picture it is what would be generated by one following rule 150.

One-way transformations [as axioms] As formulated in the main text, axioms define two-way transformations. One can also set up axiom systems based on one-way transformations (as in multiway systems).

But if one was just given a collection of initial conditions without any underlying rules one would then need to find out what underlying rules one was supposed to use in order to determine their meaning. Yet the system will always do something, whatever rules one uses. So then one is back to defining criteria for what counts as meaningful behavior in order to determine—by a kind of generalization of cryptanalysis—what rules one is supposed to use.

About one in 10,000 randomly chosen network systems seem to exhibit the kind of behavior shown here.

In this section what I will show is that with the appropriate setup just addition and subtraction turn out to be in a sense the only operations that one needs. The basic idea is to consider a sequence of numbers in which there is a definite rule for getting the next number in the sequence from previous ones. … The n th element in each sequence is denoted f[n] , and the rule specifies how this element is determined from previous ones.

In class 4, changes can also spread, but only in a sporadic way—as illustrated on the facing page and the one that follows. … In class 2, some information about initial conditions is retained in the final configuration of structures, but this information always remains completely localized, and is never in any way communicated from one part of the system to another. … Long-range communication of information is in principle possible, but it does not always occur—for any particular change is only communicated to other parts of the system if it happens to affect one of the localized structures that moves across the system.

Special Initial Conditions We have seen that cellular automata such as rule 30 generate seemingly random behavior when they are started both from random initial conditions and from simple ones. So one may wonder whether there are in fact any initial conditions that make rule 30 behave in a simple way. As a rather trivial example, one certainly knows that if its initial state is uniformly white, then rule 30 will just yield uniform white forever.

As an idealization of this process, one can consider a cellular automaton in which black cells represent regions of solid and white cells represent regions of liquid or gas. If one assumes that any cell which is adjacent to a black cell will itself become black on the next step, then one gets the patterns of growth shown below.

If one had replacements for blocks such as , or then these could overlap. … If a rule involves replacements for several distinct blocks, then to avoid the possibility of interference one must require that these blocks can never overlap either themselves or each other. … But what we have seen in this section is that there also exist systems in which rules can in effect be applied whenever and wherever one wants—but the same definite causal network always emerges.

But it is rather easy to foil this particular approach to cryptanalysis: all one need do is not sample every single cell in a given column in forming the encrypting sequence. For without every cell there does not appear to be enough information for any kind of local rule to be able to deduce one column from others. … And with one complete column given, significant patches of cells still have determined colors.